The Vlasov--Poisson--Landau System in the Weakly Collisional Regime

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1. Introduction In this paper, we study the Vlasov–Poisson–Landau system for a particle density T3 R3 T3 R3 Z 3 function F 0, x v → 0, on the 3-torus 2π , which takes the form ∶ [ ∞) × × [ ∞) ∶= ~( ) ∂tF t, x, v vi∂xi F t, x, v Ei t, x ∂vi F t, x, v νQ F, F t, x, v , (1.1a) ( ) + ( ) + ( ) ( ) = ( )( ) 1 E t, x φ t, x , φ t, x ∆− F t, x, v dv F t, x, v dv dx , (1.1b) ( ) = −∇ ( ) ( ) = − (SR3 ( ) − aT3 SR3 ( ) ) where (from now on) repeated lower case Latin indices are summed over i, j 1, 2, 3, T3 1 = ⨏ ∶= 2π 3 T3 , and Q is the Landau collision operator with Coulomb potential given by ( ) ∫

Q G, F t, x, v ∂vi Φij v v G t, x, v ∂vj F t, x, v F t, x, v ∂vj G t, x, v dv , ( )( )∶= SR3 ( − ∗){ ( ∗)( )( )− ( )( )( ∗)} ∗ where 1 zizj Φij z δij 2 , (1.2) ( )∶= z { − z } S S S S with δij being the Kronecker delta. We will work in the weakly collisional regime, i.e. we will assume ν in (1.1a) satisfies ν 1, which is relevant in physical situations (see [75]). The system (1.1a)–(1.1b) describes≪ the dynamics of electrons in a constant ion background. The electrons both undergo (weak) bilinear collisions and are subject to the mean field force generated by the electrons themselves. It is also of interest to consider the 2-species analogue of (1.1a)–(1.1b), which describes the motion of both the electrons and the ions. We will not explicitly write down that case, though the main result of this paper has an easy analogue in that setting after small modifications. It is easy to check that the global Maxwellian

v 2 µ v e−S S (1.3) ( )∶= is a steady state solution to (1.1a)–(1.1b). For any fixed ν 0, the celebrated work of Guo [54] implies that the global Maxwellian µ is asymptotically> stable. For ν 0, however, the situation is much more subtle. The seminal work of Mouhot–Villani [76]= showed that the global Maxwellians are stable in an analytic topology via a phase-mixing mechanism known as Landau damping, which causes the electric field to decay rapidly. The same was proven to hold in a sufficiently strong Gevrey topology [15]; see also a more recent proof in [46]. Nevertheless, in a Sobolev topology, Bedrossian showed in [8] that (for a different spatially homogeneous background,) a uniform statement of the stability does not hold due to so-called plasma echoes. (See however [47].) Our goal in this paper is twofold. First, we give a detailed description of the dynamics in the presence of both (collisional) entropic effect and (non-collisional) phase mixing effect. Second, we seek to understand the threshold of stability for (1.1a)–(1.1b), i.e. for an appropriate norm X (which will be chosen to be a Sobolev norm) and a β 0, we want to show > β Data X ν Ô⇒ stability. (1.4) Y Y ≪ Ideally, we would like to find a β that is optimal. The proof of Guo’s result [54] discussed above, when appropriately adapted to the weakly collisional regime, straightforwardly implies a version of (1.4) with β 1. This is summarized in the following theorem. = THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 3

Theorem 1.1 (Guo [54]). There exist a (v-weighted, L2-based, up to second order derivatives) Sobolev space X and an ǫ0 0 independent of ν such that if the initial data F0 satisfies > 1 F0 µ X ǫ0ν, Y√µ( − )Y ≤ then there is a unique global smooth solution to (1.1a)–(1.1b) arising from the given data, which converges to µ as t → . +∞ To improve the threshold in Theorem 1.1, one needs to take advantage of the following two mechanisms specific to the ν → 0 limit: (1) (Enhanced dissipation) Solutions to (1.1a) dissipate energy much faster than that given by the proof of Theorem 1.1. From (1.1a), one may expect (say, by comparison 1 with the heat equation) that the solution dissipates energy at an O ν− time scale. However, the transport part shifts the solution to high v-frequency,( which) enhances the dissipation. As a result, after subtracting the average-in-x mode, the solution in 1 3 fact dissipates energy at an O ν− ~ time scale. (2) (Landau damping) When ν (0, the) Vlasov–Poisson–Landau system reduces to the Vlasov–Poisson system, which= as discussed above has a decay mechanism of Landau damping. One expects that Landau damping persists for small ν, and gives a decay mechanism at an O 1 time, before the collisional effects enter. To understand exactly how( ) Landau damping enters requires some knowledge of nonlinear Landau damping for Sobolev data. It is by now well-understood, for instance by adapting methods of [75], that for initial data of size O δ (with δ small) in a (sufficiently regular) Sobolev topology, the linear Landau damping mechanism( ) drives the nonlinear dynamics for 1 1 3 the Vlasov–Poisson system up to a time of O δ− . It is therefore reasonable to expect O ǫν to be a natural threshold for the problem (1.4( ):) Landau damping gives a decay mechanism( ) 1 1 3 up to time O ǫ− ν− , at which point the collisional effect kicks in and dominates due to the enhanced dissipation.( ) (See discussions in [7].) This is exactly what we obtain in our main theorem. Theorem 1.2. There exist a (v-weighted, L2-based, up to ninth order derivatives) Sobolev space X and an ǫ0 0 independent of ν such that if the initial data F0 satisfies > 1 1 3 F0 µ X ǫ0ν ~ , Y√µ( − )Y ≤ then there is a unique global smooth solution to (1.1a)–(1.1b) arising from the given data, which converges to µ as t → . Moreover, the solution exhibits+∞ enhanced dissipation and uniform-in-ν Landau damping. The enhanced dissipation and uniform-in-ν Landau damping are reflected in the large-time estimates that we prove; see Section 1.1.6 and Theorem 3.1. (Notice that if we only capture enhanced dissipation without exploiting Landau damping in the proof, this would correspond 2 to a weaker theorem where the initial data could only be an O ǫ0ν 3 -perturbation of the global Maxwellian.) ( ) A very similar result was proven in a recent work of Bedrossian for the Vlasov–Poisson– Fokker–Planck system [7]. The Landau collision kernel is more complicated than Fokker– Planck collision kernel in its anisotropy and degeneracy as v → , as well as a lack of a spectral gap. More importantly from the point of view ofS thisS ∞ problem, the linearized THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 4

Vlasov–Poisson–Landau system around the global Maxwellians cannot be solved explicitly, unlike the corresponding linearized Vlasov–Poisson–Fokker–Planck system. This requires a different approach for the problem. Given the lack of explicit representation formulas for the linear solution, we rely instead on adaptations of Guo’s energy method, but we further need to design the energies so as to capture the phenomena of enhanced dissipation and Landau damping. (1) (Hypocoercivity) To capture enhanced dissipation, we use the hypocoercive energy method: this is a choice of an energy which incorporates some lower order boundary terms, which in turn generates useful coercive spacetime terms. This idea is known to be well-suited to capture the interaction of the transport and the collision terms [58, 94], and particularly to obtain sharp enhanced dissipation rate in some weakly viscous settings [9]. See Section 1.1.3. (2) (Commuting vector fields method) To capture Landau damping, we need quantita- tive estimates showing that f 1 F µ behaves like a solution of the transport = √µ equation. To achieve this, we∶ commute( − the) equation with the (t-weighted) vector ω field Yi t∂xi ∂vi and bound Y f (and its derivatives) in addition to f itself. This lets one= prove+ transport bounds in the presence of collision, and in fact to take advantage of the coercivity given by collision. Such a commutating vector field method is inspired by related techniques for treating dispersion in nonlinear wave equations and other kinetic models [28, 29, 63, 69, 83]. See Sections 1.1.4 and 1.2.6. We emphasize that our use of the commuting vector field method is not only limited to proving an energy estimates for f. As in [7, 15, 46, 76], in addition to energy estimates for f, we need an independent density estimate for the macroscopic density ρ R3 f√µ dv, proven using the Volterra equation that it satisfies. To achieve the density estimates∶= ∫ involves (1) proving a resolvent estimate to invert the linear part and (2) bounding the nonlinear contributions. Both of these can be achieved by extending the resolvent estimate and nonlinear analysis in [46] in conjunction with obtaining control of the linear Landau flow (see Sec- tion 1.1.5). The estimates we need for the linear Landau flow can in turn be achieved by a combination of the hypocoercive energy method and the commuting vector field method. We further discuss the ideas of the proof in Section 1.1. We then turn to related works in Section 1.2. Finally, we will end our introduction with an outline of the remainder of the paper in Section 1.3.

1.1. Idea of the proof.

1.1.1. Preliminaries. Let µ be the global Maxwellian (1.3). We rewrite the problem for f deﬁned by F µ √µf so that the Vlasov–Poisson–Landau system (1.1a)–(1.1b) becomes = + ∂tf v xf E vf E v f 2 E v √µ νLf νΓ f, f , (1.5) + ⋅ ∇ + ⋅ ∇ − ( ⋅ ) − ( ⋅ ) + = ( ) where E is as in (1.1b), L is the linearized Landau collision operator, which has some coercivity properties, and Γ is the nonlinear collisional terms in f (see Section 2.1 for precise deﬁnitions). The problem is now rephrased to proving boundedness and decay for f.

1.1.2. The Guo’s energy method. The starting point of our approach is Guo’s work [54] (see Theorem 1.1 above). The general strategy, ﬁrst devised by Guo and is applicable in many kinetic models of the form ∂tf v xf νLf νΓ f, f , is to ﬁnd an energy norm , a + ⋅ ∇ + = ( ) Y ⋅ YE THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 5

dissipation norm , and a suitable θ 0, η R such that Y ⋅ YD > ∈ d f 2 θνη f 2 0. (1.6) dtY YE + Y YD ≤ The control in comes from the linear Lf part, and the norms are chosen appropriately to bound the nonlinearY ⋅ YD term ν f, Γ f, f νη′ f f 2 so that one can indeed obtain (1.6) with suitable smallness onS⟨ the initial( )⟩ data.E S ≲ Y YE Y YD The proof of such an energy inequality, or even the choice of the norms and , is delicate, and depends on the kinetic model under consideration. The constructionY ⋅ YE inY general⋅ YD requires a careful choice of weight functions, as well as using an additional argument to handle the kernel of the linear operator L. This type of method is particularly useful for soft potentials (such as the Landau collision operator), since in general the norm is weaker than the norm in v-weights. Y ⋅ YD We highlightY ⋅ YE two innovations in the energy introduced by Guo [54] which are speciﬁc to the Vlasov–Poisson–Landau system: use of eφ weights in the energy, where φ is the electric potential (to handle the ● costly-in-v-moment term E v f), and use of weights in v which( ⋅ are) weaker for higher derivatives (to handle simultane- ● ously the weak coercivity⟨ ⟩ of the the dissipation energy for large v and commutator terms arising from the linear free streaming term). ⟨ ⟩ A more detailed explanation of these weights and their motivations can be found in the introduction of [54]. These will also be featured prominently in our energies.

1.1.3. Hypocoercivity and energy method. Slightly over-simplifying1 for the moment, the Guo energy in [54], when adapted to small ν, corresponds to

φ 2M 2 α 2 β 2 β α β e v − S S− S SHα,β L2 , Hα,β νS S∂x ∂v f, (1.7) Y ⟨ ⟩ Y x,v ∶= after appropriately summing up in α and β. (Note the v and eφ weights are incorporated in the energy, as discussed in Section 1.1.2.) ⟨ ⟩ Diﬀerentiating the energy (1.7) also gives an integrated decay estimate (cf. (1.6)) which controls, for Hα,β as in (1.7),

T T φ 2M 2 α 2 β 1 2 φ 2M 2 α 2 β 3 β′ 2 ν e v − S S− S S− 2 Hα,β 2 dt ν e v − S S− S S− 2 ∂ Hα,β 2 dt. (1.8) Lx,v v Lx,v S0 Y ⟨ ⟩ Y + S0 βQ′ 1 Y ⟨ ⟩ Y S S= 1 One reads oﬀ from (1.7) that each ∂v derivative costs ν− , and by comparing (1.7) and 1 (1.8) that integration in t also costs ν− . Heuristically, this means that one expects to deduce 1 from (1.7) and (1.8) that energy decays on a time scale of order ν− . In particular, this does not capture the enhanced dissipation generated by the interaction between the transport and the diﬀusive terms. (Notice that the second term in (1.8) gives a better (in ν) estimate for the ∂v derivatives, but without some corresponding estimates for the ∂x derivatives, it is unclear how that could improve the rate.)

1We have suppressed in particular the fact that (1) Guo also incorporated E in the energy (which we will not need, see beginning of Section 1.1.5) and (2) Guo has stronger weighted in v so as to obtain stretched exponential decay estimates (which we will discuss later in Section 1.1.6). THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 6

Inspired by [6, 9, 58, 94], we modify the Guo energy so as to capture enhanced dissipation. 2 1 α β Precisely, for every α, β , we define the following energy at the H level for ∂x ∂v f: α β 2 ( ) ∂x ∂v f Eα,β Y Y 2 φ 2 M α α′ β α′ 2 φ 2 M α β 1 β′ 2 A0 e v ( −S S−S S−S S)∂ Hα,β 2 ν 3 e v ( −S S−S S− )∂ Hα,β 2 x Lx,v v Lx,v ∶= αQ′ 1 Y ⟨ ⟩ Y + βQ′ 1 Y ⟨ ⟩ Y S S≤ S S= (1.9) 1 2φ 4 M α β 1 ν 3 e v ( −S S−S S− ) xHα,β vHα,β dv dx . T3 R3 + S × ⟨ ⟩ ∇ ⋅ ∇ ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ (∗) β 3 α β where now Hα,β νS S~ ∂ ∂v f. For A0 large but fixed, (1.9) is comparable to ∶= x φ 2 M α α′ β α′ 2 2 φ 2 M α β 1 β′ 2 e v ( −S S−S S−S S)∂ Hα,β 2 ν 3 e v ( −S S−S S− )∂ Hα,β 2 . (1.10) x Lx,v v Lx,v 0 Qα′ 1 Y ⟨ ⟩ Y + βQ′ 1 Y ⟨ ⟩ Y ≤S S≤ S S= The key here is that despite the equivalence of (1.9) and (1.10), when differentiating the d 1 3 φ 2 M α β 2 α′ 2 term in (1.9) by dt , a non-negative useful term ν ~ ∑ α′ 1 e v ( −S S−S S− )∂x Hα,β L2 (∗) S S= Y ⟨ ⟩ Y x,v is generated. As a result, after suppressing some terms, the d derivative of ∂α∂βf 2 dt x v α,β precisely controls Y YE T 1 3 α β 2 1 3 φ 2M 2 α 2 β 2 α′ 2 ν ~ ∂x ∂v f ν ~ e v − S S− S S− ∂x Hα,β 2 dt α,β S Q Lx,v Y YD ≳ ( 0 α′ 1 Y ⟨ ⟩ Y S S= T (1.11) 7 ′ 2 3 φ 2M 2 α 2 β 2 β 2 ν ~ e v − S S− S S− ∂v Hα,β 2 dt . S Q Lx,v + 0 β′ 1 Y ⟨ ⟩ Y ) S S≤ 1 In (1.9) and (1.11), we see that each ∂v derivative now costs ν− 3 . Moreover, comparing the α′ 1 and β′ 1 terms in (1.9) with those in (1.11) may suggest that the energy for the = = 1 S S S S 3 1 derivatives of f decay at a time scale of ν− , which is much earlier than ν− , i.e. this energy captures enhanced dissipation. This enhancement is crucial in controlling the nonlinear terms. Note, however, that if we compare the α′ 0 term in (1.9) with the β′ 0 term in (1.11), = 2 3 = we see that the term has an additionalS lossS of ν− ~ . This is a reflectionS S of the fact that enhanced dissipation only holds after removing the x-average mode.

1.1.4. Commuting vector fields and Landau damping. We capture Landau damping using the commuting vector field method, with vector fields adapted to the flow of the transport equation. The advantage of using such a commuting vector field method approach is that we can hope to prove transport estimates and largely ignore the collision term because in principle the collision term gives rise to terms which have a good sign.

Let Yi t∂xi ∂vi . We will use Yi as a commuting vector field, together with ∂xi and 1 = 3 + β 3 α β ω ν ∂vi , i.e. we control νS S~ ∂x ∂v Y f in appropriate weighted spaces (with weights allowed to depend on α,β,ω ) andS define moreS generally α,β,ω and α,β,ω spaces (see (5.11)–(5.12) for details). The( significance) of Y can be explainedE as follows:D

For a solution flin to the linear transport equation ∂tflin v xflin 0 with regular ● ω + ⋅ ∇ = data, it is easy to see that Y flin is uniformly bounded in time (since ∂t v x,Yi S S [ + ⋅∇ ] = 2 This is still not yet the actual energy we use, which also includes commutations with Yi; see Section 1.1.4. THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 7

0), despite Y being a t-weighted vector ﬁeld. Thus controlling the Y derivatives of f can be viewed as proving an asymptotic transport-like estimate. Controlling Y f also implies decay of averaged quantities of f, thus capturing phase ● mixing. For instance, Poincar´e’s inequality gives that for ρ ∫R3 f√µ dv and for ⨏T3 the average over T3, we have ∶=

2 2 2 ρ ρ dx 2 xρ 2 ∂x f µ dv dx aT3 Lx Lx Q ST3 SR3 i √ Y − Y ≲ Y∇ Y ≲ i ( ) 2 2 2 2 2 t− Yif ∂v f µ dv dx t− Y f 2 f 2 , Q ST3 SR3 i √ Lx,v Lx,v ≲ i ( ( − ) ) ≲ (Y Y + Y Y ) where we integrated by parts in the last estimate. Importantly for understanding phase mixing in the presence of collision, capturing ● phase mixing by the vector ﬁeld Y allows one to still take advantage of the coercivity of the collision term while proving phase mixing. More precisely, a term such as νL Y f (where L is the linear Landau collision operator) that arises in the argument for( bounding) Y f is not treated as an error, but instead we take advantage of the coercivity of L and make use of this term. (There are associated commutator terms, which we will show to be of a lower order.)

1.1.5. Density estimates. The above ideas would in principle be sufficient to prove enhanced dissipation and Landau damping for the Landau equation (i.e. without the Poisson part) in the weakly collisional regime. However, the Vlasov–Poisson–Landau system has terms in E (see E vf, E vf, and 2 E v √µ in (1.5)), which require an additional idea. The⋅ linear ∇ E⋅term was handled( ⋅ ) by Guo [54] using a cleverly designed energy which incorporates E so that this linear E term is cancelled in the derivation of the energy estimates. Such a strategy seems difficult to implement when at the same time carrying out ideas in Sections 1.1.3 and 1.1.4. The nonlinear E vf, if treated using the energy estimates alone, ⋅ ∇1 3 would give a worse threshold compared to ǫν ~ . Instead, we follow the general strategy [4, 15, 46, 76] and prove an independent estimate for the density that does not depend on the energy estimate. These density estimates rely on resolvent bounds on the kernel of the linearized density, which we now explain. In the Vlasov–Poisson case, the k-th Fourier mode of the density ρ satisfies a Volterra equation t VP VP ρˆk t S Kk t τ ρˆk τ dτ k t , ( ) + 0 ( − ) ( ) = N ( ) VP 2 ik⋅vt VP where the kernel Kk t k 2 R3 i k v e− µ dv, and k t is an error term containing = S S ∫ N the contributions from( the) initial data( ⋅ and) the nonlinear terms.( ) In the Vlasov–Poisson–Landau case, the Volterra equation is less explicit, and the kernel takes the form 2 Kk t ik Sk t v µ µ dv, 2 S 3 √ √ ( ) = k R ⋅ ( )[ ] S S where Sk t denotes the linear Landau semigroup generated by the fixed mode linear Landau equation(∂t)h ik vh νLh 0. To solve the+ nonlinear⋅ + Volterra= equation, we take the following steps: THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 8

We first derive pointwise resolvent estimates (cf. [46, 55, 56]), showing that there is ● a kernel Gk which is rapidly decaying (and thus negligible) such that t ρˆk t k t S Gk t s k s ds. (1.12) ( ) = N ( ) + 0 ( − )N ( ) Relying on the resolvent estimate proven for the Vlasov–Poisson case in [46], it es- VP sentially suffices to show that lim → K K 1 0. This in turn can be ν 0 k k Lt → obtained by energy and vector field methodsY (⋅) − for the(⋅ linear)Y Landau flow for all small ν. We then need to control k t in (1.12) (see (7.4), (8.4) for the precise terms). The ● N most difficult term here comes( ) from the nonlinear contribution E vf associated with the Poisson part, which takes the form ⋅ ∇ t El τ Sk t τ vf k−l τ µ dv dτ. Q S SR3 √ l 0 0 ̂ ( ) ⋅ ( − )[∇Ä ( )] ~= (The nonlinear collisional terms are slightly easier.) As above, we control Sk t τ using the hypocoercive energy method and the commuting vector field method.( − The) 1 3 bounds we prove give (1) rapid decay in ν ~ t τ , and (2) bounds associated with the Y vector field, which can be viewed⟨ as transport-like( − )⟩ bounds. Precisely because we obtain transport-like bounds, this gives hope of controlling the nonlinear term by extending ideas from the density estimates for the Vlasov–Poisson system.

1.1.6. Decay estimates. Once we close the energy estimates, we adapt the methods of Strain– Guo [87, 88] to exchange v-weights in the energy with decay in the variable νt. More c v 2 precisely, following [88], we additionally introduce e S S weights in the energy (1.9) so as to obtain energy decay with a stretched exponential rate. In order to avoid the technicalities c v 2 c v 2 associated with simultaneously using e S S weights and commuting with Y , we only use e S S weights when there are no Y commutations. At ﬁrst this only gives decay of energy without Y commutations, yet a full decay statement can then be achieved by interpolation. This allows us to obtain the following decay results (see precise statements in Theorem 3.1): (1) Essentially arguing as Strain–Guo, but taking into account the dependence on ν, we 2 prove that the energy decays with an exp δ νt 3 rate (for δ 0 small). (2) As discussed earlier, there is an enhanced(− dissipation( ) ) (which> operates at the time − 1 − scale of O ν 3 instead of O ν 1 ) after removing the zeroth spatial Fourier mode. Instead of( explicitly) removing( the) zeroth mode, we prove an enhanced decay estimate by considering an energy in which f has at least one ∂x derivative. For such an energy, 1 1 2 we prove energy decay with a rate min exp δ ν 3 t 3 , exp δ νt 3 . { (− ( ) ) (− ( ) )} In order to obtain the decay estimates, in addition to deriving weighted energy estimates, we also need to propagate the stretched exponential decay in the density estimates (recall Section 1.1.5). This requires (1) a precise estimate for the resolvent, which incorporates the stretched exponential decay, and (2) a more careful nonlinear analysis. This more precise nonlinear analysis (see for instance the decomposition of the density in (11.3)–(11.4)) is devised so that one does not see an analogue of the top-order loss in the energy boundedness argument (see Section 1.1.8), which is now possible because we are only propagating the stretched exponential decay estimate for the low-order derivatives. (See the beginning of Section 11 for further remarks on the nonlinear density estimates.) THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 9

Once we obtain the enhanced decay rate for the nonzero modes, the density estimate implies that the Fourier modes ρk obey uniform Landau damping estimates

1 3 −N −δ ν1~3 t 1~3 −δ νt 2~3 ρk ǫν ~ 1 k kt min e ( ) , e ( ) . S S ≲ ( + S S + S S) { } 1.1.7. Structure of the energy estimates. In order to carry out the full scheme described above, we implicitly need that under suitable bootstrap assumptions, we can bound various energies which use only some subsets of commutators. For instance, for the linear Landau energy estimates used in the density estimates (see Section 1.1.5), we need to commute the linear Landau equation for each ﬁxed mode with a large number of Y derivatives, but with at most one ∂x or ∂v derivatives. This is important for obtaining the correct constants in the estimates. On the other hand, for the stretched exponential decay (see Section 1.1.6), we need an energy without any Y commutations (since, as discussed above, we do not put Y commu- c v 2 tations together with e S S weights). For the decay of the full solution, we use only the hypocoercive energy without any additional commutations. For the enhanced dissipation, we need to remove the k 0 x-Fourier mode. For this purpose, we consider the hypocoercive = energy with exactly one additional ∂x commutation. To propagate the boundedness of energies with only suitable subsets of commutators,

E low we deﬁne an energy Nα ,Nα,β,Nβ,Nω , which is a sum of appropriate α,β,ω energies. The low E parameters Nα , Nα,β, Nβ, Nω describe the commutators used: they depend not only on the total number of commutators, but also on various upper and lower bounds on each type of commutators used; see (5.15).

1.1.8. Additional difficulties. While we have already described the main conceptual difficulties, the even more interesting difficulties lie in the technicalities. We highlight a few technical issues here.

Asymmetric use of commutators. At the top order of energy, we do not allow for an arbitrary combination of the commutator vector ﬁelds. Instead, we only allow for α β ω α β ω α β ω 2 α β ω ∂x ∂v Y f, ∂xi ∂x ∂v Y f, ∂vi ∂x ∂v Y f, ∂vivj ∂x ∂v Y f (1.13)

for 0 α β ω Nmax. (See the α,β.ω and α,β,ω norms in (5.13)–(5.14).) Put diﬀerently, ≤ S S+S S+S S ≤ Ẽ D̃ at the top level (with Nmax 2 derivatives), at least two commutators have to be ∂v; ● + at the penultimate level (with Nmax 1 derivatives), at least one commutator has to ● + be ∂x or ∂v; at lower levels (with Nmax derivatives or fewer), the commutators can be arbitrary ● combinations of ∂x, ∂v and Y . On the one hand, this is needed because the nonlinear density estimates (unlike the energy α ω α β ω estimates) lose derivatives, and thus to bound ∂x Y ρ requires estimates for ∂x ∂v Y f for β 3, which can only be obtained by commuting with two additional ∂v derivatives. On S S ≤ 2 2 the other hand, this is possible because commuting ∂v does not generate terms like ∂xf.

Growth of top-order energy. When controlling the energy for the terms (1.13) with − 1 α β ω Nmax 1 or Nmax, we allow the energy to grow either in t or ν 3 . The underlying reasonS S+S S+ isS thatS = the decay− of E by Landau damping is determined by the regularity. The decay THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 10

at the highest level is thus slower, and ultimately the terms 2 E v √µ and E vf in (1.5) cause the top-order energy to grow. ( ⋅ ) ⋅ ∇ Nevertheless, importantly, even though the energies at the top two orders grow, the nonlinear analysis in the density estimates (see Section 1.1.5) still allows one to prove a desired density estimate without loss at the top level. This allows the bootstrap argument to close.

Diﬀerent decay rates for the k 0 and k 0 modes. As we have already discussed above (see Section 1.1.3), enhanced dissipation= is=~ only seen for the spatial Fourier modes k 0, i.e. the k 0 mode decays slower. This in particular means that in the nonlinear analysis,=~ we need= to be careful of terms without derivatives, as they could potentially be more slowly decaying. In all cases, we show that there is an integration by parts giving bounds with the right decay; see Lemmas 9.5 and 9.6.

Handling some lowest order terms. Finally, recall that the linearized Landau operator has a non-trivial kernel, which was dealt with in [54] by analyzing a separate system for the macroscopic quantities. Amusingly, this is bypassed in our analysis due to the hypocoercive energy; see a related observation in [20]. More precisely, the hypocoercive energy gives better bounds on the ∂x derivatives, so that we only need to control the x-mean of the contribution from the kernel, which in turn can be treated trivially using the conservation laws. 1.2. Related works. 1.2.1. Landau damping for the Vlasov–Poisson system. Linear Landau damping for the Vlasov–Poisson system was first observed in Landau’s seminal work [64]. A mathematical breakthrough was achieved by Mouhot–Villani [76], justifying Landau damping in a nonlinear setting under analyticity assumptions. This has been extended and simplified in [15, 46]. More recently, the effect of plasma echoes have been further explored in [8, 47]. See also [24, 60] for earlier constructions of some Landau damped solutions, [16, 17, 42, 43, 55, 56] for works on the whole space (instead of the torus), and [103] for the relativistic case. 1.2.2. Nonlinear stability of global Maxwellians. In the ν 1 case of (1.1a)–(1.1b) (or its two- species analogue), the nonlinear asymptotic stability of global= Maxwellians was first proven in Guo’s [54] in a periodic box; see also [32, 35]. The corresponding stability result on R3 was proven in [89] (with alternative proofs in [57, 66, 99]). See also the more recent [36] for stability of local Maxwellians representing rarefaction waves. The work [54] can be viewed in the context of a larger program of stability of Maxwellians result using energy methods. This began with Guo’s seminal work [50] for the Landau equation, and inspired many subsequent works; see [1, 2, 3, 25, 26, 27, 49, 51, 53, 52, 59, 86, 87, 88] and the references therein for further discussions. 1.2.3. Related works in the physics literature. There have been many works in the physics literature studying the interaction of Landau damping and weak collisions, see [23, 37, 45, 61, 67, 71, 72, 77, 78, 79, 81, 82, 85, 90, 93, 104, 105] and the references therein. 1.2.4. Weakly collisional regimes for kinetic equations. Despite its physical importance, there are very few mathematical works on weakly collisional regimes for kinetic equations. This type of questions were raised in the mathematics literature for instance in [33, IV.25.8.2] and [95, Section 8]. The only nonlinear result is the work of Bedrossian [7] on the Vlasov–Poisson– Fokker–Planck system that we already mentioned. This was predated slightly earlier by a THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 11

linear analysis in [92]. More recently, the linear analysis was extended to include eﬀects ofa uniform background magnetic ﬁeld [20].

1.2.5. Related models with vanishing dissipation. Even though there are not many works on weakly collisional regimes for kinetic equations, there are closely related models, problems and results in ﬂuid dynamics. See [9, 10, 11, 12, 13, 14, 18, 19, 30, 31, 34, 40, 48, 62, 65, 70, 73, 74, 80, 100, 101, 106] and the references therein for a sample of results.

1.2.6. Commutating vector field method for kinetic models. The commutating vector field method, pioneered in [63] for quasilinear wave equations, has been very successful in capturing dispersion to prove global stability for nonlinear evolution equations. Recently, it has likewise found many applications for collisionless kinetic equations. In particular, the stability of vacuum has been established in many different settings [21, 39, 83, 102], and the stability of the Minkowski spacetime for the Einstein–Vlasov system in general relativity has also been resolved [22, 38, 68, 84, 91]. (See also [5, 4, 44, 41, 98, 97, 96] for related works on stability of vacuum type results for collisionless models.) For collisional models, recent works using the commutating vector field method give — for the first time — stability of vacuum results for collisional models with a long range interaction, first for the Landau equation [69, 29], and more recently for Boltzmann equation without angular cutoff [28].

1.3. Outline of the paper. The remainder of the paper is structured as follows. In Section 2, we introduce the notation that will be in eﬀect for the rest of the paper. ● In Section 3, we give a precise statement of the main theorem. ● In Section 4, we collect some facts about the Landau collisional operator. ● In Section 5, we set up the main energy estimate for the whole Vlasov–Poisson– ● Landau system. In particular, the precise energy and dissipation norms will be introduced.

In Section 6, we perform energy estimates for the linear Landau ﬂow that are needed ● for closing the density estimates.

In Section 7, we provide pointwise resolvent bounds on density of the linearized ● Vlasov–Poisson–Landau system.

In Section 8, we establish the nonlinear density estimates under the bootstrap as- ● sumptions on the energy.

In Section 9, we close the main nonlinear energy estimates for the Vlasov–Poisson– ● Landau system.

In Section 10, we prove global existence of solutions via a continuity argument. ● In Section 11, we prove stretched exponential decay for the density at lower order. ● THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 12

In Section 12, we prove stretched exponential decay for lower order energy. ● In Section 13, we put everything together and prove the main conclusions of the ● paper including the global existence, the stretched exponential decay, as well as the uniform Landau damping for the density.

Finally, in Appendix A, we give two versions of Strain–Guo lemmas adapted to our ● setting.

Acknowledgments. S. Chaturvedi and J. Luk are supported by the NSF grant DMS- 2005435. J. Luk also gratefully acknowledges the support of a Terman Fellowship. T. Nguyen is partly supported by the NSF under grant DMS-1764119, an AMS Centennial fellowship, and a Simons fellowship.

2. Notation We ﬁrst introduce a reformulation of the problem in terms of f 1 F µ , and then ∶= √µ − introduce some notations that will be used throughout the paper. ( )

2.1. Reformulation in terms of f. For the remainder of the paper, it is convenient to ﬁrst rewrite the problem in terms of f (see Section 1.1.1). Deﬁne f via F µ µf. (2.1) = + √

In the remainder of the paper, we will solve (1.1a)–(1.1b) with initial data f t=0 f0 that in S = particular satisﬁes 3 3 f0 µ dv dx 0. The conservation of mass ensures that ⨏T ∫R √ = f t, x, v »µ v dv dx 0. a 3 S 3 T R ( ) ( ) = Under this mean zero condition, it can be deduced that the Vlasov–Poisson–Landau system (1.1a)–(1.1b) is equivalent to the following system for f:

∂tf v xf E vf E vf 2 E v √µ νLf νΓ f, f , (2.2a) + ⋅ ∇ + ⋅ ∇ − ⋅ − ( ⋅ ) + = ( ) E t, x φ t, x , ∆φ f t, x, v »µ v dv, (2.2b) S 3 ( ) = −∇ ( ) − = R ( ) ( ) where, following [50, Lemma 1], the linear Landau operator L admits a decomposition ● L A, (2.3) = −K − where A and K are given respectively by

Ag ∂vi σij∂vj g σijvivjg ∂vi σig, (2.4) ∶= ( ) − + − 1 ′ ′ ′ ′ ′ ′ g µ 2 v ∂v µ v Φij v v µ v ∂v g v v g v dv , (2.5) K i S 3 √ j j ∶= − ( ) ( ) R ( − ) ( )[ ( ) + ( )] with

σij Φij µ, σi Φij vjµ σijvj, (2.6) ∶= ⋆ ∶= ⋆ ( ) = for Φij as in (1.2), and being the v-convolution, ⋆ THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 13

and the nonlinear Landau collisional term Γ f, f is given by ● ( ) 1 2 1 2 Γ g1,g2 ∂vi Φij µ ~ g1 ∂vj g2 Φij viµ ~ g1 ∂vj g2 ( ) ∶= ⋆ ( ) − ⋆ (2.7) 1 2 1 2 ∂v Φij µ ~ ∂v g1 g2 Φij viµ ~ ∂v g1 g2. − i ⋆ ( j ) + ⋆ j The rest of the paper deals with solutions f to (2.2a)-(2.2b).

2.2. Notations. Vector Field Y . For any t 0 and i 1, 2, 3 , we introduce the time- dependent vector ﬁeld ≥ ∈ { }

Yi ∂v t∂x . = i + i

N 3 α α1 α2 α3 Multi-indices. Given a multi-index α α1,α2,α3 0 , we deﬁne ∂x ∂x1 ∂x2 ∂x3 β β1 β2 β3 ω ω=1 ( ω2 ω3 ) ∈ ( ∪{ }) = and similarly, ∂v ∂v1 ∂v2 ∂v3 and Y Y1 Y2 Y3 . Multi-indices are added according to the rule that if α′ =α′ ,α′ ,α′ and α′′ = α′′,α′′,α′′ , then α′ α′′ α′ α′′,α′ α′′,α′ α′′ . = ( 1 2 3) = ( 1 2 3 ) + = ( 1 + 1 2 + 2 3 + 3 ) We also set α α1 α2 α3. S S = + + 1 Japanese brackets. Given w Rn, n N, deﬁne w 1 w 2 2 . ∈ ∈ ⟨ ⟩∶= ( +S S )

Velocity weights. Fix Nmax 9, M Nmax 30 and q0 0, 1 (cf. Theorem 3.1). For ≥ = + ∈ ( ) any ϑ 0, 2 and any triple of multi-indices α,β,ω such that α β ω Nmax, we introduce∈ { velocity} weights ( ) S S+S S+S S ≤

qSvSϑ ℓα,β,ω wα,β,ω v e 2 (2.8) = ⟨ ⟩ q0 if ϑ 2 for q = , and for the polynomially weighted index = 0 if ϑ 0 =

ℓα,β,ω 2M 2 α 2 β 2 ω (2.9) = − ( S S + S S + S S) to be used throughout in the analysis. These velocity weights will be appropriately asso- α β ω ciated with norms for derivatives ∂x ∂v Y of the Vlasov–Poisson–Landau solutions. Note that in the applications below, when ω 0, we take ϑ 0 in (2.8): namely, only polynomial velocity weights will be used. =~ =

p p p p L spaces. We will work with L spaces with standard norm Lx or Lv for functions depending on x or v, respectively. We also use mixed norms Y ⋅ Y Y ⋅ Y

p 1 q q p h Lp Lq h x, v dv dx Y Y x v ∶= (ST3 (SR3 S S ( ) ) ) which reduce to Lp in the case when p q. Y ⋅ Y x,v =

Weighted norms. Fix q0 0, 1 for the remainder of the paper (cf. Theorem 3.1). For ℓ R and 1 p , we∈ deﬁne ( ) the following weighted norms ∈ ≤ ≤ ∞ 2 ℓ ℓ q0SvS p p p 2 p h Lv ℓ,0 v h Lv , h Lv ℓ,2 v e h Lv , Y Y ( ) ∶= Y⟨ ⟩ Y Y Y ( ) ∶= Y⟨ ⟩ Y THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 14 where q0 0, 1 is the ﬁxed constant above. Analogously, we introduce the following dissipation norms∈ ( ) v v h 2 v 2ℓ ∂ g σ ∂ g σ i j g2 v, ∆v ℓ,0 vi ij vj ij d ( ) = SR3 2 2 Y Y ∶ ⟨ ⟩ ( ) + (2.10) 2 2 2ℓ q0 v vi vj 2 h v e S S ∂v g σij ∂v g σij g dv, ∆v ℓ,2 S 3 i j Y Y ( ) ∶= R ⟨ ⟩ ( ) + 2 2

p1 p2 p for q0 0, 1 as above, and σij as in (2.6). We also use mixed norms h L L ℓ,ϑ , h Lx∆v ℓ,ϑ ∈ x v ( ) ( ) ( ) 2 Y Y Y Y and h ∆x,v ℓ,ϑ h L ∆v ℓ,ϑ in an obvious manner, for ϑ 0, 2 . Using the properties of ( ) ∶= x ( ) ∈ σij (seeY Y Lemma 4.1Y below),Y we note that { }

2 − 2 − h Lv ℓ 1 2,ϑ vh Lv ℓ 3 2,ϑ h ∆v ℓ,ϑ . (2.11) Y Y ( ~ ) + Y∇ Y ( ~ ) ≲ Y Y ( ) For all of the above norms, we also define analogous norms, specified with a ′, so that when ϑ 2, they have a weaker Gaussian weight in v, with q0 replaced by q0 2. More precisely, we define= ~

p ′ p ′ h Lv ℓ,0 h Lv ℓ,0 , h ∆v ℓ,0 h ∆v ℓ,0 (2.12) Y Y ( ) ∶= Y Y ( ) Y Y ( ) ∶= Y Y ( ) and 2 2 ℓ q0SvS 2ℓ q0SvS vi vj 4 2 2 2 h Lp ℓ,2 ′ v e h Lp , h ′ v e σij∂v g∂v g σij g dv. v v ∆v ℓ,2 S 3 i j Y Y ( ) ∶= Y⟨ ⟩ Y Y Y ( ) ∶= R ⟨ ⟩ + 2 2 (2.13)

3. Statement of the main theorem The following is the precise version of our main theorem.

Theorem 3.1. Let q0 0, 1 and Nmax N with Nmax 9. Deﬁne M Nmax 30. There ∈ ∈ ≥ = + exist ǫ0 ǫ0 q0, Nmax 0(and)ν0 ν0 q0, Nmax 0 such that the following hold. Consider= ( the Vlasov–Poisson–Landau) > = ( system) > (2.2a)–(2.2b) with collision parameter ν ∈ 0, ν0 . Suppose that the initial function f0 is smooth and satisﬁes ( ] 2 f0 µ dv dx f0vj µ dv dx f0 v µ dv dx 0, (3.1) S 3 3 √ S 3 3 √ S 3 3 √ T ×R = T ×R = T ×R S S = and for some ǫ 0, ǫ0 , f0 obeys the smallness bound ∈ ( ] 2 q0 v 2M α β 1 3 e S S v ∂ ∂ f 2 ǫν ~ . (3.2) Q x v 0 Lx,v α + β Nmax+4 Y ⟨ ⟩ Y ≤ S S S S≤

Then there exists a global-in-time smooth solution f to (2.2a)–(2.2b) with f t 0 f0. More- S = = over, there exist constants C 0 and δ 0 (depending only on q0 and Nmax, and in particular independent of ǫ and ν) such> that the following> estimates hold for all t 0, : ∈ [ ∞) (1) (Boundedness of weighted energy)

2 β 3 q0 v 2M−2 α −2 β α β 1 3 νS S~ e S S v S S S S∂ ∂ f 2 t Cǫν ~ , Q x v Lx,v (3.3a) α + β Nmax−1 Y ⟨ ⟩ Y ( ) ≤ S S S S≤ β 3 2M−2 α −2 β α β ω 1 3 νS S~ v S S S S∂ ∂ Y f 2 t Cǫν ~ . (3.3b) Q x v Lx,v α + β + ω Nmax−2 Y⟨ ⟩ Y ( ) ≤ S S S S S S≤ THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 15

(2) (Energy decay) 2 β 3 α β ω 1 3 −δ νt 3 νS S~ ∂ ∂ Y f 2 t Cǫν ~ e ( ) . (3.4) Q x v Lx,v α + β + ω Nmax−2 Y Y ( ) ≤ S S S S S S≤ (3) (Enhanced dissipation) For f 0 t, x, v f t, x, v T3 f t, x, v dx, ~= ⨏ ( ) ∶= ( ) − ( 1 )1 2 β 3 α β ω 1 3 −δ ν 3 t 3 −δ νt 3 νS S~ ∂x ∂v Y f 0 L2 t Cǫν ~ min e ( ) , e ( ) . (3.5) Q ~= x,v α + β + ω Nmax−2 Y Y ( ) ≤ { } S S S S S S≤ ik⋅x (4) (Uniform Landau damping) For ρ t, x k Z3 ρk t e , it holds that ∑ ∈ ( ) = ( ) 1 1 2 1 3 −Nmax+1 −δ ν 3 t 3 −δ νt 3 ρk t Cǫν ~ 1 k kt min e ( ) , e ( ) (3.6) S S( ) ≤ ( + S S + S S) { } for every k N 0 . ∈ ∖ { } A few remarks are in order. Remark 3.2 (Some top-order bounds not stated). Notice that some of the top-order bounds 1 3 are not stated. In fact, the highest order energies will not be shown to be bounded by Cǫν ~ , −1 3 but instead has a loss in ν ~ or t ; see Theorem 9.1. ⟨ ⟩ 2 q0 v Remark 3.3 (Exponential v-weights). We only propagate the exponential weight e S S when there are no Y derivatives, i.e. when ω 0. Note that the techniques of [88] require using the exponential weights in order to obtainS S = the stretched exponential decay in (3.4) and (3.5). We will therefore ﬁrst obtain the stretched exponential decay statement for ω 0, and then deduce the full statement by interpolation. S S =

1 3 Remark 3.4 (ν ~ weights for ∂v derivatives). Notice that in all estimates (3.3a)–(3.5), every −1 3 ∂v derivative loses a power of ν ~ . These estimates can be improved for short times so that −1 3 −1 3 ν ~ is replaced by min ν ~ , t . For this one only needs to perform the corresponding change in the energy estimates.{ ⟨ We⟩)} will not pursue the details. The remainder of the paper will be devoted to the proof of Theorem 3.1. From now on, we work under the assumptions of Theorem 3.1. We will use the convention that, unless otherwise stated, all constants C or implicit constants in will be ≲ allowed to depend on q0 and Nmax, but are not allowed to depend on ǫ or ν, as long as ǫ0 and ν0 are suﬃciently small.

4. Landau collision operator In this section, we recall basic properties of the linear and quadratic Landau collision operators Lf and Γ f, f (recall (2.3)–(2.7)). Most of these results are proven by Guo [50, 54] or Strain–Guo( [88)]. 4.1. Basic properties.

Lemma 4.1 (Lemma 3 in [50]). The functions σij v and σi v (see (2.6)) are smooth and satisfy ( ) ( ) β β −1− β ∂v σij v ∂v σi v Cβ v S S , S ( )S + S 2( )S ≤ ⟨ ⟩ 2 σijgigj λ1 v Pvgi λ2 v I Pv gi , = + − and ( ){ } ( ){[ ] } 2 2 2 2 σij v vivjg Φij vivjµ g λ1 v v g , ( ) = ∗ { } = ( )S S THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 16

v⋅g v where Pvg ( v 2) , the projection of vector g onto v. The spectrum of σij v consists of a = S S ( ) simple eigenvalue λ1 v 0 associated with the vector v and a double eigenvalue λ2 v 0 ⊥ > > associated with v . Moreover,( ) there are constants c1 0 and c2 0 such that asymptotically,( ) > > as v → , we have ∞ S S −3 −1 λ1 v → c1 v , λ2 v → c2 v . ( ) ⟨ ⟩ ( ) ⟨ ⟩ 4.2. Lower bounds for the linear Landau operator. In this subsection we will collect estimates which show that the linear Landau operator L is coercive up to lower order terms. While most bounds can be found in [50, 54, 88], we need some small modiﬁcations when the vector ﬁeld commutator Y is involved.

Lower bounds from [50, 54, 88]. We give three lowers bounds for L: (1) a weighted lower bound with ∂v derivatives (Corollary 4.3), (2) a weighted lower bound without ∂v derivatives (Lemma 4.4), (3) an unweighted lower bound without ∂v derivatives (Lemma 4.5). From now on, let us deﬁne

v 1 if z 1 χm v χ , where χ 0, → 0, is smooth, χ z ≤ . (4.1) = SmS ∶ ∞ ∞ = 0 if z 2 ( ) ( ) [ ) [ ) ( ) ≥ In order to give our first lower bound for L, we estimate each piece in the decomposition in (2.3)–(2.5). The estimates (4.2)–(4.3) were proven in [88, Lemma 8], while (4.4) follows from an easy adaptation of the proof of (4.3). (We note that the exact statement in [88] may look slightly different: in [88], only polynomial weights with negative powers are used, though the actual proof applies more generally to our setting. In fact, this slightly modified version was used in [54, (93), (94)].)

Lemma 4.2 (Lemma 8 in [88]). Let β 0, ℓ R, ϑ 0, 2 and ﬁx 0 q0 1. Deﬁne ϑ q0SvS > ∈ ∈ < < ℓ S S { } w ℓ, ϑ v e 2 . Then for any small η 0, there exists Cη 0 such that ( ) = ⟨ ⟩ > >

2 ′ β ⎧ β 2 ⎫ w ℓ, ϑ ∂v g1 g2 dv ⎪η ∂v g1 ∆v ℓ,0 Cη χC g1 L ℓ,0 ⎪ g2 ∆v ℓ,ϑ . (4.2) SR3 K ⎪ Q η v ⎪ V ( ) [ ] V ≤ ⎨ β′ β Y Y ( ) + Y Y ( )⎬ Y Y ( ) ⎪ S S≤S S ⎪ ⎩⎪ ⎭⎪ Further,

w2∂β g ∂βg dv S 3 v A v − R [ ] β 2 β′ 2 β′′ 2 (4.3) ∂ g η ∂ g C ∂ g ′′ . v ∆v ℓ,ϑ Q v ∆v ℓ,ϑ η Q v ∆v ℓ+2 β −2 β ,ϑ ≥ Y Y ( ) − β′ β Y Y ( ) − β′′ β Y Y ( S S S S ) S S=S S S S

2 β β β′ β ′ w ∂v g1 ∂v g2 dv ∂v g1 ∆v ℓ+2 β −2 β ,ϑ ∂v g2 ∆v ℓ,ϑ . SR3 A Q ( S S S S ) ( ) (4.4) U [ ] U ≲ ( β′ β Y Y )Y Y S S≤S S Using (2.3)–(2.5), the inequalities (4.2)–(4.3) easily imply the lower bound given in the following corollary. This is the content of the ﬁrst part of [88, Lemma 9]. THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 17

Corollary 4.3 (Lemma 9 in [88]). Let β 0, ℓ R, ϑ 0, 2 and ﬁx 0 q0 1. Deﬁne ϑ q0SvS > ∈ ∈ < < ℓ S S { } w ℓ, ϑ v e 2 . Then for any small η 0, there exists Cη 0 such that ( ) = ⟨ ⟩ > > 2 β β β 2 β′ 2 w ℓ, ϑ ∂v Lg ∂v g dv ∂v g η ∂v g SR3 ∆v ℓ,ϑ Q ∆v ℓ,ϑ V ( ) [ ] V ≥ Y Y ( ) − β′ β Y Y ( ) S S=S S β′′ 2 C ∂ g ′′ . η Q v ∆v ℓ+2 β −2 β ,ϑ − β′′ β Y Y ( S S S S ) S S > 2 2 2 2 2 w ℓ, ϑ Lg g dv 1 q0 η g Cη χC g 2 . S 3 ∆v ℓ,ϑ η Lv R ( )[ ] ≥ ( − − ) Y Y ( ) − Y Y Next, we state a lower bound without derivatives in an unweighted space, which can be viewed as a scalar version of the positivity lemma for L in [54, Lemma 2].

Lemma 4.5 (Lemma 2 in [54]). We have ∫R3 Lg h dv ∫R3 Lh g dv, ∫R3 Lg g dv 0 and [ 2 R] 3 = [ ] [ ] ≥ 2 Lg 0 if and only if g Πg where Π is the Lv projection with respect to the Lv in- = = ( ) 2 ner product onto the null space of L, given by span √µ, vi√µ, v √µ , where i 1, 2, 3 . Furthermore, S S ∈ { } Lg g dv I Π g 2 . S 3 ∆v 0,0 R [ ] ≳ Y( − ) Y ( ) Lower bounds for the linear Landau operator when Y commutations are involved. We now turn to the analogue of Corollary 4.3 when the vector ﬁeld commutator Y v t x is involved. The main estimate is given in Corollary 4.7 below. = ∇ + ∇ Just as Corollary 4.3 is based on Lemma 4.2, the lower bound in Corollary 4.3 is based on a similar lemma (see Lemma 4.6). One diﬀerence between the bounds we prove here in Lemma 4.6 and the previous bounds where Y commutations are not involved is that we do not use exponential weights in Lemma 4.6. The proof of Lemma 4.6 is an adaptation of the ideas in the proof of Lemma 4.2. Lemma 4.6. Let ℓ N. Then ∈ 2ℓ β ω β ω β′ ω′ ω′ β′ v ∂v Y g1 ∂v Y g2 dv dx ∂v Y g1 ∆x,v ℓ,0 Y ∂v g2 ∆x,v ℓ,0 , ST3 SR3 A Q ( ) ( ) U ⟨ ⟩ [ ] U ≲ ( β′ β Y Y )Y Y Sω′S≤SωS S S≤S S and, for any small η 0, there exists C 0 such that > η > 2ℓ ′ ′ β ω β ω ω 2 v ∂v Y g1 g2 dv η ∂v Y g1 ∆v ℓ,0 Cη µY g1 L ℓ,0 g2 ∆v ℓ,0 . SR3 K Q ( ) Q v ( ) V ⟨ ⟩ [ ] V ≤ β′ β Y Y + ω′ ω Y Y ( )Y Y S S≤S S S S≤S S In addition, for any small η 0, there exists C 0 such that > η > 2ℓ β ω β ω β ω 2 β′ ω′ 2 v ∂v Y g ∂v Y g dv dx ∂v Y g η ∂v Y g ST3 SR3 A ∆x,v ℓ,0 Q ∆x,v ℓ,0 − ⟨ ⟩ [ ] ≥ Y Y ( ) − β′ β Y Y ( ) Sω′S≤SωS S S≤S S β′ ω′ 2 ω′ 2 (4.5) Cη ∂ Y g µY g 2 . Q v ∆x,v ℓ,0 Q Lx,v − (β′ β Y Y ( ) −ω′ ωY Y ) Sω′S≤SωS S S≤S S β′ + ωS ′ S≤Sβ +S ω −1 S S S S≤S S S S THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 18

Proof. We will only prove (4.5) as the other estimates are similar, if not simpler. ℓ 2 2 Let w v . To lighten the notation, we write Lv ℓ Lv ℓ, 0 , etc. in this proof. Using= (2.4⟨ ⟩) we get, ( ) = ( )

w2∂βY ω g ∂βY ωg dv dx S 3 S 3 v A v − T R ( ) ∂βY ωg 2 v ∆x,v ℓ,ϑ ≥ Y Y ( ) 2 β′ ω′ β′′ ω′′ β ω C w ∂v Y σij ∂v Y ∂vj g ∂v Y ∂vi g dv dx (4.6) Q ST3 SR3 − β′ + β′′ β V ( ) V Sω′S+Sω′′S≤SωS S βS′ +S ω′S≤S1 S S S S S≥ 2 β′ ω′ β′′ ω′′ β ω C ∂vi w ∂v Y σij ∂v Y ∂vj g ∂v Y g dv dx (4.7) Q ST3 SR3 − β′ + β′′ β V ( ) ( ) V Sω′S+Sω′′S≤SωS S S S S≤S S 2 β′ ω′ β′′ ω′′ β ω C w ∂v Y σijvivj ∂v Y g ∂v Y g dv dx (4.8) Q ST3 SR3 − β′ + β′′ β V ( )( ) V Sω′S+Sω′′S≤SωS S βS′ +S ω′S≤S1 S S S S S≥ 2 β′ ω′ β′′ ω′′ β ω C w ∂v Y ∂vi σi ∂v Y g ∂v Y g dv dx . (4.9) Q ST3 SR3 − β′ + β′′ β V ( ) V Sω′S+Sω′′S≤SωS S S S S≤S S

Estimates for (4.8) and (4.9). Since σij is independent of x, Y acts as a purely velocity derivative. It follows from Lemma 4.1 that 2 −2 β′′ ω′′ β ω 4.8 4.9 w v ∂v Y g ∂v Y g dv dx. (4.10) Q ST3 SR3 S( )S + S( )S ≲ β′′ β ⟨ ⟩ S SS S Sω′′S≤SωS S S≤S S For every m 1, let χ be as in (4.1). Then, ≥ m −1 β′′ ω′′ 2 β′′ ω′′ 2 −1 β′′ ω′′ 2 v ∂v Y g L2 ℓ χm∂v Y g L2 ℓ 1 χm v ∂v Y g L2 ℓ . Y⟨ ⟩ Y x,v( ) ≤ Y Y x,v( ) + Y( − ) ⟨ ⟩ Y x,v( ) For the large velocity part, we use the extra v weights to get ⟨ ⟩ −1 β′′ ω′′ 2 1 −1 2 β′′ ω′′ 2 1 β′′ ω′′ 2 χ v ∂ Y g 2 v ~ ∂ Y g 2 ∂ Y g 1 m v L ℓ v L ℓ v ∆x,v ℓ Y( − ) ⟨ ⟩ Y x,v( ) ≲ mY⟨ ⟩ Y x,v( ) ≲ mY Y ( ) For the small velocity part, we interpolate between Sobolev spaces to get that for any m 1, ′ > η 0, there is Cη′,m 0 so that > > 3 ′′ ′′ − ′′ ′′ ′′ β ω 2 ′ 2 β ω 2 ω 2 ′ χm∂v Y g L2 ℓ η v ∂vi ∂v Y g L2 ℓ Cη ,m µY g L2 Y Y x,v( ) ≤ Y⟨ ⟩ Y x,v( ) + Y Y x,v ′ β′′ ω′′ 2 ω′′ 2 η ∂ Y g C ′ µY g 2 . v ∆v ℓ η ,m L ≲ Y Y ( ) + Y Y x,v Hence, in total we have,

−1 β′′ ω′′ 2 ′ 1 β′′ ω′′ 2 ω′′ 2 v ∂ Y g 2 η ∂ Y g C ′ µY g 2 . v L ℓ v ∆x,v ℓ η ,m L Y⟨ ⟩ Y x,v( ) ≲ ( + m)Y Y ( ) + Y Y x,v Choosing 1 m and η′ suﬃciently small in terms of η, we obtain ~ η β′ ω′ 2 ω′ 2 4.8 4.9 ∂ Y g Cη µY g 2 . Q v ∆x,v ℓ Q Lx,v S( )S + S( )S ≤ 10 β′ β Y Y ( ) + ω′ ω Y Y Sω′S≤SωS S S≤S S S S≤S S THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 19

Estimates for (4.7). If β′ ω′ 1 in (4.7), the corresponding terms are bounded by S S + S S ≥ 2 −3 β′′ ω′′ β ω w v ∂vj ∂v Y g ∂v Y g dv dx, Q ST3 SR3 β′ β ⟨ ⟩ S SS S Sω′S≤SωS S S≤S S which has enough v decay for the argument above (with easy modiﬁcations) for (4.8), (4.9). ⟨ ⟩ ′ ′ We thus only need to consider β ω 0, for which we integrate by parts in ∂vj to get S S + S S = 2 β ω β ω ∂v w σij ∂ Y ∂v g ∂ Y g dv dx S 3 S 3 i v j v T R ( ) ( ) 1 2 2 2 β ω 2 ∂ w σij ∂v w ∂v σij ∂ Y g dv dx. S 3 S 3 vj vi i j v = −2 T R [ ( ) + ( ) ]( ) Now, by Lemma 4.1, this term is bounded above by w2 v −3 ∂βY ωg ∂βY ωg dv dx. S 3 S 3 v v T R ⟨ ⟩ S SS S This is better than the term in (4.10), which can therefore be controlled in the same way. Estimates for (4.6). If β′ ω′ 2, then using the Cauchy–Schwarz and the Young inequalities, we can bound S S + S S ≥

2 −3 β′′ ω′′ β ω w v ∂v Y ∂vj g ∂v Y ∂vi g dv dx Q ST3 SR3 β′′ β ⟨ ⟩ S SS S Sω′′S≤SωS β′′ + ωS ′′ S≤Sβ +S ω −2 S S S S≤S S S S ′ β ω 2 β′′ ω′′ 2 η ∂ Y g C ′ ∂ Y g . v ∆x,v ℓ η Q v ∆x,v ℓ ≲ Y Y ( ) + β′′ β Y Y ( ) Sω′′S≤SωS β′′ + ωS ′′ S≤Sβ +S ω −2 S S S S≤S S S S If β′ ω′ 1, then we have two cases: ′′ S S + S′ S = β ω β ω′′ Case 1: β 1. In this case ∂v Y ∂vl ∂v Y . Then integrating by parts in ∂vl , we get, S S = = 2 β′′ ω β′′ ω w ∂v σij ∂ Y ∂v g ∂v ∂v ∂ Y g dv dx S 3 S 3 l v j l i v T R ( )( ) 1 2 β′′ ω 2 ∂v w ∂v σij ∂ Y ∂v g dv dx. S 3 S 3 l l v j = −2 T R ( )( ) 2 2 −3 Since ∂vl w ∂vl σij w v (by Lemma 4.1), we have S ( )S ≤ ⟨ ⟩ 2 β′′ ω 2 β′′ ω 2 ∂v w ∂v σij ∂v Y ∂vj g ∂v Y g . ST3 SR3 l l Q ∆x,v ℓ ( )( ) ≲ β′′ β Y Y ( ) S S

2 β ω′′ β ω′′ w ∂v σij ∂ Y ∂v g Yl∂v ∂ Y g dv dx S 3 S 3 l v j i v T R ( )( ) 1 2 β ω′′ 2 ∂vl w ∂vl σij ∂v Y ∂vj g dv dx. = −2 ST3 SR3 ( )( ) We again get the required bound as above. Combining the estimates for (4.6)–(4.9), and choosing η′ small enough in terms of η, we obtain (4.5). THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 20

Using the decomposition (2.3)–(2.5), the previous lemma immediately implies the following lower bound for L: Corollary 4.7. Fix ℓ N. For any small η 0, there exists C 0 such that ∈ > η > v 2ℓ ∂βY ω Lg Y ω∂βg dv dx S 3 S 3 v v T R ⟨ ⟩ [ ] ω β 2 β′ ω′ 2 β′ ω′ 2 ω′ 2 Y ∂ g η ∂ Y g Cη ∂ Y g µY g 2 . v ∆x,v ℓ,0 Q v ∆x,v ℓ,0 Q v ∆x,v ℓ,0 Q Lx,v ≥ Y Y ( ) − β′ β Y Y ( ) − (β′ β Y Y ( ) +ω′ ωY Y ) Sω′S≤SωS Sω′S≤SωS S S≤S S S S≤S S β′ + ωS ′ S≤Sβ +S ω −1 S S S S≤S S S S 4.3. Upper bounds for the linear Landau operator. Using Lemma 4.2 and Lemma 4.6, we also obtain the following upper bounds for the linear Landau operator.

ϑ q0SvS Corollary 4.8. (1) For ℓ N, ϑ 0, 2 , and w ℓ, ϑ v ℓe 2 , ∈ ∈ { } ( ) = ⟨ ⟩ 2 β β′ w ℓ, ϑ ∂v Lg1 g2 dv dx ∂v g1 ∆x,v ℓ,ϑ g2 ∆x,v ℓ,ϑ . ST3 SR3 Q ( ) ( ) U ( ) [ ] U ≲ (β′ βY Y )Y Y S S≤S S (2) For any ℓ N, ∈ 2ℓ β ω β′ ω′ v ∂v Y Lg1 g2 dv dx ∂v Y g1 ∆x,v ℓ,0 g2 ∆x,v ℓ,0 . ST3 SR3 Q ( ) ( ) U ⟨ ⟩ [ ] U ≲ (β′ βY Y )Y Y Sω′S≤SωS S S≤S S Proof. Recalling (2.3)–(2.5), the ﬁrst estimate follows from Lemma 4.2, while the second estimate follows from Lemma 4.6. 4.4. Bounds for the nonlinear Landau operator. We close this section with bounds for the nonlinear Landau operator (see (2.7)). We begin with the following estimate from [88].

Lemma 4.9 (Lemma 10 in [88]). Let ϑ 0, 2 , ℓ 0 and ﬁx 0 q0 1. Deﬁne w ℓ, ϑ ϑ ∈ ≥ < < = q0SvS ′ { } ( ) v ℓe 2 . Then for any ℓ R we have, ⟨ ⟩ ∈ 2 β ω β ω w ℓ, ϑ ∂ Y Γ g1,g2 ∂ Y g3 dv S 3 v v V R ( ) ( ) V β ω β′ ω′ β′′ ω′′ (4.11) ∂ Y g ∂ Y g 2 ∂ Y g Q v 3 ∆v ℓ,ϑ v 1 Lv v 2 ∆v ℓ,ϑ ≲ ′ + ′′ ( ) ( ) β β β Y Y Y β′ ω′Y Y β′′ Y ω′′ S ′S+S ′′S≤S S 2 ω ω ω ∂v Y g1 ∆v 0,0 ∂v Y g2 Lv ℓ,ϑ . S S S S≤S S +Y Y ( )Y Y ( ) We also need the following more pessimistic estimates for Γ g1,g2 , in which we do not exploit the divergence structure. (They will be relevant for contr( olling) the inhomogeneous terms in the density estimates; see (8.37) and (11.31).)

ϑ q0SvS ℓ Lemma 4.10. Let ϑ 0, 2 , ℓ 0 and ﬁx 0 q0 1. Deﬁne w ℓ, ϑ v e 2 . Then ∈ ≥ < < = β ω { } β′+β′ ω′ ( ) ⟨ ⟩ β′′+β′′ ω′′ w ℓ, ϑ ∂ Y g ,g 2 ∂ ̃ Y g 2 w ℓ, ϑ ∂ ̃ Y g 2 . v Γ 1 2 Lv Q Q v 1 Lv v 2 Lv Y ( ) ( )Y ≲ β′ + β′′ β ′ + ′′ Y Y Y ( ) Y β̃ β̃ 2 Sω′S+Sω′′S≤SωS S S S S≤ Proof. Recalling (2.7), we knowS S thatS S≤S S

1 2 1 2 Γ g1,g2 ∂vi Φij µ ~ g1 ∂vj g2 Φij viµ ~ g1 ∂vj g2 ( ) = ⋆ ( ) − ⋆ (4.12) 1 2 1 2 ∂v Φij µ ~ ∂v g1 g2 Φij viµ ~ ∂v g1 g2. − i ⋆ ( j ) + ⋆ j THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 21

−1 1 3 2 3 Now, it is easy to check that v h L∞ h ~1 h ~2 (for instance by adapting the proof v Lv Lv YS S ⋆ Y ≲ Y Y 1 Y2 Y − ′ ′ ℓ ∞ ℓ 2 of [69, Lemma 5.1]). It follows that Φij v µ ~ h Lv v h Lv for any ℓ 0. Therefore, using H¨older’s inequalityY and⋆ (⟨ apply⟩ the)Y above≲ Y⟨ observat⟩ Y ion for h being≥ derivatives of g1 or g2, we obtain the required result.

5. Setting up the energy estimates In this section, we set up the main energy estimates as well as introduce the global energy and dissipation norms for the full nonlinear Vlasov–Poisson–Landau system (2.2a)–(2.2b). Precisely, for a given electric ﬁeld E xφ, we shall derive energy estimates for smooth solutions f to the following Vlasov–Landau= −∇ equation

~~Dtf E vf νLf (5.1) − ⋅ + = Q where Dt denotes the transport operator~~

Dt ∂t v x E v. (5.2) = + ⋅ ∇ + ⋅ ∇ The transport-diffusion structure of (5.1) is clear, being transported by the electric field in the phase space and diffused through the Landau collision operator L. We note that a similar α β ω structure also holds for derivatives of ∂x ∂v Y f for any triple of multi-indices α,β,ω . The main result of this section will be given in Subsection 5.4 below. ( ) Remark 5.1. The equation (5.1) is exactly the Vlasov–Poisson–Landau equation (2.2a) with 2E v√µ νΓ f, f . (5.3) Q = ⋅ + ( ) Note that the first term in is linear in f and thus it cannot in principle be treated as a 2 remainder. However, this linearQ term is very localized both in velocity v (through µ e− v ) = S S and in frequency ∂v (through the Poisson equation), a fact that will play a role in our nonlinear analysis. 5.1. Basic energy estimates. We start with basic energy estimates for the transport- diffusion equation (5.1).

q0 if ϑ 2 Lemma 5.2. Let ℓ R, 0 q0 1, and ϑ 0, 2 . Deﬁne q = . Then, there is a ∈ < < ∈ = 0 if ϑ 0 { } = positive constant θ θ ϑ, q0 so that smooth solutions to (5.1) satisfy = ( ) d q+1 φ 2 q+1 φ 2 L,ℓ T,ℓ Q,ℓ e f 2 θν e f ν , (5.4) ( ) L ℓ,ϑ ( ) ∆x,v ℓ,ϑ 0 0 0 dtY Y x,v( ) + Y Y ( ) ≲ R + R + R where the remainders are deﬁned by L,ℓ q+1 φ 2 0 µe( ) f L2 , R = Y Y x,v T,ℓ q+1 φ 2 ∞ ∞ 2 0 ∂tφ Lx E Lx e( ) f L ℓ,ϑ , R = (Y Y + Y Y )Y Y x,v( ) Q,ℓ 2 q+1 φ 2 e ( ) w f dx dv . 0 U 3 3 R = U T ×R Q U Remark 5.3. Observe that there are three contributions to the energy production of (5.1): namely, the remainders from the transport dynamics Dt, the Landau operator L, and the source . Q THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 22

~~Proof. Directly from the transport structure of (5.1), we compute~~

~~1 d q+1 φ 2 1 2 2 q+1 φ 2 e( ) wf 2 f Dt 2v E e ( ) w dv dx Lx,v U 3 3 2 dtY Y = 2 T ×R S S ( + ⋅ )[ ] 2 q+1 φ 2 νLf e ( ) w f dv dx. + UT3×R3 − +Q~~

ϑ ℓ qSvS Recalling that E xφ and w v e 2 , we compute = −∇ = ⟨ ⟩ 1 2 q+1 φ 2 2 q+1 φ 2 Dt 2v E e ( ) w q 1 ∂t v x φ E v log w v E e ( ) w 2( + ⋅ )[ ] = ( + )( + ⋅ ∇ ) + ⋅ ∇ + ⋅ (5.5) q ϑ−2 −2 2 q+1 φ 2 ϑ v 2 v E q 1 ∂tφ ℓ v v E e ( ) w , = 2( S S − ) ⋅ + ( + ) + ⟨ ⟩ ⋅ in which we note that the ﬁrst term vanishes, since either ϑ 2 or q 0 (when ϑ 0). This proves = = =

2 2 q+1 φ 2 q+1 φ 2 f Dt 2v E e ( ) w dv dx 2 ∂tφ L∞ ℓ E L∞ e( ) f 2 . U 3 3 x Lx,v ℓ,ϑ T ×R S S ( + ⋅ )[ ] ≤ ( Y Y + Y Y )Y Y ( ) − 2 Finally, using Lemma 4.4 with η 1 q and noting φ is independent of v, we get = 2 2 2 q+1 φ 2 1 q q+1 φ 2 q+1 φ 2 e ( ) w fLf dv dx − e( ) f Cq χC e( ) f 2 U 3 3 ∆x,v ℓ,ϑ q Lx,v ℓ,0 T ×R ≥ 2 Y Y ( ) − Y Y ( ) where χCq is a cut oﬀ function near the origin. The lemma follows. Remark 5.4. Note that the basic energy estimate derived in Lemma 5.2 uses only the equation (5.1) for a given electric ﬁeld E (i.e. the Poisson equation was not used).

~~5.2. Derivative energy estimates. Next, we obtain the following energy estimates for derivatives.~~

Lemma 5.5. Let ℓ R, 0 q0 1, ϑ 0, 2 , and α,β,ω be any triple of multi-indices. If ∈ < < ∈ { } ( ) q0 if ϑ 2 ω 0, we take ϑ 0. Deﬁne q = . > = = 0 if ϑ 0 S S = Then, there is a positive constant θ so that smooth solutions to (5.1) satisfy

d q+1 φ α β ω 2 q+1 φ α β ω 2 T,ℓ L,ℓ Q,ℓ e ∂ ∂ Y f 2 θν e ∂ ∂ Y f ν , ( ) x v L ℓ,ϑ ( ) x v ∆x,v ℓ,ϑ α,β,ω α,β,ω α,β,ω dtY Y x,v( ) + Y Y ( ) ≲ R + R + R (5.6) in which we have collected the remainders T,ℓ due to the transport dynamics: ● Rα,β,ω T,ℓ q+1 φ α+β′ β′′ ω q+1 φ α β ω e( ) ∂ ∂ Y f 2 e( ) ∂ ∂ Y f 2 α,β,ω Q x v Lx,v ℓ,ϑ x v Lx,v ℓ,ϑ R = β′′ β −1 Y Y ( )Y Y ( ) S βS=S′ 1S S S= q+1 φ α β ω 2 ∞ ∞ 2 ∂tφ Lx E Lx e( ) ∂x ∂v Y f L ℓ,ϑ , + (Y Y + Y Y )Y Y x,v( ) the remainders L,ℓ due to the linear Landau operator: ● Rα,β,ω THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 23

L,ℓ q+1 φ α β′ ω′ 2 q+1 φ α β′ ω′ 2 η e( ) ∂ ∂ Y f C e( ) ∂ ∂ Y f α,β,ω Q x v ∆x,v ℓ,ϑ η Q x v ∆x,v ℓ,ϑ R = β′ β Y Y ( ) + β′ β Y Y ( ) Sω′S≤SωS Sω′S≤SωS S S≤S S β′ + ωS ′ S≤Sβ +S ω −1 S S S S≤S S S S q+1 φ α ω′ 2 Cη µe( ) ∂ Y f 2 , Q x Lx,v + ω′ ωY Y S S≤S S for any small η 0, > Q,ℓ the remainders due to the source term (of the equations for derivatives): ● Rα,β,ω Q,ℓ 2 q+1 φ 2 α β ω e ( ) w ∂ ∂ Y f α,β,ω dv dx α,β,ω U 3 3 x v R = U T ×R Q U ϑ ℓ qSvS where w v e 2 , and = ⟨ ⟩ α β ω α β ω α,β,ω ∂x ∂v Y E v E v, ∂x ∂v Y f. (5.7) Q ∶= Q − [ ⋅ ∇ − ⋅ ] α β ω Proof. Directly from (5.1), we observe that derivatives ∂x ∂v Y solve

α β ω β ω α α β ω α β ω Dt E v ∂x ∂v Y f ν∂v Y L∂x f ∂x ∂v Y Dt E v, ∂x ∂v Y f. − ⋅ + [ ] = Q + [ − ⋅ ] Note that ∂t v x, ∂x 0 and ∂t v x,Y 0. Hence, for β 0, we compute [ + ⋅ ∇ ] = [ + ⋅ ∇ ] = S S > α β ω β′ β′′ α ω ∂t v x, ∂x ∂v Y Q ∂x ∂v ∂x Y . (5.8) [ + ⋅ ∇ ] = − β′′ β −1 S βS=S′ 1S S S= Thus, the lemma follows directly from performing a similar energy estimate as done in the previous lemma and using Corollary 4.3 and Corollary 4.7 (which contribute precisely into the remainder L,ℓ ). Rα,β,ω T,ℓ Remark 5.6. Note that the first term in α,β,ω is linear (due to (5.8)), which reflects precisely the linear growth in t of v-derivativesR in the regime where the transport dynamics in (5.1) is dominant. 5.3. Hypocoercivity estimates. We next derive hypocoercivity estimates that capture precisely the transport-diffusion structure of (5.1). Precisely, we obtain the following key lemma.

Lemma 5.7. Let ℓ R, 0 q0 1, ϑ 0, 2 , and α,β,ω be any triple of multi-indices. ∈ < < ∈ { } ( ) q0 if ϑ 2 Deﬁne q = as before. If ω 0, we take q 0 and ϑ 0. = 0 if ϑ 0 S S > = = = ϑ ℓ qSvS Then, for w v e 2 , smooth solutions to (5.1) satisfy = ⟨ ⟩ d 2 q+1 φ 2 α β ω α β ω q+1 φ α β ω 2 e ( ) w ∂xj ∂x ∂v Y f∂vj ∂x ∂v Y f dv dx e( ) ∂xj ∂x ∂v Y f 2 U 3 3 Lx,v ℓ,ϑ dt T ×R + Y Y ( ) (5.9) T,ℓ L,ℓ Q,ℓ ≲Zα,β,ω +Zα,β,ω +Zα,β,ω in which we have denoted THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 24

by T,ℓ the contribution from the transport dynamics: ● Zα,β,ω T,ℓ q+1 φ α β ω q+1 φ α β ω ∞ ∞ 2 2 α,β,ω ∂tφ Lx E Lx e( ) ∂x∂x ∂v Y f Lx,v ℓ,ϑ e( ) ∂v∂x ∂v Y f Lx,v ℓ,ϑ Z = (Y Y + Y Y )Y Y ( )Y Y ( ) 2 q+1 φ 2 α β ω α+β′ β′′ ω e ( ) w ∂xj ∂x ∂v Y f∂vj ∂x ∂v Y f dv dx Q UT3×R3 + β′′ β −1 U U S βS=S′ 1S S S= 2 q+1 φ 2 α β ω α+β′ β′′ ω e ( ) w ∂vj ∂x ∂v Y f∂xj ∂x ∂v Y f dv dx , Q UT3×R3 + β′′ β −1 U U S βS=S′ 1S S S= by L,ℓ the contribution from the linear Landau operator: ● Zα,β,ω

L,ℓ 2 q+1 φ 2 α β ω α β ω ν e ( ) w ∂x ∂ ∂ Y f∂v ∂ ∂ Y Lf dv dx α,β,ω U 3 3 j x v j x v Z = U T ×R [ ] U 2 q+1 φ 2 α β ω α β ω ν e ( ) w ∂vj ∂x ∂v Y f∂xj ∂x ∂v Y Lf dv dx , + U UT3×R3 [ ] U by Q,ℓ the contribution from the source: ● Zα,β,ω

Q,ℓ 2 q+1 φ 2 α β ω α β ω e ( ) w ∂x ∂ ∂ Y f α,β+e ,ω ∂v ∂ ∂ Y f α+e ,β,ω dv dx , α,β,ω U 3 3 j x v j j x v j Z = U T ×R Q + Q U ej ej recalling α,β,ω deﬁned as in (5.7), with ∂x ∂x and ∂v ∂v . Q = j = j T,ℓ Remark 5.8. Note that the last integral term in the above remainders α,β,ω are of the q+1 φ α β ω 2 Z same order as the good term e( ) ∂xj ∂x ∂v Y f L2 ℓ,ϑ on the left hand side! A crucial Y Y x,v( ) point here is that these last remainder terms vanish for β 0, while for β 0 they are controlled by the good terms for β 0 and the dissipationS norms;S = see (5.30S) below.S > S S = Proof. Recall that the derivatives satisfy

Dt E v ∂xj f ν∂xj Lf ∂xj E v E v, ∂xj f ( − ⋅ ) + [ ] = Q + [ ⋅ ∇ − ⋅ ] (5.10) Dt E v ∂vj f ν∂vj Lf ∂xj f ∂vj E v E v, ∂vj f, ( − ⋅ ) + [ ] = − + Q + [ ⋅ ∇ − ⋅ ] in which we note that the ﬁrst term on the right in the second equation plays a crucial role. Indeed, we compute

Dt 2E v ∂xj f∂vj f ν ∂xj f∂vj Lf ∂vj f∂xj Lf ( − ⋅ )(2 ) + ( [ ] + [ ]) ∂xj f ∂xj f∂vj ∂vj f∂xj ∂vj f E v E v, ∂xj f ∂xj f E v E v, ∂vj f. = −S S + Q + Q + [ ⋅ ∇ − ⋅ ] + [ ⋅ ∇ − ⋅ ] 2 q+1 φ 2 Therefore, multiplying the above equation by e ( ) w and integrating the result, we get

d 2 q+1 φ 2 2 q+1 φ 2 2 e ( ) w ∂x f∂v f dv dx e ( ) w ∂x f dv dx U 3 3 j j U 3 3 j dt T ×R + T ×R S S 2 q+1 φ 2 ∂x f∂v f Dt 2E v e ( ) w dv dx U 3 3 j j = T ×R ( )( + ⋅ )[ ] 2 q+1 φ 2 ν ∂x f∂v Lf ∂v f∂x Lf ∂x f∂v ∂v f∂x e ( ) w dv dx U 3 3 j j j j j j j j + T ×R − ( [ ] + [ ]) + Q + Q 2 q+1 φ 2 ∂vj f E v E v, ∂xj f ∂xj f E v E v, ∂vj f e ( ) w dv dx. + UT3×R3 [ ⋅ ∇ − ⋅ ] + [ ⋅ ∇ − ⋅ ] THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 25

~~In view of (5.5), we note~~

2 q+1 φ 2 ∂x f∂v f Dt 2E v e ( ) w dv dx U 3 3 j j T ×R ( )( + ⋅ )[ ] q+1 φ q+1 φ ∞ ∞ 2 2 ∂tφ Lx E Lx e( ) ∂xf Lx,v ℓ,ϑ e( ) ∂vf Lx,v ℓ,ϑ . ≲ (Y Y + Y Y )Y Y ( )Y Y ( ) This yields the lemma for α,β,ω 0. For any triple of α,β,ω , we simply observe that α β ω ( ) = ( ) the derivatives ∂x ∂v Y f satisfy similar transport-diffusion equations to (5.10), upon noting that β α ω α+β′ β′′ ω ∂t v x, ∂xj ∂v ∂x Y Q ∂xj ∂x ∂v Y [ + ⋅ ∇ ] = − β′′ β −1 S βS=S′ 1S S S= β α ω α+β′ β′′ ω β α ω ∂t v x, ∂vj ∂v ∂x Y Q ∂vj ∂x ∂v Y ∂xj ∂v ∂x Y . [ + ⋅ ∇ ] = − β′′ β −1 − S βS=S′ 1S S S= α β ω 2 The last term in the second equation above yields the crucial bound on ∂xj ∂x ∂v Y f in (5.9). Collecting terms, we obtain the lemma. S S 5.4. The hypocoercive energies. We are now ready to introduce the main energy esti- α β ω mates, which are an intricate combination of the energy estimates derived for ∂x ∂v Y f in the previous sections. In addition to the ν-dependence that respects the hypocoercivity scaling of the Landau equations, the norms also reflect the weight loss in v due to the Landau collision operator. The partial energy and dissipation norms. For each triple of multi-indices α,β,ω , we introduce the partial energy and dissipation norms ( ) 2 q+1 φ α′ 2 H (ϑ) A0 e( ) ∂ H 2 − ′ Q x Lx,v ℓα,β,ω 2 α ,ϑ Y YEα,β,ω ∶= α′ 1 Y Y ( S S ) S S≤ (5.11) 1 3 2 q+1 φ 2 3 q+1 φ 2 2 − 2 ν ~ e ( ) xH, vH Lx,v ℓα,β,ω 2,ϑ ν ~ e( ) vH L ℓ −2,ϑ , + ⟨ ∇ ∇ ⟩ ( ) + Y ∇ Y x,v( α,β,ω ) 2 2 3 q+1 φ α′ 2 q+1 φ 2 H (ϑ) ν ~ A0 e( ) ∂ H − ′ e( ) xH 2 − Q x ∆x,v ℓα,β,ω 2 α ,ϑ Lx,v ℓα,β,ω 2,ϑ Y YDα,β,ω ∶= α′ 1 Y Y ( S S ) + Y ∇ Y ( ) S S≤ (5.12) ν4 3 e q+1 φ H 2 , ~ ( ) v ∆x,v ℓ −2,ϑ + Y ∇ Y ( α,β,ω ) β 3 α β ω which are used for derivatives H ν ∂ ∂v Y f, with ℓ as in (2.9), and A0 a (large) = S S~ x α,β,ω q0 if ϑ 2 constant to be determined. Here, q = as above. Moreover, when ω 0, only = 0 if ϑ 0 S S > the ϑ 0 case will be considered. = = q+1 φ Note that these norms are weighted by e( ) , for a given electric potential φ. In the nonlinear analysis, we shall bootstrap the nonlinear solution so that φ remains sufficiently ∞ small in Lx (and in fact decays rapidly in time). Therefore, the weight is harmless. The top-order partial energy and dissipation norms. We need a variation of the ϑ ϑ ϑ partial energy and dissipation norms α,β,ω( ) and α,β,ω( ) norms, which we denote by α,β,ω( ) ϑ E D Ẽ and ( ) . The difference is that they include one more ∂ derivative, which is useful to ̃α,β,ω v handlingD the loss of derivative from the density estimates; see Section 1.1.8. More precisely, β 3 α β ω for H ν ∂ ∂v Y f as before, define = S S~ x 2 2 −1 2 β′ 3 q+1 φ β′ 2 H (ϑ) H (ϑ) A0 ν S S~ e( ) ∂v H 2 − , Q Lx,v ℓα,β,ω 4,ϑ (5.13) Y YẼα,β,ω ∶= Y YEα,β,ω + β′ 2 Y Y ( ) S S= THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 26

2 2 −1 2+2 β′ 3 q+1 φ β′ 2 H (ϑ) H (ϑ) A ν( S S)~ e( ) ∂ H − . 0 Q v ∆x,v ℓα,β,ω 4,ϑ (5.14) Y YD̃α,β,ω ∶= Y YDα,β,ω + β′ 2 Y Y ( ) S S= The combined energy and dissipation norms. Given any ϑ 0, 2 and any quadruple low N 4 low ∈ { } Nα , Nα,β, Nβ, Nω 0 with Nα,β Nω Nmax, Nα , Nβ Nα,β, define the norms (E ϑ ) D∈ (ϑ ∪ { }) + ≤ ≤ ( low) and ( low) by Nα ,Nα,β,Nβ,Nω Nα ,Nα,β ,Nβ,Nω 2 α β 2 α β ω 2 f E(ϑ) Q ∂x ∂v f (ϑ) Q ∂x ∂v Y f (0) , low = low α,β,0 low α,β,ω Y Y Nα ,Nα,β ,Nβ,Nω ∶ α N , βY N YE + α N , βY N YE S S≥ α S S≤ β S S≥ α S S≤ β α + β N α + β N , 1 ω Nω S S S S≤ α,β S S S S≤ α,β ≤S S≤ 2 α β 2 α β ω 2 (5.15) f D(ϑ) Q ∂x ∂v f (ϑ) Q ∂x ∂v Y f (0) . low = low α,β,0 low α,β,ω Y Y Nα ,Nα,β ,Nβ,Nω ∶ α N , βY N YD + α N , βY N YD S S≥ α S S≤ β S S≥ α S S≤ β α + β N α + β N , 1 ω Nω S S S S≤ α,β S S S S≤ α,β ≤S S≤ We emphasize two points about the definition (5.15): low (1) Nα is a lower bound, while the other are upper bounds. E ϑ D ϑ (2) Even though we may use ϑ 2 in the ( low) and ( low) norms, the = Nα ,Nα,β,Nβ,Nω Nα ,Nα,β,Nβ,Nω exponential v-weight is only present when ω 0. S S = low The top-order combined energy and dissipations norms. Given ϑ, Nα , Nα,β, Nβ, Nω as above, we also define corresponding combined norms which include the extra ∂v derivatives as in (5.13), (5.14), 2 α β 2 α β ω 2 f E(ϑ) Q ∂x ∂v f (ϑ) Q ∂x ∂v Y f (0) , low = low α,β,0 low α,β,ω Y ỸNα ,Nα,β ,Nβ ,Nω ∶ α N , βY N YẼ + α N , βY N YẼ S S≥ α S S≤ β S S≥ α S S≤ β α + β N α + β N , 1 ω Nω S S S S≤ α,β S S S S≤ α,β ≤S S≤ 2 α β 2 α β ω 2 (5.16) f D(ϑ) Q ∂x ∂v f (ϑ) Q ∂x ∂v Y f (0) . low = low α,β,0 low α,β,ω Y ỸNα ,Nα,β,Nβ ,Nω ∶ α N , βY N YD̃ + α N , βY N YD̃ S S≥ α S S≤ β S S≥ α S S≤ β α + β N α + β N , 1 ω Nω S S S S≤ α,β S S S S≤ α,β ≤S S≤ For brevity, we also introduce 2 2 2 2 f E(ϑ) Q f E(ϑ) , f D(ϑ) Q f D(ϑ) . (5.17) N 0,N ,N ,N N 0,N ,N ,N Y Ỹ ∶= Nα,β+Nω N Y Ỹ α,β α,β ω Y Ỹ ∶= Nα,β +Nω N Y Ỹ α,β α,β ω ≤ ≤ The primed energy and dissipation norms. Finally, for each of the norms defined above, we introduce an analogous norm, labelled by ϑ ′ instead of ϑ , which is defined q v 2 ( ) ( q)′ v 2 ′ 1 so that when ϑ 2, the exponential v-weights e S S are replaced by e S S , where q 2 q; cf. (2.12)–(2.13).= In other words, starting from (5.11), we define = 2 q+1 φ α′ 2 H (ϑ)′ A0 e( ) ∂ H 2 − ′ ′ Q x Lx,v ℓα,β,ω 2 α ,ϑ Y YEα,β,ω ∶= α′ 1 Y Y ( S S ) S S≤ (5.18) 1 3 2 q+1 φ 2 3 q+1 φ 2 2 − ′ 2 ν ~ e ( ) xH, vH Lx,v ℓα,β,ω 2,ϑ ν ~ e( ) vH L ℓ −2,ϑ ′ , + ⟨ ∇ ∇ ⟩ ( ) + Y ∇ Y x,v( α,β,ω ) β 3 α β ω for H ν ∂ ∂v Y f, and make similar definitions for = S S~ x β 3 α β ω 2 2 2 ν ∂ ∂ Y f ′ , f ′ , f ′ , S S~ x v (ϑ) E(ϑ) D(ϑ) α,β,ω low low Y YD̃ Y Y Nα ,Nα,β,Nβ ,Nω Y Y Nα ,Nα,β,Nβ ,Nω 2 2 2 2 (5.19) f ′ , f ′ , f ′ , f ′ , E(ϑ) D(ϑ) E(ϑ) D(ϑ) low low N N Y ỸNα ,Nα,β ,Nβ,Nω Y ỸNα ,Nα,β ,Nβ,Nω Y Ỹ Y Ỹ by modifying (5.12), (5.13), (5.14), (5.15) and (5.17). THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 27

5.5. The main energy estimates. In the following proposition, we estimate all the remainder terms in Lemma 5.5 and Lemma 5.7 except for the Q and Q terms. For the full nonlinear solution, those terms will be treated in Sections 9 andR 12. Z Proposition 5.9. Let ϑ 0, 2 , and N low, N , N , N N 0 4 with N N N , ∈ α α,β β ω ∈ α,β ω ≤ max low { } ( E ϑ ) ( ∪D{ϑ }) + Nα , Nβ Nα,β. Recall the deﬁnitions of ( low) and ( low) in (5.15). ≤ Nα ,Nα,β,Nβ,Nω Nα ,Nα,β,Nβ,Nω There is a positive constant θ so that for N low 1, smooth solutions to (5.1) satisfy α ≥ d 2 1 3 2 f E(ϑ) θν ~ f D(ϑ) dt low low Y Y Nα ,Nα,β ,Nβ,Nω + Y Y Nα ,Nα,β,Nβ ,Nω (5.20) 1 2 ∂tφ L∞ φ ,∞ f (ϑ) α,β,ω, x Wx E Q low low ≲ (Y Y + Y Y )Y Y Nα ,Nα,β ,Nβ,Nω + α N , βR N S S≥ α S S≤ β α + β N , ω Nω S S S S≤ α,β S S≤ and for N low 0, smooth solutions to (5.1) satisfy α = d 2 1 3 2 f (ϑ) θν ~ f (ϑ) E0 D0 dtY Y ,Nα,β ,Nβ,Nω + Y Y ,Nα,β ,Nβ,Nω 2 2 (5.21) ∞ 1,∞ ν a, b, c ∂tφ Lx φ W f (ϑ) α,β,ω, x E0 Q ≲ S( )S + (Y Y + Y Y )Y Y ,Nα,β ,Nβ,Nω + β NR S S≤ β α + β N , ω Nω S S S S≤ α,β S S≤ where the remainders are calculated by 2 β + β′ 3 Q,ℓ−2 α′ −2 β′ 1 3 2 β 3 Q,ℓ−2 α,β,ω Q ν (S S S S)~ α+α′,βS +βS ′,ωS S ν ~ ν S S~ α,β,ω (5.22) R = α′ + β′ 1 R + Z S S S S≤ Q,ℓ Q,ℓ 2 2 with α,β,ω and α,β,ω as introduced in Lemma 5.5 and Lemma 5.7, and a, b, c a 3 R 2 2 Z S( )S ∶= S S + ∑j 1 bj c , where = S S + S S 2 a f µ dv dx, bj fvj µ dv dx, c f v µ dv dx. U 3 3 √ U 3 3 √ U 3 3 √ ∶= T ×R ∶= T ×R ∶= T ×R S S Remark 5.10. We note that the “remainders” α,β,ω do contain linear terms (due to from (5.3)), the control of which by the energy and dissipationR norms is certainly not immediate;Q see Section 9 for the full treatment of these and the other nonlinear terms.

Proof. Let α,β,ω be any triple of multi-indices, with α β ω Nmax, and let ℓ ℓα,β,ω + + ≤ = and w wα,β,ω( be the) weight functions deﬁned as in (2.8S)–(S 2.9S ).S RecallingS S the partial energy and dissipation= norms and appropriately combining Lemma 5.5 and Lemma 5.7, we obtain

~~d α β ω 2 1 3 α β ω 2 ′ ∂x ∂v Y f (ϑ) θν ~ ∂x ∂v Y f (ϑ) α,β,ω, (5.23) dtY YEα,β,ω + Y YDα,β,ω ≲ R where the remainders are calculated by~~

′ 2 β 3 T,ℓ−2 α′ L,ℓ−2 α′ Q,ℓ−2 α′ α,β,ω A0ν S S~ Q α+α′,β,ωS S ν α+α′,β,ωS S α+α′,β,ωS S R = α′ 1 R + R + R S S≤ 2 β 3 2 β′ 3 T,ℓ−2 β′ L,ℓ−2 β′ Q,ℓ−2 β′ ν S S~ Q ν S S~ α,β+βS′,ωS ν α,β+βS′,ωS α,β+βS′,ωS (5.24) + β′ 1 R + R + R S S≤ 1 3 2 β 3 T,ℓ−2 L,ℓ−2 Q,ℓ−2 ν ~ ν S S~ + Zα,β,ω +Zα,β,ω +Zα,β,ω where the remainders were introduced previously in Lemma 5.5 and Lemma 5.7. THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 28

T,ℓ Estimates on α,β,ω. Let us take care of the remainders arising due to the transport R ′ ′ dynamics. We ﬁrst consider the T terms in (5.24) with α 0 and β 0. We claim that R S S = S S = 2 β 3 T,ℓ −1 2 1 3 α β′ ω 2 α β ω 2 ν S S~ A ~ ν ~ ∂ ∂ Y f ∂ φ ∞ E ∞ ∂ ∂ Y f . α,β,ω 0 Q x v (ϑ) t Lx Lx x v (ϑ) (5.25) R ≲ β′ β Y YDα,β′,ω + (Y Y + Y Y )Y YEα,β,ω S S

2 β 3 T,ℓ−2 α′ 2 β 3 2 β′ 3 T,ℓ−2 β′ A0ν S S~ Q α+α′,β,ωS S ν S S~ Q ν S S~ α,β+βS′,ωS α′ 1 R + β′ 1 R S S≤ S S≤ α β ω 2 1 2 1 3 α β′ ω 2 A ∂ φ ∞ E ∞ ∂ ∂ Y f A ~ ν ∂ ∂ Y f 0 t Lx Lx x v (ϑ) 0 ~ Q x v (ϑ) ≲ (Y Y + Y Y )Y YEα,β,ω + β′ β Y YDα,β′,ω S S

T,ℓ T,ℓ−2 Estimates on α,β,ω. Next, let us give bounds on α,β,ω deﬁned as in Lemma 5.7. We claim that Z Z 1 3 2 β 3 T,ℓ−2 −1 2 1 3 α′ β′′ ω α β ω ν ~ ν S S~ A0 ~ ν ~ ∂x ∂v Y f (ϑ) ∂x ∂v Y f (ϑ) α,β,ω Q ′ ′ Z ≲ α′ α +1 Y YDα ,β ,ω Y YDα,β,ω S βS=S′ βS (5.30) S S

~~The third terms require slightly more work. First, we bound~~

2 β 3 L,ℓ−2 α′ ,3 2 β 3 2 β′ 3 L,ℓ−2 β′ ,3 A0νν S S~ Q α+α′,β,ωS S νν S S~ Q ν S S~ α,β+βS′,ωS α′ 1 R + β′ 1 R S S≤ S S≤ 2 β 3 q+1 φ α+α′ ω′ 2 (5.33) CηA0νν S S~ µe( ) ∂ Y f 2 Q x Lx,v ≲ α′ 1Y Y ωS ′′ S≤ω S S≤S S

We will analyze the RHS of (5.33) further. The issue here is that if we control it directly with the dissipation norms, we would not have enough smallness. For any function g, decompose ′ ′ g g 0 g 0, where g 0 t, v T3 g t, x, v dx. For α 1 and ω ω , we bound = = + ~= = ( ) ∶= ∫ ( ) S S ≤ S S ≤ S S

q+1 φ α+α′ ω′ 2 A0ν µe( ) ∂x Y f L2 Y Y x,v (5.34) α+α′ ω′ 2 α+α′ ω′ 2 q+1 φ α+α′ ω′ 2 A0ν µ ∂x Y f 0 L2 µ ∂x Y f 0 L2 µ e( ) 1 ∂x Y f L2 . = (Y ( )~= Y x,v + Y ( )= Y x,v + Y ( − ) Y x,v )

′ 2 ′ ∞ α ω The last term is clearly bounded by ν φ L ∂x Y f (ϑ) . As for the ﬁrst term, if α 1, Y Y Y YEα,0,ω′ S S = 2 3 1 3 α ω′ 2 2 3 then we bound it directly by ν ~ ν ~ ∂x Y f (ϑ) (noting the extra factor of ν ~ ); while Y YDα,0,ω′ if α′ 0, we use the Poincar´e’s inequality to obtain S S =

~~′ ′ 2 3 1 3 ′ 2 α ω 2 α ω 2 α ω A0ν µ ∂x Y f 0 L A0ν µ x∂x Y f L ν ~ ν ~ ∂x Y f (ϑ) . ~= x,v x,v Y ( ) Y ≲ Y ∇ Y ≲ Y YDα,0,ω′~~

Next, note that the zeroth mode (i.e. second term in (5.34)) is only non-vanishing when α α′ 0. Fixing α α′ 0, we further have two cases: if ω′ 0, we simply bound the S S = S S = 2 S S = S S = ′ ω′ ω′′ S S = ′′ term by A0ν µf0 2 ; while if ω 0, we write Y YjY for some j and ω , and use the Y YLx,v S S > = ω′ ω′ ω′′ ω′′ fact Y f 0 Y f 0 t∂xj ∂vj Y f 0 ∂vj Y f 0 to deduce ( )= = ( = ) = ( + ) = = =

~~′ ′′ 1 ′′ ω 2 ω 2 3 ω 2 A0ν µ Y f 0 L2 A0ν µ∂vj Y f 0 L2 ν Y f (ϑ) . (5.35) = x,v = x,v Y ( ) Y ≲ Y Y ≲ Y YDα,0,ω′′~~

~~Hence, combining all the cases above, we obtain~~

RHS of (5.33) 2 3 1 3 α β′ ω′ 2 α β′ ω′ 2 ⎧Cη Q ν ~ ν ~ ∂x ∂v Y f (ϑ) ν φ L∞ ∂x ∂v Y f (ϑ) if α 0 ⎪ ′ ′ ′ ′ ⎪ β′ β( Y YDα,β ,ω + Y Y Y YEα,β ,ω ) S S > ⎪ Sω′S≤SωS ⎪ S S≤S S ⎪ 2 3 1 3 α β′ ω′ 2 α β′ ω′ 2 ⎪ ∞ ⎪Cη Q ν ~ ν ~ ∂x ∂v Y f (ϑ) ν φ L ∂x ∂v Y f (ϑ) (5.36) ⎪ β′ β( Y YDα,β′,ω′ + Y Y Y YEα,β′,ω′ ) ≲ ⎨ Sω′S≤SωS ⎪ S S≤S S ′ ′ ⎪ 1 3 α β ω 2 2 ⎪ Cη Q ν ~ ∂x ∂v Y f (ϑ) Cην µf0 L2 if α 0. ⎪ ′ ′ x,v ⎪ + β′ β Y YDα,β ,ω + Y Y S S = ⎪ S ′S≤S S ⎪ ω ω ⎩⎪ S S

~~Putting together (5.31), (5.32) and (5.36), we obtain~~

2 β 3 L,ℓ−2 α′ 2 β 3 2 β′ 3 L,ℓ−2 β′ A0νν S S~ Q α+α′,β,ωS S νν S S~ Q ν S S~ α,β+βS′,ωS α′ 1 R + β′ 1 R S S≤ S S≤ 2 3 1 3 α β′′ ω′′ 2 1 3 α β′′ ω′′ 2 ⎧ η Cην ~ ν ~ Q ∂x ∂v Y f (ϑ) Cην ~ Q ∂x ∂v Y f (ϑ) ⎪( + ) β′′ βY YDα,β′′,ω′′ + β′′ βY YDα,β′′,ω′′ ⎪ S ′′S≤S S S ′′S≤S S ⎪ ω ω ω ω ⎪ S S≤S S β′′ + ωS ′′ S≤Sβ +S ω −1 ⎪ S S S S≤S S S S ⎪ α β′′ ω′′ 2 ⎪ ∞ ⎪ Cην φ L Q ∂x ∂v Y f (ϑ) if α 0 ⎪ ′′ ′′ ′′ ⎪ + Y Y β βY YEα,β ,ω S S > ⎪ Sω′′S≤SωS ⎪ S S≤S S ⎪ 2 3 1 3 α β′′ ω′′ 2 1 3 α β′′ ω′′ 2 ≲ ⎨ η Cην ~ ν ~ Q ∂x ∂v Y f (ϑ) Cην ~ Q ∂x ∂v Y f (ϑ) ⎪( + ) β′′ βY YDα,β′′,ω′′ + β′′ βY YDα,β′′,ω′′ ⎪ S ′′S≤S S S ′′S≤S S ⎪ ω ω ω ω ⎪ S S≤S S β′′ + ωS ′′ S≤Sβ +S ω −1 ⎪ S S S S≤S S S S ⎪ α β′′ ω′′ 2 2 ⎪ Cην φ L∞ ∂ ∂ Y f (ϑ) Cην µf0 2 if α 0. ⎪ Q x v Lx,v ⎪ ′′ ′′ ′′ ⎪ + Y Y β βY YEα,β ,ω + Y Y S S = ⎪ Sω′′S≤SωS ⎪ S S≤S S ⎩⎪ (5.37)

~~L,ℓ L,ℓ Estimates on α,β,ω. We now bound the remainder α,β,ω introduced in Lemma 5.7 and appeared on theZ right hand side of (5.23). Using CorollaryZ 4.8, we bound~~

−4 2 2 q+1 φ β ω α α β ω v w e ( ) ∂ Y L ∂x ∂ f ∂v ∂ ∂ Y f dv S 3 v i x i x v S R ⟨ ⟩ [ ] S α β ω α β′ ω′ ∂vi ∂x ∂v Y f ∆v ℓ −2,ϑ ∂xi ∂x ∂v Y f ∆v ℓ −2,ϑ Q ( α,β,ω ) ( α,β,ω ) ≲β′ βY Y Y Y Sω′S≤SωS S S≤S S and

−4 2 2 q+1 φ β ω α α β ω v w e ( ) ∂v ∂ Y L ∂ f ∂x ∂ ∂ Y f dv S 3 i v x i x v S R ⟨ ⟩ [ ] S α β ω α β′ ω′ ∂xi ∂x ∂v Y f ∆v ℓ −2,ϑ ∂x ∂v Y f ∆v ℓ −2,ϑ . Q ( α,β,ω ) ( α,β,ω ) ≲β′ βY+1 Y Y Y S ωS≤S′ ωS S S≤S S

4 3 2 β 3 Therefore, multiplying by ν ~ ν S S~ and integrating over x, we get 4 3 2 β 3 L,ℓ−2 ν ~ ν S S~ Zα,β,ω 4 3 2 β 3 α β ω α β′ ω′ ν ~ ν S S~ v∂x ∂v Y f ∆x,v ℓ −2,ϑ x∂x ∂v Y f ∆x,v ℓ −2,ϑ ( α,β,ω ) Q ( α,β,ω ) ≲ Y∇ Y (β′ βY∇ Y ) Sω′S≤SωS S S≤S S 4 3 2 β 3 α β ω α β′ ω′ ν ~ ν S S~ x∂x ∂v Y f ∆x,v ℓ −2,ϑ ∂x ∂v Y f ∆x,v ℓ −2,ϑ (5.38) ( α,β,ω ) Q ( α,β,ω ) + Y∇ Y β(′ βY+1 Y ) S ωS≤S′ ωS S S≤S S 1 3 −1 2 α β ω α β′ ω′ ν ~ A0 ~ ∂x ∂v Y f (ϑ) ∂x ∂v Y f (ϑ) . Q ′ ′ ≲ Y YDα,β,ω β′ β Y YDα,β ,ω Sω′S≤SωS S S≤S S Proof of (5.20). We now put together the above estimates. The α 0 and α 0 cases are treated slightly diﬀerently. Consider ﬁrst α 0. Combining (5.23S ),S > (5.24) withS S = the bounds S S > THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 32

~~(5.29), (5.30), (5.37) and (5.38) for the remainder terms,~~

d α β ω 2 1 3 α β ω 2 ∂x ∂v Y f (ϑ) θν ~ ∂x ∂v Y f (ϑ) dtY YEα,β,ω + Y YDα,β,ω 2 3 −1 2 1 3 α β′ ω′ 2 η Cην ~ A0 ~ ν ~ Q ∂x ∂v Y f (ϑ) ≲ ( + + ) β′ βY YDα,β′,ω′ Sω′S≤SωS S S≤S S 1 3 α β′ ω′ 2 1 2 1 3 α′ β′ ω 2 ~ (5.39) Cην ~ Q ∂x ∂v Y f (ϑ) A0 ν ~ Q ∂x ∂v Y f (ϑ) + β′ βY YDα,β′′,ω′′ + α′ α , βY′ β YDα,β′,ω Sω′S≤SωS Sα′ S≥S+ β′S S αS

d 2 1 3 2 f E(ϑ) θν ~ f D(ϑ) dt low low Y Y Nα ,Nα,β ,Nβ,Nω + Y Y Nα ,Nα,β ,Nβ ,Nω 1 2 1 3 2 2 ~ Cη A0 ν ~ f D(ϑ) f D(ϑ) low 1 low 1 (5.41) ≲ ( + ) (Y Y Nα ,Nα,β,Nβ − ,Nω + Y Y Nα ,Nα,β ,Nβ,Nω− ) 2 1 CηA0 ∂tφ L∞ φ ,∞ f (ϑ) α,β,ω. x Wx E Q low low + (Y Y + Y Y )Y Y Nα ,Nα,β ,Nβ,Nω + α N , βR N S S≥ α S S≤ β α + β N , ω Nω S S S S≤ α,β S S≤ For fixed N N low 1, we now perform an induction in N and N . The base case α,β ≥ α ≥ β ω is Nβ Nω 0: since f D(ϑ) f D(ϑ) 0, the desired conclusion is low 1 low 1 = = Y Y Nα ,Nα,β ,− ,Nω = Y Y Nα ,Nα,β,Nβ ,− = immediate. A simple induction, say, first in Nω, and then in Nβ, finishes the proof of the proposition in the case N low 0. α >

low Proof of (5.21). Finally, we consider the case Nα 0. Notice that when repeating the = 2 argument in the proof of (5.20), the only diﬀerence is that we obtain an extra term ν µf 0 L2 Y = Y x,v THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 33

(coming from (5.36) in the α 0 case). Thus S S = d 2 1 3 2 f (ϑ) θν ~ f (ϑ) E0 D0 dtY Y ,Nα,β ,Nβ ,Nω + Y Y ,Nα,β ,Nβ ,Nω 2 2 (5.42) ν µf 2 ∂ φ ∞ φ 1,∞ f . 0 L t Lx W E(ϑ) α,β,ω = x,v x 0 Q ≲ Y Y + (Y Y + Y Y )Y Y ,Nα,β ,Nβ ,Nω + β NR S S≤ β α + β N , ω Nω S S S S≤ α,β S S≤ 2 To proceed, we write f t, x, v a t, x √µ bj t, x vj √µ c t, x v √µ I Π f, where I is the identity, and Π is( the projection) = ( ) as in+ Lemma( ) 4.5.+ Repeating( )S S now+ the( − basic) energy estimate in Lemma 5.4 with ℓ ϑ 0, we obtain = = 1 d φ 2 φ φ T,0 Q,0 e f 2 ν e fL e f dv dx . Lx,v U 3 3 0,0,0 0,0,0 2 dtY Y + T ×R ( ) ≲ R + R φ φ T,0 Applying Lemma 4.5 for e f, L e f and controlling 0,0,0 by (5.25), we thus obtain ⟨ ( )⟩ R d φ 2 2 φ 2 2 Q,0 e f 2 δ ν e I Π f ∂tφ L∞ E L∞ f (0) . (5.43) Lx,v ∆x,v x x 0,0,0 dtY Y + Y ( − ) Y ≲ (Y Y + Y Y )Y YE0,0,0 + R Notice that (5.43) implies 2 2 φ 2 2 2 ∞ 2 µf 0 L a, b, c e I Π f ∆x,v φ Lx µf 0 L Y = Y x,v ≲ S( )S + Y ( − ) Y + Y Y Y = Y x,v 2 2 Q,0 (5.44) ∞ ∞ a, b, c ∂tφ Lx φ Lx f (0) 0,0,0. E0 ≲ S( )S + (Y Y + Y Y )Y Y ,Nα,β ,Nβ ,Nω + R Plugging this into (5.42) yields the desired conclusion. 5.6. The main energy estimates including the top-order energy. Proposition 5.11. Fix ϑ 0, 2 . The estimates (5.20) and (5.21) in Proposition 5.9 both low low∈ { } E ϑ D ϑ hold (for Nα 0 and Nα 0 respectively) with ( low) , ( low) and α,β,ω > = Nα ,Nα,β,Nβ,Nω Nα ,Nα,β,Nβ,Nω R E ϑ D ϑ E ϑ replaced by ( low) , ( low) and α,β,ω respectively, where ( low) ̃Nα ,Nα,β,Nβ,Nω ̃Nα ,Nα,β,Nβ,Nω R̃ ̃Nα ,Nα,β,Nβ,Nω D ϑ and ( low) are as in (5.16), and ̃Nα ,Nα,β,Nβ ,Nω 2 β + β′ 3 Q,ℓ−2 β′ α,β,ω α,β,ω Q ν (S S S S)~ α,β+βS′,ωS, R̃ = R + β′ 2 R S S= Q,ℓ where and α,β,ω are as introduced in Lemma 5.5 and Proposition 5.9. Rα,β,ω R Proof. We repeat the argument in Proposition 5.9, except that we now derive the energy estimates for α,β,ω and α,β,ω instead of α,β,ω and α,β,ω. For this, we need to handle the additional termsẼ D̃ E D −1 2 β 3 2 β′ 3 T,ℓ−2 β′ L,ℓ−2 β′ Q,ℓ−2 β′ A0 ν S S~ Q ν S S~ α,β+βS′,ωS ν α,β+βS′,ωS α,β+βS′,ωS . β′ 2 R + R + R S S= Q,ℓ−2 β′ Now the ∑ β′ 2 α,β+βS′,ωS term is part of ̃α,β,ω and does not need to be estimated for the purpose of thisS S= proposition.R R As for the other two terms, notice that while they contain one more ∂v derivative compared to their counterparts in Proposition 5.9, the norms ̃α,β,ω and ̃α,β,ω also control the additional terms as indicated in (5.13) and (5.14). It canE be checkedD that the same energy estimates as in Proposition 5.9 can be obtained, as long as the , norms are replaced (E D) THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 34

T,ℓ−2 β′ by the , . We only consider in detail the following term from S′ S which requires (̃ ̃) α,β+β ,ω modiﬁcationsE D that are not completely obvious: R 2 β +4 3 q+1 φ α+β′′′ β′′+β′ ω q+1 φ α β+β′ ω ν( S S )~ e( ) ∂ ∂ Y f 2 − e( ) ∂ ∂ Y f 2 − Q Q x v Lx,v ℓα,β,ω 4,ϑ x v Lx,v ℓα,β,ω 4,ϑ β′ 2 β′′ β −1 Y Y ( )Y Y ( ) S S= S βS=S′′′ S1 S S= 1 3 β′′ 3+2 3 q+1 φ β′ α′′ β′′ ω ν ~ νS S~ ~ e( ) ∂ ∂ ∂ Y f 2 − Q v x v Lx,v ℓα,β,ω 4,ϑ ≲ ( α′′ αY+1 Y ( )) β′ S 2,S=Sβ′′ S β −1 S S= S S=S S β 3+2 3 q+1 φ α β+β′ ω νS S~ ~ e( ) ∂ ∂ Y f 2 − Q x v Lx,v ℓα,β,ω 4,ϑ × ( β′ 2Y Y ( )) S S= 1 3 β′′ 3+2 3 q+1 φ α′′ β′′ ω ν ~ νS S~ ~ e( ) v∂x ∂v Y f ∆x,v ℓ −2,ϑ Q ( α,β,ω ) ≲ ( α′′ αY+1 ∇ Y ) Sβ′′S=SβS−1 S S=S S β 3+2 3 q+1 φ β′′′ α β′′ ω νS S~ ~ e( ) ∂v ∂x ∂v Y f ∆x,v ℓ ′′ −4,ϑ Q ( α,β ,ω ) × ( β′′ β −1 Y Y ) S βS=S′′′ S2 S S= 1 3 α′′ β′′ ω 1 2 α β′′ ω ν ~ ∂x ∂v Y f (ϑ) A0~ ∂x ∂v Y f (ϑ) . Q ′′ ′′ Q ′′ ≲ α(′′ αY+1 YD̃α ,β ,ω ) × ( β′′ β −1 Y YD̃α,β ,ω ) Sβ′′S=SβS−1 S S=S S S S=S S As a result, we can then complete the argument following the proof of Proposition 5.9.

~~6. Linear Landau equation In this section, we derive estimates on the semigroup of the linear Landau equation~~

∂tf v xf νLf 0 (6.1) + ⋅ ∇ + = 3 3 on T R , with initial data f 0, x, v f0 x, v , where L denotes the leading linear Landau operator× as in (2.3), (2.4) and( (2.5).) Let = S( t )be the semigroup associated to (6.1), that is, ( ) for each f0 x, v , we set ( ) S t f0 x, v f t, x, v (6.2) ( )[ ]( ) ∶= ( ) where f t, x, v is the unique solution to (6.1) with initial data f0 x, v . As L is independent of x, the( problem) (6.1) can be solved via the Fourier transform.( Indeed,) we can write ik⋅x ˆ S t f0 x, v Q e Sk t f0k v ( )[ ]( ) = k Z3 ( )[ ]( ) ∈ 3 in which, for each k Z , fˆ0 v is the Fourier transform of f x, v in variable x, and ∈ k Sk t h0 denotes the corresponding( ) semigroup to the Fourier transform( ) of (6.1): namely, h t( )[ Sk] t h0 solves the following ﬁxed mode linear Landau equation ( ) = ( )[ ] ∂th ik vh νLh 0 (6.3) + ⋅ + = with initial data h 0, v h0 v . ( ) = ( ) This section is devoted to deriving estimates for ∫R3 h√µ dv . We will prove both ﬁnite U U time bounds (Proposition 6.5) and decay estimates (Proposition 6.4). We prove two types of decay estimates: Uniform phase mixing: decay in the variable kt , uniformly in ν 0. ● 1 3 ≥ Enhanced dissipation: decay in the variable ν⟨ ~ ⟩t . ● ⟨ ⟩ THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 35

The precise decay estimates can be found in Proposition 6.4 below. When ν 0, (6.3) −ikt⋅v = becomes the free transport equation, whose semigroup reads Sk t h e h. In that case the decay estimates in kt are thus direct. We shall prove the( phase)[ ] = mixing for the linear Landau equations (6.1)⟨ uniformly⟩ in ν 0. (In fact, we also prove a “twisted” estimate with decay in kt η for η R3, which will≥ be useful in the nonlinear density estimate.) Next, using⟨ + methods⟩ ∈ of [54], it follows that the Landau diffusion dissipates energy at −δ νt 2~3 least at a rate of order e ( ) , which in particular becomes relevant at time of order 1 ν. Making use of the transport-diffusion structure of the Landau operator, we shall prove the~ 1 3 −1 3 enhanced dissipation in ν ~ t , which takes place at a much earlier time of order ν ~ , as ν is sufficiently small. ⟨ ⟩ 6.1. Phase mixing and vector field bounds. In this subsection, we prove that control of Yk,η derivatives (defined below) implies decay estimates for velocity averages. Z3 R3 R3 R Proposition 6.1. For k , η , set Yk,η v i η kt , and g → . Then, for ′ ∈ ∈ = ∇ + + ∶ any N 0 and any ℓ 0, there is a positive constant C( ′ so) that ≥ ≥ N,ℓ −N −ℓ′ ω g µ dv CN,ℓ′ kt η v Y g L2 R3 . (6.4) SR3 √ Q k,η V V ≤ ⟨ + ⟩ ω N Y⟨ ⟩ Y ( ) S S≤ Proof. If kt η 1, the desired estimate follows directly from the Cauchy–Schwarz inequality. S + S ≤ 1 Suppose that kt η 1. Take j such that kjt ηj √ kt η . Then, writing i kjt ηj S + S > S + S ≥ 3 S + S ( + ) = Yk ,η ∂v , we bound j j − j N N N kt η g µ dv 3 2 kjt ηj g µ dv S 3 √ S 3 √ S + S V R V ≤ V R ( + ) V N1 N2 −ℓ′ ω ′ 2 R3 N Yk ,η ∂vj g µ dv N,ℓ v Yk,ηg L , Q SR3 j j √ Q ( ) ≲ N1+N2 N V V ≲ ω N Y⟨ ⟩ Y = S S≤ where the final inequality is achieved by integrating by parts N2 times in ∂vj . 6.2. Enhanced dissipation. In this subsection, we prove the following enhanced dissipation estimates for the linear Landau equation (6.1), which are a direct consequence of energy estimates. 3 3 Proposition 6.2. For k Z 0 and η R , set Yk,η v i η kt and let Sk t be the semigroup of (6.3). Then,∈ there∖ { exists} δ′ ∈0 so that = ∇ + ( + ) ( ) > ω 1 3 −3 2 2 (10,0)′ Yk,ηSk t h0 Lv ν ~ t ~ h0 E (6.5) Y ( )[ ]Y ≲ ⟨ ⟩ Y Y Landau,k,η,SωS and −δ′ ν1~3t 1~3 −δ′ νt 2~3 2 (2,2)′ Sk t h0 Lv min e ( ) , e ( ) h0 E (6.6) Y ( )[ ]Y ≲ { }Y Y Landau,k,η,0 uniformly in k Z3 0 , η R3, and ν 0, where for ℓ∗ R and ϑ 0, 2 , the linear Landau ∈ { } ∈ ≥ ∈ ∈ { } energy norm h0 E(ℓ∗,ϑ)′ is defined by Y Y Landau,k,η,SωS

h0 E(ℓ∗,ϑ)′ Y Y Landau,k,η,N ′ ϑ 2ℓ q SvS 2ℓ ω v ∗ e 2 h 2 v ∗ Y h 2 0 Lv Q 0,η 0 Lv ∶= Y⟨ ⟩ Y + 1 ω N Y⟨ ⟩ Y (6.7) ≤S S≤ ′ ϑ q SvS ′ ′ 1 3 −1 2ℓ∗−2 β 2ℓ∗−2 β ω ν ~ k v e 2 ∂ h 2 v ∂ Y h 2 , Q v 0 Lv Q v 0,η 0 Lv + S S β′ 1 Y⟨ ⟩ Y + 1 ω N Y⟨ ⟩ Y S S= ≤S S≤ THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 36

1 q0 if ϑ 2 and q′ is defined by q′ 2 = . = 0 if ϑ 0 = Remark 6.3. Note that q′ 1 q with q defined as in Section 5. That is, the linear Landau = 2 energy norm E(ℓ∗,ϑ)′ has slower Gaussian v-weights than do the corresponding energy Y ⋅ Y Landau,k,η,N and dissipation norms. In addition, it involves precisely the stationary vector field Y0 ,η = v iη (i.e. independent of t). ∇ + Proof. Basic energy estimates. Let h t S t h0 . We note that h t solves the linear = k Landau equation (6.3) with initial data h(0.) As(6.1( )[) is a] particular version( ) of the full Landau equation (2.2a) without the electric field and nonlinear terms, we can thus apply to (6.3) the low same energy estimates developed in Proposition 5.9 for Nα Nα,β Nβ 0. Indeed, we claim that = = = d 2 1 3 2 h t ′ θν h t ′ . E(ℓ∗,ϑ) ~ D(ℓ∗,ϑ) 0 (6.8) dtY ( )Y 0,0,0,Nω + Y ( )Y 0,0,0,Nω ≤ R for any ℓ∗ and ϑ 0, 2 . Here, in (6.8), the energy norm h t E(ℓ∗,ϑ)′ and the dissipation ∈ ∈ { } Y ( )Y 0,0,0,Nω norm h t D(ℓ∗,ϑ)′ are defined by Y ( )Y 0,0,0,Nω 2 2 ω 2 h t ′ h t ′ Y h t 0 ′ , E(ℓ∗,ϑ) (ℓ∗,ϑ) Q k,η (ℓ∗, ) Y ( )Y 0,0,0,Nω = Y ( )YE0,0,0 + 1 ω N Y ( )YE0,0,ω ≤S S≤ 2 2 ω 2 h t ′ h t ′ Y h t 0 ′ , D(ℓ∗,ϑ) (ℓ∗,ϑ) Q k,η (ℓ∗, ) Y ( )Y 0,0,0,Nω = Y ( )YD0,0,0 + 1 ω N Y ( )YD0,0,ω ≤S S≤ where for H Y ω h, we set = k,η 2 α′ 2 1 3 2 2 3 2 H (ℓ ,ϑ)′ A0 k H 2 − ′ ′ ν ~ R iw k vH H dv ν ~ vH 2 − ′ , ∗ Q Lv ℓ∗ 2 α ,ϑ SR3 Lv ℓ∗ 2,ϑ Y YE0,0,ω = α′ 1Y Y ( S S ) + ⋅ (∇ ) + Y∇ Y ( ) S S≤ 2 2 3 α′ 2 2 4 3 2 H (ℓ ,ϑ)′ A0ν ~ k H − ′ ′ k H 2 − ν ~ vH − ′ , ∗ Q ∆v ℓ∗ 2 α ,ϑ Lv ℓ∗ 2,ϑ ∆v ℓ∗ 2,ϑ Y YD0,0,ω = α′ 1 Y Y ( S S ) + YS S Y ( ) + Y∇ Y ( ) S S≤ q′SvSϑ 2 ′ ′ ℓ∗−2 2 for Lv ℓ, ϑ , ∆v ℓ, ϑ as in (2.12)–(2.13), A0 as in (5.11)–(5.12), w v e , with 1( ) ( ) = ⟨ ⟩ q0 if ϑ 2 q′ 2 = . Observe that these are exactly the norms in Proposition 5.9 adapted = 0 if ϑ 0 = to the current setting, except with (1) φ 0, (2) Y is replaced by Yk,η, (3) 2M replaced by qSvSϑ ≡ q′SvSϑ ℓ∗, (4) the Gaussian weights e 2 are replaced by e 2 , and (5) the polynomial v weights depend only on the k weights and ∂v derivatives, but not the Yk,η derivatives. ⟨ ⟩ Now make the following observations:

Yk,0 v ikt corresponds to the vector field Y v t x in the physical space. ● = ∇ + = ∇ + ∇ Hence, the energy estimates for Yk,0 are a Fourier transformed version of those in Section 5. Now, for η 0, we observe that the commutator of Y with the linear = k,η Landau equation is identical~ to that of Yk,0. The argument in Proposition 5.9 goes through identically with q replaced by q′ in ● the Gaussian v-weights, and with ℓα,β,ω ℓ∗ 2 α 2 β . (The choice of ℓα,β,ω 2M 2 α 2 β 2 ω will only be relevant= in Section− S S −9; seeS S for instance Lemma 9.6.)= − − − Therefore, theS estimateS S S (6.8S S) can be obtained as in Proposition 5.9, read off specifically for the linear Landau equations. THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 37

Polynomial decay. Deﬁne 2 4−4 α′ α′ ω 2 1 3 ω ω 2 3 ω 2 g t, v Q A0 Q v S S k Yk,ηh ν ~ R ik jYk,ηh Yk,ηh ν ~ vYk,ηh (6.9) ( ) = ω N( α′ 1⟨ ⟩ S S + ( ⋅ (∇ ) ) + S∇ S ) S S≤ S S≤ − 2 2 3 2 2 so that h 2 0 ′ 3 g dv. Notice that (using in particular v ∆ ) E( , ) ∫R ~ v Lv v Y Y 0,0,0,N = Y⟨ ⟩ (∇ ⋅)Y ≤ Y ⋅ Y 2 2 ω 2 2 3 −3 2 ω 2 −4 2 h (2,0)′ k Yk,ηh 2 ν ~ v ~ vYk,ηh 2 v g dv. (6.10) D Q Lv 0,0 Lv 0,0 SR3 Y Y 0,0,0,N ≳ ω N(S S Y Y ( ) + Y⟨ ⟩ ∇ Y ( )) ≳ ⟨ ⟩ S S≤ That is, using (6.8) with ℓ∗, ϑ 2, 0 , we get ( ) = ( ) d 2 ′ 1 3 −4 2 g dv θ ν ~ v g dv 0, (6.11) S 3 S 3 dt R + R ⟨ ⟩ ≤ for some positive constant θ′. Moreover, using (6.8) again, we have

~~16 2 2 2 2 2 sup v g t, v dv sup h t 10 0 ′ h 0 10,0 k h0 10 0 ′ , S 3 E( , ) E( ) E( , ) t [0,∞) R ≲ t [0,∞) 0,0,0,N ≲ 0,0,0,N ≲ Landau,k,η,N ∈ ⟨ ⟩ ( ) ∈ Y ( )Y Y ( )Y S S Y Y~~

recalling the definition of E(10,0)′ in (6.7) and noting Yk,ηh 0 Y0,ηh0. Therefore, ⋅ Landau,k,η,N = Y Y 1~3 2 2 ( ) applying Lemma A.2 to (6.11) (with c ν , C k h ′ and m 4), we obtain 0 E(10,0) ≳ ≲ Landau,k,η,N = 1 S S Y Y 2 3 −3 2 2 2 2 ω 2 2 3 g t, v dv ν t k h ′ . Noticing that g k Y h and dividing by k , ∫R 0 E(10,0) k,η ( ) ≲ ⟨ ⟩ S S Y Y Landau,k,η,N ≳ S S S S S S we obtain (6.5). Stretched exponential decay. This requires only little modification from the previous 1 3 1 3 2 3 case, except that we need to prove both e−δ′(ν ~ t) ~ and e−δ′(νt) ~ decay. Define g as in (6.9) but only for N 0, i.e. = 2 4−4Sα′S α′ 2 1~3 2~3 2 g t, v A0 Q v k h ν R ik jh h ν vh . ( ) = Sα′S 1⟨ ⟩ S S + ( ⋅ (∇ ) ) + S∇ S ≤ The equation (6.10) holds in the particular case N 0. This time, moreover, the initial = bound on h0 E(2,2)′ and (6.8) (with ℓ∗, ϑ 2, 2 ) give uniform in t bounds for the Y Y Landau,k,η,0 ( ) = ( ) Gaussian moments for g2, i.e.

1 2 2 q0SvS 2 2 e g dv k h0 E(2,2)′ . S 3 0 R ≲ S S Y Y Landau,k,η, 1~3 2 2 Therefore, applying Lemma A.1 to (6.11) (with c ν , C k h ′ , m 4), we E(2,2) ≳ ≲ S S Y Y Landau,k,η,0 = obtain 2 2 2 −δ(ν1~3 t)1~3 2 2 k h 2 t g dv e k h (2,2)′ . Lv SR3 E S S Y Y ( ) ≲ ≲ S S Y Y Landau,k,η,0 2 3 Finally, to obtain the other, i.e. the e−δ(νt) ~ , stretched exponential decay, note that we − 1~2 2 have (using v Lv ∆v ) the following bound, in addition to (6.10): Y⟨ ⟩ ⋅ Y ≤ Y ⋅ Y 2 2~3 2Sα′S 2 2~3 −1~2 2 2~3 −1 2 h (2,0)′ ν k h 2 − ′ ν v vh 2 ν v g dv. D Q Lv (3~2 2Sα S,0) Lv (0,0) SR3 Y Y 0,0,0,0 ≳ [Sα′S 1 S S Y Y + Y⟨ ⟩ ∇ Y ≳ ⟨ ⟩ ≤ Remark that this features both the improved v −1 weight and the extra ν2~3 factor when compared to (6.10). Thus, an application of (6.8⟨ )⟩ (with ℓ∗, ϑ 2, 0 and Nω 0) yields ( ) = ( ) = d g2 dv θ′ν v −1g2 dv 0, dt SR3 + SR3 ⟨ ⟩ ≤ THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 38

′ 2 2 for some θ 0. Using Lemma A.1 (with c ν, C k h ′ , m 1) thus gives E(2,2) > ≳ ≲ S S Y Y Landau,k,η,0 = 2 2 2 −δ(νt)2~3 2 2 k h 2 t g dv e k h (2,2)′ . Lv SR3 E S S Y Y ( ) ≲ ≲ S S Y Y Landau,k,η,0 Combining the two stretched exponential decay estimates above, and dividing by k 2, yield (6.6). S S 6.3. Mixed decay estimates. In the nonlinear analysis, we also need the following proposition, which is a direct combination of Proposition 6.1 and Proposition 6.2.

3 3 Proposition 6.4. Fix k Z and η R , and let Sk t be the semigroup of (6.3). Then, for any N 0, there exist C ∈ 0 and δ ∈ 0 such that ( ) ≥ N > N > −N 1~3 −3~2 Sk t h0 µ dv CN kt η ν t h0 (10,0)′ , (6.12) S 3 √ E V R ( )[ ] V ≤ ⟨ + ⟩ ⟨ ⟩ Y Y Landau,k,η,N

~~1 3 1 3 2 3 −δ0(ν ~ t) ~ −δ0(νt) ~ Sk t h0 µ dv C0 min e , e h0 E(2,2)′ , (6.13) S 3 √ 0 V R ( )[ ] V ≤ { }Y Y Landau,k,η, and~~

Sk t h0 µ dv SR3 √ V ( )[ ] V (6.14) 1~3 1~3 2~3 −N −δN (ν t) −δN (νt) CN kt η min e , e h0 E(10,0)′ h0 E(2,2)′ , ≤ ⟨ + ⟩ { }Y Y Landau,k,η,N+1 + Y Y Landau,k,η,0 Z3 R3 uniformly in k 0 , η , and ν 0, where the norm E(ℓ∗,ϑ)′ is deﬁned as in ∈ { } ∈ ≥ Y ⋅ Y Landau,k,η,N Proposition 6.2.

Proof. Let h t S t h0. We combine Propositions 6.1 and 6.2 to obtain ( ) = k( ) −N ω −N 1~3 −3~2 h t √µ dv kt η Y h L2 kt η ν t h0 (10,0)′ . SR3 Q k,η v E V ( ) V ≲ ⟨ + ⟩ SωS N Y Y ≲ ⟨ + ⟩ ⟨ ⟩ Y Y Landau,k,η,N ≤ The proof of the exponential decay in (6.13) is similar (there recalling N 0). For (6.14) with N 0, we use Propositions 6.1, 6.2 and interpolate (using Plancherel’s= theorem and H¨older’s> inequality), namely

−N ω h t √µ dv kt η Y h L2 SR3 Q k,η v V ( ) V ≲ ⟨ + ⟩ SωS N Y Y ≤ 1 − N N ω 1 N+1 kt η Y h L2 N+ h t 2 Q k,η v Lv ≲ ⟨ + ⟩ (SωS N+1 Y Y ) Y ( )Y ≤ − − 1~3 1~3 − 2~3 N δN (ν t) δN (νt) kt η min e , e h0 E(10,0)′ h0 E(2,2)′ . ≲ ⟨ + ⟩ { }Y Y Landau,k,η,N+1 + Y Y Landau,k,η,0 6.4. Finite time energy estimates and the ν → 0 limit. In this subsection, we study the ν → 0 limit of Sk t h0 , where t ranges over a ﬁnite time interval. To clarify the notations, ( )[ ] (ν) in this subsection we use Sk t to denote the semigroup associated to equation (6.3). ( ) R3 Proposition 6.5. Fix T 0, . For any h0 S , ∈ ( +∞) ∈ ( ) (ν) (0) lim sup S t h0 µ dv S t h0 µ dv 0. → 3 k √ 3 k √ ν 0 t [0,T ] SR − SR = ∈ V ( )[ ] ( )[ ] V THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 39

(ν) (ν) Proof. Let f Sk t h0 . For every ν 0, 1 , standard finite time energy estimates for (6.3) give that for= C ( )[0 (depending] on T ∈but[ not] on ν) T > α β (ν) α β k ∂ f 2 C k ∂ h 2 . sup Q v Lv T Q v 0 Lv (6.15) t [0,T ] SαS+SβS 2 Y Y ≤ SαS+SβS 2 Y Y ∈ ≤ ≤ (For finite time energy estimates, we simply use Grönwall’s inequality for many commutator terms, making them much easier than those in Proposition 5.9.) (ν) (0) (ν) (0) (ν) Consider now the equation ∂t f f ik v f f νLf . Multiplying by f (ν) f (0), integrating in v and then( using− ()6.15+ )⋅ gives( − ) = − − d (ν) (0) 2 (ν) (0) (ν) (ν) (0) α β f f 2 ν f f L2 Lf L2 CT ν f f L2 k ∂ h0 L2 . Lv v v v Q v v dtY − Y ≲ Y − Y Y Y ≲ Y − Y SαS+SβS 2 Y Y ≤ d (ν) (0) This implies f f 2 Cν, for some constant depending on T and h0. Noticing dt Lv ≤ (ν) Y (0) − 2 Y (ν) (0) 2 now that f f Lv 0 0, we then deduce that supt [0,T ] f f Lv t CT ν. The conclusionY then− followsY ( from) = the Cauchy–Schwarz inequality∈ inY v. − Y ( ) ≤

~~7. Linear density estimates The goal of this section is to derive decay estimates for the density of the following linear Vlasov–Poisson–Landau equation~~

∂tf v xf νLf 2E v√µ N t, x, v (7.1) + ⋅ ∇ + = ⋅ + ( ) for the linear Landau operator L deﬁned as in (2.3), (2.4) and (2.5). The equation is solved with initial data f 0, x, v f0 x, v and a source N t, x, v , coupled with the Poisson −1 = equation E x ∆x (ρ, where) the( density) is deﬁned by( ) = −∇ (− ) ρ t, x f t, x, v µ dv. S 3 √ ( ) = R ( ) 3 For k Z 0 , letρ ˆk t be the Fourier transform of ρ t, x with respect to variable x. We also ∈ { } ( )ˆ ( ) denote by f0k v and Nk t, v the Fourier transform of f0 x, v and N t, x, v , respectively. ̂ ( ) ( ) ( ) ( ) The main result of this section is the following proposition.

2 3 3 Proposition 7.1. For any initial data f0 x, v and any source term N t, x, v in L R R , the unique density solution ρ t, x to (7.1() satisﬁes) the following representation( ) ( × ) ( ) t ρˆk t Nk t S Gk t s Nk s ds (7.2) ( ) = ( ) + 0 ( − ) ( ) 3 ′′ for each Fourier mode k Z 0 , where for any N0 2, there are C 0 0 and δ 0 such ∈ ≥ N > N0 > that the kernel Gk t satisﬁes{ } ( ) − − + − ′′ 1~3 1~3 − ′′ 2~3 1 N0 2 δN0 (ν t) δN0 (νt) Gk t CN0 k kt min e , e , ∀ t 0, (7.3) S ( )S ≤ S S ⟨ ⟩ { } ≥ uniformly in k 0 and ν 0, and the source t is given by = ≥ Nk ~ (t ) k t Sk t f0 v µ dv Sk t τ Nˆ k τ, v µ dv dτ (7.4) N S 3 k √ + S S 3 − √ ( ) = R ( )[ ̂ ( )] 0 R ( )[ ( )] where Sk t is the semigroup of the linear Landau equation (6.3). ( ) THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 40

Remark 7.2. Proposition 7.1 in particular shows uniform linear Landau damping for the 2 linearized Vlasov–Poisson–Landau equation near the global Maxwellian µ e−SvS . Indeed, = combining with Proposition 7.1 with Proposition 6.4, one deduces that for h t S t hˆ0 , k ∶= k k the corresponding density functionρ ˆk satisﬁes ( ) ( )

1~3 1~3 2~3 −N −δN (ν t) −δN (νt) ρˆk t ≲N kt min e , e , S ( )S ⟨ ⟩ { } uniformly in ν 0, for sufficiently regular initial h0. ≥ Remark 7.3. By comparison with the ν 0 case (see e.g., [46]), one may expect that (7.3) −1 −δ0SktS = even holds with Gk t ≲ k e . Proving this seems to require the technically involved task of deriving PropositionS ( )S S S6.4 for Yk,η derivatives of all orders with almost-sharp constants, and has not been carried out. 7.1. Equation for the density. We first derive an equation for the density from which the estimates are obtained. Lemma 7.4. Introduce the kernel 2 Kk t ik Sk t v µ µ dv, (7.5) 2 S 3 ⋅ √ √ ( ) = k R ( )[ ] S S 3 where Sk t is the solution operator of (6.3). Then, for each k Z 0 , the density ρˆk t satisfies the( ) following Volterra equation ∈ { } ( ) t ρˆk t + S Kk t − τ ρˆk τ dτ Nk t , (7.6) ( ) 0 ( ) ( ) = ( ) where the nonlinear source term Nk t is as in (7.4). ( ) Proof. The lemma is direct. Indeed, taking the Fourier transform in x of the linear Vlasov– Poisson–Landau equation (7.1), we get ˆ ˆ ˆ ˆ ˆ ∂tfk + ik ⋅ vfk + νLfk 2Ek ⋅ v√µ + Nk t, v . (7.7) = ( ) Let Sk t be the semigroup of the linear Landau operator ∂t + ik ⋅ v + νL. Applying the Duhamel’s( ) principle to (7.7), we obtain

~~t t ˆ ˆ ˆ (7.8) fk t, v S t f0k v + 2 S Sk t − τ Ek τ ⋅ v√µ dτ + S Sk t − τ Nk τ, v dτ. ( ) = ( )[ ̂ ( )] 0 ( )[ ( ) ] 0 ( )[ ( )]~~

Note that Eˆk t is independent of v, and so ( ) t t ˆ ˆ S Sk t − τ Ek τ ⋅ v√µ dτ S Ek τ ⋅ Sk t − τ v√µ dτ. 0 ( )[ ( ) ] = 0 ( ) ( )[ ] ˆ −2 Recall that Ek t −ik k ρˆk t . Therefore, multiplying the equation (7.8) by √µ and integrating it over( ) R= 3, weS obtainS ( ) the density equation t t ρˆk t Kk t τ ρˆk τ dτ Sk t f0 v µ dv Sk t τ Nˆ k τ, v µ dv dτ + S − S 3 k √ + S S 3 − √ ( ) 0 ( ) ( ) = R ( )[ ̂ ( )] 0 R ( )[ ( )] where the kernel Kk t is deﬁned as in (7.5). Setting the right hand side to be Nk t , which is the expression (7.4(),) the lemma follows. ( ) THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 41

7.2. Kernel Kk t . To solve the density equation (7.6), let us first study the kernel Kk t defined as in (7.5().) We obtain the following. ( ) Lemma 7.5. For any n, N 0, there exist constants C 0 and δ 0 so that ≥ N,n > N > 1~3 1~3 2~3 n n−1 −N −δN (ν t) −δN (νt) ∂t Kk t CN,n k kt min e , e , ∀ t 0, (7.9) S ( )S ≤ S S ⟨ ⟩ { } ≥ uniformly in ν 0 and k 0. ≥ =~ −2 Proof. Let hk t, v 2 k ik ⋅ Sk t v√µ , i.e. that hk t, v solves the linear fixed mode ( ) = S S ( )[ ] −2( ) Landau equation (6.3) with initial data h 0, v 2 k ik ⋅ v√µ. By definition (see (7.5)), ( ) = S S Kk t R3 hk t, v √µ dv. Hence, by (6.14) in Proposition 6.4 with η 0, we have ( ) = ∫ ( ) = 1~3 1~3 2~3 −N −δN (ν t) −δN (νt) −2 Kk t ≲ CN kt min e , e k k ⋅ v√µ E(2,2)′ S ( )S ⟨ ⟩ { }YS S Y Landau,k,0,N+1 1~3 1~3 2~3 −1 −N −δN (ν t) −δN (νt) ≲ CN k kt min e , e S S ⟨ ⟩ { 2 } for k 0, upon recalling that µ e−SvS . As for derivatives, using (6.3) and integrating by parts=~ in v, we compute =

∂tKk t ∂thk t, v µ dv hk t, v ikj vj νL µ dv. S 3 √ − S 3 + √ ( ) = R ( ) = R ( )( ) Inductively, for n 1, we have ≥ n n n ∂ Kk t 1 hk t, v ikj vj νL µ dv. t − S 3 + √ ( ) = ( ) R ( )( ) The estimates for derivatives thus follow similarly, upon noting the loss of one factor of k for each time derivative. This ends the proof of the lemma. SS

7.3. Resolvent estimates. We are now ready to solve the linear Volterra equation (7.6) for the density t ρˆk t + S Kk t − τ ρˆk τ dτ Nk t (7.10) ( ) 0 ( ) ( ) = ( ) for the source term Nk t as in (7.4), and thus give the proof of Proposition 7.1. ( ) Proof of Proposition 7.1. Taking the Laplace transform. The linear Volterra equation (7.10) is solved through its resolvent solution. Precisely, for any F L2 R+ , let us introduce the Laplace transform ∈ ( ) ∞ −λt L F λ S e F t dt [ ]( ) = 0 ( ) which is well-deﬁned for any complex value λ with Rλ 0. Thus, taking the Laplace transform of (7.10), we obtain the resolvent solution > 1 L ρˆk λ L Nk λ . (7.11) [ ]( ) = 1 +L Kk λ [ ]( ) [ ]( ) The representation (7.2) follows from taking the inverse Laplace transform of (7.11) with the kernel Gk t being the inverse Laplace transform of ( ) L Kk λ Gk λ − [ ]( ) . (7.12) ̃ ( ) = 1 +L Kk λ [ ]( ) THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 42

Basic estimates for L Kk λ . It remains to give estimates on the resolvent kernel 1~3 1~3 2~3 [ ]( ) −δN (ν t) −δN (νt) Gk λ . To simplify the exposition, we only prove the e decay in (7.3); the e decaỹ ( ) can be proven in a completely analogous manner. By definition, we have ∞ −λt L Kk λ S e Kk t dt [ ]( ) = 0 ( ) which is well-defined for any complex value λ with Rλ 0. Fix N0 1. Using Lemma 7.5 > > with N N0, we bound = ∞ K λ C k −1 kt −N0 dt C k −2 L k N0 S N0 (7.13) S [ ]( )S ≤ S S 0 ⟨ ⟩ ≤ S S uniformly for any Rλ 0. Similarly, for any 0 N N0 − 1, we have ≥ ≤∞ < ∂N K λ C k −N−1 kt N−N0 dt C k −N−2 λ L k N0 S N0 (7.14) S [ ]( )S ≤ S S 0 ⟨ ⟩ ≤ S S uniformly in k 0 and Rλ 0. =~ ≥ For N N0 − 1, we use also the stretched exponential decay in Lemma 7.5 to obtain ≥ ∞ − + − + − 1~3 1~3 − ∂N K λ C k N0 1 tN N0 2e δN0 (ν t) kt 2 dt λ L k N0 S (7.15) S [ ]( )S ≤ S S 0 ⟨ ⟩ M −x1~3 3M with a constant independent of N. Noticing that supx [0,∞) x e 3M , we have ∈ ≤ ( ) N −N0+1 −3(N−N0+2) −(1~3)(N−N0 +2) 3(N−N0+2) ∂λ L Kk λ CN0 k δN0 ν 3 N − N0 + 2 ≤ (7.16) S [ ]( )S S S−N0+1( ) −3 −1~3 3 N ( ( )) CN0 k 27 δN0 ν N , ≤ S S [ ( ) ] assuming, without loss of generality, δN0 ν 1. Checking the Penrose condition. ≤We now check the Penrose condition (see (7.18) (ν) below) by comparing with the ν 0 case. To highlight the dependence on ν, write Kk Kk. K= (ν) 1 K = First, by (7.13), there exists large such that 1 +L Kk λ 2 for for k and ν 0. On the other hand, it is classical [76] that the Penrose[ stability]( ) ≥ conditionS holdsS ≥ at ν ≥ 0, i.e. for any positive radial equilibria in R3, which in particular includes the Gaussian µ=v , there is κ0 0, 1 such that ( ) ∈ ( ) (0) inf inf 1 Kk λ κ0 0. (7.17) Rλ 0 k R3 +L ≥ > ≥ ∈ S [ ]( )S Now by the estimates in Lemma 7.5, it follows that there exists large T 0 such that ∞ K t t κ0 ν k T > ∫T k d 4 uniformly in 0 and 0. Moreover, fixing this , Proposition 6.5 S S( ) ≤ T (ν) (0)≥ =~ → + K K t t k implies that limν 0 ∫0 k − k d 0 for every 0. It therefore follows from (7.17) S S( ) = =~ κ0 that there exists ν0 0 such that infRλ 0 infSkS K 1 Kk λ 4 for all ν 0, ν0 . Together > ≥ ≤ +L ≥ ∈ with the large k estimates above, we have, forS ν [0, ν0](, )S [ ] S S ∈ [ ] (ν) κ0 inf inf 1 Kk λ 0 (7.18) Rλ 0 k R3 +L ≥ 4 > ≥ ∈ S [ ]( )S Basic estimates for G λ . Combining (7.14) and (7.18), we obtain derivative bounds ̃k on the resolvent kernel, for 0( )N N0 1, ≤ < − N −N−2 ∂λ Gk λ CN k (7.19) S ̃ ( )S ≤ S S uniformly in k 0 and Rλ 0. =~ ≥ THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 43

~~x Moreover, since x ↦ 1−x is real analytic on κ0, ∞ , using (7.16), (7.18) and considering a power series expansion, we obtain that with B[N0 independent) of N,~~

N −N0−2 −1~3 3 N ∂λ Gk λ CN0 k BN0 ν N (7.20) S ̃ ( )S ≤ S S [ ] uniformly in k 0, Rλ 0 and N N0 1. = ≥ ≥ − Improved estimates~ for G λ . We need an improvement of (7.19) and (7.20) which ̃k incorporates decay in λ. More precisely,( ) the kernel Gk t is obtained through the inverse Laplace transform formula ( )

1 λt Gk t e Gk λ dλ (7.21) = 2πi S{Rλ γ0} ̃ ( ) = ( ) for any γ0 0. We stress that the estimates in (7.19) hold for Rλ 0. Thus, to obtain > = decay in time, we need decay in Iλ independently of γ0. To this end, for any λ γ0 + iτ, we compute = ∞ 2 2 2 2 −λt k − λ L Kk λ S k − ∂t e Kk t dt (S S ) [ ]( ) = 0 (S S )[ ] ( ) t ∞ ∞ −λt = −λt 2 2 e λKk t ∂tKk t e k ∂t Kk t dt. = ( ) + ( )Ut 0 + S0 − = (S S ) ( ) In the above, the boundary term at t ∞ vanishes, since (by Lemma 7.5) Kk t and its derivatives decay rapidly in time. On the= other hand, a direct calculation yields ( )

~~−2 Kk 0 2 k ik vµ dv 0 ( ) = S S SR3 ⋅ = −2 2 ∂tKk 0 2 k i i k v √µ νkjL vj√µ √µ dv. ( ) = S S SR3 − ( ⋅ ) − ( )~~

Hence, ∂tKk 0 C, uniformly in k for ν 1. Finally, using bounds from Lemma 7.5, we obtain S ( )S ≤ ≤ ∞ 2 2 −2 k − λ L Kk λ C + C k S kt dt C, S(S S ) [ ]( )S ≤ S S 0 ⟨ ⟩ ≤ for λ γ0 + iτ and for some constant C that is independent of k,γ0, τ. This proves that = C K γ iτ 2 2 2 , giving L k 0 + SkS +SτS −Sγ0S S [ ]( )S ≤ C G γ iτ k 0 + 2 2 S ̃ ( )S ≤ k + τ S S S S uniformly for all γ0 0, 1 2 . Similarly, repeating the proof leading to (7.19) and (7.20), but incorporating the above∈ ( integration~ ) by parts argument for additional τ 2 decay, we obtain S S C k −N ∂N G γ iτ N λ k 0 + 2 S S 2 (7.22) S ̃ ( )S ≤ k + τ S S S S for any N N0 1, and, taking B 0 larger (but still independent of N) if necessary, < − N C k −N0 B ν−1~3N 3 N ∂N G γ iτ N0 N0 λ k 0 + S S [2 2 ] (7.23) S ̃ ( )S ≤ k + τ S S S S R for any N N0 − 1, where both estimates hold for any k 0, γ0 0, 1 2 and τ . ≥ =~ ∈ ( ~ ) + ∈ Estimating Gk t . Thanks to the decay in τ, we can take the γ0 → 0 limit in (7.21) with the dominated( ) convergence theorem and perform repeated integrations by parts in τ, THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 44 yielding 1 λt 1 iτt Gk t e Gk λ dλ e Gk iτ dτ = 2πi S{Rλ 0} ̃ = 2π SR ̃ ( ) = ( ) ( ) 1 1 1 iτt − iτt (7.24) S ∂τ e Gk iτ dτ S e ∂λGk iτ dτ = 2π R it ( ) ̃ ( ) = 2πt R ̃ ( ) 1 N − eiτt∂N G iτ dτ. ( )N S λ k = 2πt R ̃ ( ) First, consider the case t 103B eν−1~3 with B as in (7.23). Using (7.22) with N 0 and ≤ N0 N0 = N N0 2, and plugging into (7.24) (with the same N), we have = − dτ kt −N0+2 dτ G t k −1, G t k −1 kt −N0+2. k ≲ S 2 2 ≲ k ≲N0 S S S 2 2 ≲N0 S ( )S R k + τ S S S ( )S R k + τ S S S S S S S S − S S S S Therefore, for any δ′′ 10−1B 1 e−1, ≤ N0 ′′ 1~3 1~3 G t k −1 kt −N0+2e−δ (ν t) for t 103B eν−1~3. (7.25) k ≲N0 ≤ N0 S ( )S S S 3⟨ ⟩ −1~3 On the other hand, for t 10 BN0 eν , we first use (7.23) and (7.24) with N N0 − 1 to obtain ≥ ≥ k −N0+1 dτ G t B ν−1~3N 3 N t−N k −N0+1 B ν−1~3N 3 N t−N . k ≲N0 N0 S S S 2 2 ≲N0 N0 S ( )S ( ) R k + τ S S ( ) 3 −1~3 −1 −S1 S1~3 S 1S~3 −1~3 3 −1 1 Given t 10 BN0 eν , take N BN0 e ν t so that BN0 ν N t e and N −≥ = ⌊( ′′ −1~)3 ⌋ ( ) ≤ ≥ 1 2 B 1 ν1~3t 1~3. This implies, for δ 1 B , N0 N0 ≤ 2 N0 ( ~ )( ) 1 3 − + − − + − 1 2 B− ~ ν1~3t 1~3 − + − ′′ 1~3 1~3 N0 1 N N0 1 ( ~ ) N0 ( ) N0 1 δN0 (ν t) Gk t ≲N0 k e ≲N0 k e ≲N0 k e . S ( )S S S S S ′′ S S A similar computation gives, after taking δN0 smaller − − + − − + − − − + − ′′ 1~3 1~3 1 N0 2 1~3 3 N N N0 2 1 N0 2 δN0 (ν t) Gk t ≲N0 k kt BN0 ν N t ≲N0 k kt e . S ( )S S S S S ( ) S S S S Combining the two estimates above gives − − + − ′′ 1~3 1~3 1 N0 2 δN0 (ν t) 3 −1~3 Gk t ≲N0 k kt e for t 10 BN0 eν . (7.26) S ( )S S S ⟨ ⟩ 1 3 1 3 ≥ 2 3 −δ′′ (ν ~ t) ~ −δ′′ (νt) ~ Combining (7.25) and (7.26) yields the e N0 decay estimate in (7.3); the e N0 decay can be proven similarly and is omitted. 8. Nonlinear density estimates: bounds for all derivatives In this section, we derive density estimates for the full nonlinear Vlasov–Poisson–Landau equation (2.2a)–(2.2b) under the bootstrap assumptions on 0, T for N N : B ≤ max For all t 0, T , the nonlinear solution f to (2.2a)–([2.2b)) satisfies ● ∈ B [ ) t − − + f t 2 ν1~3 f τ 2 dτ ǫν2~3 min ν 1~3, t max{0,N Nmax 2} (8.1) Ẽ(ϑ) + S D̃(ϑ) Y ( )Y N 0 Y ( )Y N ≤ { ⟨ ⟩} for ϑ 0, 2 , where f 2 , f 2 are defined in (5.17). Ẽ(ϑ) D̃(ϑ) ∈ N N { } Y Y Y Y −1 The following holds for all t 0, TB for φ ∆x ρ~0 and E xφ: ● ∈ ∶= − = ∶= −∇ [ ) α ω 1~2 1~3 −2 ∂tφ t L∞ φ t 5,∞ ∂ Y E t L∞ ǫ ν t . (8.2) x + Wx + Q x x Y ( )Y Y ( )Y SαS+SωS 4 Y ( )Y ≤ ⟨ ⟩ ≤ The main result of this section is the following. THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 45

Theorem 8.1. Consider data as in Theorem 3.1. Suppose there exists TB 0 such that the T3 R3 > solution f to (2.2a)–(2.2b) remains smooth in 0, TB × × and satisﬁes the bootstrap assumptions (8.1) and (8.2). [ ) Then, ρ~0 t, x satisﬁes = ( ) TB α ω 2 1~3 α ω 2 2 2~3 Q sup ∂x Y ρ~0 t L2 + ν ∂x Y ρ~0 τ L2 dτ ≲ ǫ ν . (8.3) 0 = x S0 = x SαS+SωS Nmax( t TB Y ( )Y Y ( )Y ) ≤ ≤ < The proof of Theorem 8.1 proceeds as follows. We write the nonlinear equation (2.2a) in the form of (7.1), which we recall

∂tf + v ⋅ ∇xf + νLf 2E ⋅ v√µ + N t, x, v = ( ) where the nonlinear source term N t, x, v is computed by ( ) N t, x, v ∶ E ⋅ vf − E ⋅ ∇vf + νΓ f, f . (8.4) ( ) = ( ) We can thus apply the linear theory developed in Proposition 7.1 to compute the density through the density representation (7.2). Let us ﬁrst give estimates on the source Nk t computed by (7.4): ( )

t k t Sk t f0 v µ dv Sk t τ Nˆ k τ, v µ dv dτ N S 3 k √ + S S 3 − √ ( ) = R ( )[ ̂ ( )] 0 R ( )[ ( )] ˆ where f0k v and Nk t, v are the Fourier transform of f0 x, v and N t, x, v , respectively. Precisely,̂ ( ) we will prove( ) the following proposition. ( ) ( ) Proposition 8.2. Deﬁne t 2N1 2N2 2 1~3 2N1 2N2 2 ζ t Q sup Q l lτ ρˆl τ ν Q l lτ ρˆl τ dτ . (8.5) ∶ 0 + S0 ( ) = N1+N2 Nmax τ t l~0 S S ⟨ ⟩ S ( )S l~0 S S ⟨ ⟩ S ( )S ≤ ≤ ≤ = = Then, under the assumptions of Theorem 8.1, there holds t 2N1 2N2 2 1~3 2N1 2N2 2 k kt k t ν k kτ k τ dτ Q Q N + S0 Q N N1+N2 Nmax(k~0 S S ⟨ ⟩ S ( )S k~0 S S ⟨ ⟩ S ( )S ) (8.6) ≤ = = ǫ2ν2~3 ǫζ t . ≲ + ( ) We ﬁrst prove that Proposition 8.2 gives Theorem 8.1. Recalling the density representation −1 −N0+2 −1 −2N −2 (7.2), with Gk t k kt k kt max (choosing N0 2Nmax 4), we have the S ( )S ≲ S S ⟨ ⟩ = S S ⟨ ⟩ = + following bound for any N1 N2 N + ≤ max t 2 2N1 2N2 2 2N1 2N2 2 2N1 2N2 Q k kt ρˆk t Q k kt Nk t + Q k kt S Gk t − s Nk s ds k~0 S S ⟨ ⟩ S ( )S ≤ k~0 S S ⟨ ⟩ S ( )S k~0 S S ⟨ ⟩ U 0 ( ) ( ) U = = = t t 2N1 2N2 2 2N1 2N2 2 ≲ Q k kt Nk t + Q k kt S Gk t − s Nk s ds S Gk t − s ds k~0 S S ⟨ ⟩ S ( )S k~0 S S ⟨ ⟩ 0 S ( )SS ( )S 0 S ( )S = = t 2N1 2N2 2 −3 −2 2N1 2N2 2 ≲ Q k kt Nk t + Q k S k t − s k ks Nk s ds, k~0 S S ⟨ ⟩ S ( )S k~0 S S 0 ⟨ ( )⟩ S S ⟨ ⟩ S ( )S = = t −2Nmax−2 2N2 where at the very end we used 2 max t − s,s so that k t − s kt ≲ k t − s −2 ks 2N2 . ≤ { } ⟨ ( )⟩ ⟨ ⟩ ⟨ ( )⟩ ⟨ ⟩ THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 46

Now using (8.6) and the fact that ζ t is monotone in t, we get ( ) t 2N1 2N2 2 2 2~3 −3 −2 2 2~3 Q k kt ρˆk t ≲ ǫ ν + ǫζ t + Q k S k t − s ǫ ν + ǫζ s ds k~0 S S ⟨ ⟩ S ( )S ( ) k~0 S S 0 ⟨ ( )⟩ ( ) = = 2 2~3 −4 2 2~3 ≲ ǫ ν + ǫζ t + Q k ǫ ν + ǫζ t ( ) k~0 S S ( ) = ǫ2ν2~3 ǫζ t , ≲ + ( ) Z3 −4 2 noting the summation over ∖ 0 of k is finite. Similarly, using Lt bounds in (8.6), we compute { } S S t 1~3 2N1 2N2 2 ν S Q k kτ ρˆk τ dτ 0 k~0 S S ⟨ ⟩ S ( )S = t 1~3 2N1 2N2 2 ≲ ν S Q k kτ Nk τ dτ 0 k~0 S S ⟨ ⟩ S ( )S = t τ 1~3 −3 −2 2N1 2N2 2 + ν Q k S S k τ − s k ks Nk s ds dτ k~0 S S 0 0 ⟨ ( )⟩ S S ⟨ ⟩ S ( )S = 2 2~3 −4 2 2~3 2 2~3 ≲ ǫ ν + ǫζ t + Q k ǫ ν + ǫζ t ≲ ǫ ν + ǫζ t . ( ) k~0 S S ( ) ( ) = Combining and recalling (8.5), we obtain ζ t ǫ2ν2~3 ǫζ t ( ) ≲ + ( ) which immediately yields Theorem 8.1, upon taking ǫ sufficiently small and recalling that α ω α ω the Fourier transform of ∂x Y ρ t is precisely ik ikt ρˆk t . The remaining subsections are( ) thus entirely devoted( ) ( to) prove( ) Proposition 8.2. In view of (8.4), we write t k t Sk t f0 µdv Sk t τ E vf k E vf k τ µdvdτ N S 3 k √ + S S 3 − ⋅ − ⋅ ∇ √ ( ) = R ( )[ ̂ ] 0 R ( )[(Æ) ( Æ ) ]( ) t (8.7) ν Sk t τ Γ f, f k τ µ dv dτ Ik t IIk t IIIk t . + S S 3 − √ ∶ + + 0 R ( )[( Æ( )) ( )] = ( ) ( ) ( ) We shall now prove (8.6) for each term in the following subsections. For the remainder of the section, fix N1, N2 such that N1 N2 N . + ≤ max 8.1. Initial data contribution. In this section, we give estimates on

Ik t Sk t f0 v µ dv. S 3 k √ ( ) = R ( )[ ̂ ( )] By (6.14) in Proposition 6.4 with η k, = − − − ′ 1~3 1~3 − ′ 2~3 I Nmax 1 (δ ~2)(ν t) (δ ~2)(νt) k t ≲ k t + 1 min e , e f0k E(2,2)′ . S ( )S ⟨ ( )⟩ { }Y ̂ Y Landau,k,k,Nmax+2 Summing over k, and using the assumption (3.2) for the initial data,

1 3 1 3 2 3 2N1 2N2 2 −2 −δ′(ν ~ t) ~ −δ′(νt) ~ 2 k kt I t t e , e f 2 2 ′ Q k ≲ min Q 0k E( , ) k~0 S S ⟨ ⟩ S ( )S ⟨ ⟩ { } k Z3 Y ̂ Y Landau,k,k,Nmax+2 = ∈ (8.8) 2 2~3 −2 −δ′(ν1~3t)1~3 −δ′(νt)2~3 ≲ ǫ ν t min e , e . ⟨ ⟩ { } ∞ 2 which in particular satisﬁes both the Lt and Lt bounds required in Proposition 8.2. THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 47

8.2. Nonlinear interaction I. In this section, we bound t IIk t Sk t τ E vf k τ E vf k τ µ dv dτ IIk,1 t IIk,2 t (8.9) S S 3 − ⋅ − ⋅ ∇ √ ∶ + ( ) = 0 R ( )[(Æ) ( ) ( Æ ) ( )] = ( ) ( ) under the bootstrap assumption (8.1) on f. Precisely, we will prove that t 2N1 2N2 II 2 1~3 2N1 2N2 II 2 Q k kt k t + ν S Q k kt k τ dτ ≲ ǫζ t , (8.10) k~0S S ⟨ ⟩ S ( )S 0 k~0 S S ⟨ ⟩ S ( )S ( ) = = where ζ t is deﬁned as in (8.5). ( ) Clearly, the ﬁrst term in (8.9) involving E ⋅ vf can be treated similarly as E ⋅ ∇vf. (In fact, it is better due to the absence of ∂v derivatives). We focus only the proof of the bounds involving the last term. Note that the semigroup Sk t − s commutes with El s , as it is independent of v. Therefore, we have ( ) ̂ ( ) t II k,2 t El τ Sk t τ vf k−l τ µ dv dτ. (8.11) − Q S ⋅ SR3 − ∇ √ ( ) = l~0 0 ̂ ( ) ( )[ Ä ( )] = To prove (8.10), we use (6.12) in Proposition 6.4 for the semigroup Sk t−τ with η k −l τ. Thus, for any N ′, N ′ 0 with N ′ N ′ N, we bound ( ) = ( ) 1 2 ≥ 1 + 2 ≤

Sk t τ vf − τ µ dv S 3 − ∇ k l √ U R ( )[ Ä ( )] U ′ −N2 1~3 −3~2 kt lτ ν t τ vf k−l τ E(10,0)′ ≲ − − ∇ ′ ⟨ ⟩ ⟨ ( )⟩ Y Ä ( )Y Landau,k,(k−l)τ,N2 (8.12) −N ′ −N ′ 1~3 −3~2 (SβS−1)~3 10 α β ω k l 1 kt lτ 2 ν t τ ν v ∂ ∂ Y f τ 2 , ≲ − − − Q x v k−l Lv ′ ′ ⟨ ⟩ ⟨ ⟩ ⟨ ( )⟩SαS N1, SωS N2 Y⟨ ⟩ ( )̂ ( )Y ≤ ≤ 1 SβS 2 ≤ ≤ ˆ where we have used Y0,(k−l)τ fk−l τ Y f k−l τ , recalling the vector fields Y0,(k−l)τ ∇v + ( ) = ( Ã ) ( ) = i k − l τ and Y t∇x + ∇v. ( To lighten) the= notation, define, for any N N and k Z3, ∈ ∈ (SβS−1)~3 10 α β ω fˆ τ ̃ ν v ∂ ∂ Y f τ 2 . (8.13) k GN ∶ Q x v k Lv Y ( )Y Sα=S+SωS N Y⟨ ⟩ ( )̂( )Y 1 SβS ≤2 ≤ ≤ G E(0) −1~3 Note that the N norm can be controlled by the N norm (with an ν weight) because ̃ ̃ 1~3 it controls up to two ∂v derivatives (taking into account the ν power), and we have a lot of extra v -weights. In other words, for any t 0, T , ∈ B ⟨ ⟩ − [ −) − + ˆ 2 2~3 2 1~3 max{0,N Nmax 2} fk t ν f t (0) ǫ min ν , t , (8.14) Q G̃N ≲ Ẽ ≲ k Y ( )Y Y ( )Y N { ⟨ ⟩} where at the end we used the bootstrap assumption (8.1). ˆ −2 We now plug the estimate (8.12) into (8.11), and recall that El −il l ρˆl, to deduce = S S N1 N2 k kt IIk,2 t S S ⟨ t ⟩ S ( )S ′ ′ −1 N1 N2 −N1 −N2 1~3 −3~2 l k kt k l kt lτ ν t τ ρˆl τ fˆk−l τ G̃ dτ ≲ Q S − − − N l~0 0 S S S S ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ( )⟩ S ( )SY ( )Y (8.15) = t 1 2 ′ ′ −1 N ,N ,N 1,N 2,N1,N2 N 1 N 2 l C t, τ l lτ ρˆl τ fˆk−l τ G̃ dτ, ≲ Q S k,l N l~0 0 S S ( )S S ⟨ ⟩ S ( )SY ( )Y = THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 48 where we have set N1,N2,N ,N ,N ′ ,N ′ − − − ′ − ′ − C 1 2 1 2 t, τ k N1 l N 1 kt N2 lτ N 2 k l N1 kt lτ N2 ν1~3 t τ 3~2, (8.16) k,l ∶= − − − ( ′ )′ S S S S ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ( ′ )⟩ ′ Here, N 1, N 2 , N1, N2 and N are arbitrary, as long as N 1 + N 2 Nmax, N1 + N2 N. ( ) ( ) ≤ ≤ ′ ′ ′ ′ N1,N2,N 1,N 2,N1,N2 N1,N2,N1,N 2,N1,N2 Estimates for Ck,l t, τ . Our next step is to estimate Ck,l t, τ . We divide up the integration region( ) in τ into lτ kt 2 and lτ kt 2. In the former( ) case, we further split up the sum in l to l k 2S andS ≤ Sl S~k 2. InS eachS > S case,S~ we obtain the following bound: S S ≤ S S~ S S > S S~ ● Case 1: lτ kt 2 and l k 2. In this case, kt − lτ kt 2 and k − l k 2. We S ′S ≤ S′ S~ S S ≤ S S~ S S ≥ S S~ S S ≥ S S~ choose N1, N2 N1, N2 , N Nmax and N 1, N 2 2, 3 . Then ( ) = ( ) = ( ) = ( ) N2 −N −N ′ −3 N1 −N −N ′ −2 kt lτ 2 kt − lτ 2 ≲ lτ , k l 1 k − l 1 ≲ l , ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ S S S S ⟨ ⟩ S S giving ′ ′ N1,N2,N 1,N 2,N1,N2 −3 −2 1~3 −3~2 Ck,l t, τ ≲ lτ l ν t − τ . (8.17) ( ) ⟨ ⟩ S S ⟨ ( )⟩ ● Case 2: lτ kt 2 and l k 2. In this case kt − lτ kt 2. We choose N Nmax, S S ≤ S S~ S S > S S~ S S ≥ S S~ = N1, N2 ′ ′ 2, N2 + 3 if N1 N2 N 1, N 2 ( ) N1, N2 ( ) ≥ ( ) = N1 + 2, 3 , ( ) = 0, N2 if N1 N2. ( ) ( ) < Notice that our choice satisfies N N N and N ′ N ′ N N (since 1 + 2 ≤ max 1 + 2 ≤ = max N 9). Whether N1 N2 or N1 N2, it is straightforward to check that max ≥ ≥ < N2 −N −N ′ −3 N1 −N −N ′ −2 −2 kt lτ 2 kt − lτ 2 ≲ lτ , k l 1 k − l 1 ≲ max l , k − l , ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ S S S S ⟨ ⟩ {S S ⟨ ⟩ } giving ′ ′ N1,N2,N 1,N2,N1,N2 −3 1~3 −3~2 −2 −2 Ck,l t, τ ≲ lτ ν t − τ max l , k − l . (8.18) ( ) ⟨ ⟩ ⟨ ( )⟩ {S S ⟨ ⟩ } Case 3: lτ kt 2. In this case we must have l k 2 (since τ t). Taking ● > ′ ′ > ≤ N , N S SN1,S N2S~ , N , N 4, 2 and N N S S 2 S S~6, we have 1 2 = 1 2 = = max − ≥ ( ) ( ) ( ) ′ ( ′ ) N1,N2,N 1,N2,N1,N2 −4 −2 Ck,l t, τ ≲ k − l kt − lτ . (8.19) ( ) ⟨ ⟩ ⟨ ⟩ Define C1, C2 and C3 by −3 −2 1~3 −3~2 −3 −2 1~3 −3~2 1 lτ l ν t τ , 2 lτ k l ν t τ , C ∶= − C ∶= − − ⟨ ⟩ S S ⟨ ( )⟩ −4 ⟨ −⟩2 ⟨ ⟩ ⟨ ( )⟩ (8.20) C3 ∶ k − l kt − lτ , = ⟨ ⟩ ⟨ ⟩ and define 2N1 2N 2 2 rl t ∶ Q l lt ρˆl t , ( ) = N 1+N 2 Nmax S S ⟨ ⟩ S ( )S ≤ ζ t r τ ν1~3 t r τ τ ζ t (so that ≲ sup0 τ t ∑l~0 l + ∑l~0 ∫0 l d ≲ ). Set ( ) ≤ ≤ = ( ) = ( ) ( ) t 2 −1 1~2 j t l jr τ fˆk−l τ G̃ dτ , j 1, 2, (8.21) I ∶ Q Q S C l Nmax ( ) = k~0 l~0 0 S S ( )Y ( )Y = = = and set t 2 −1 1~2 3 t l 3r τ fˆk−l τ G̃ dτ . (8.22) I ∶ Q Q S C l Nmax−2 ( ) = k~0 l~0 0 S S ( )Y ( )Y = = THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 49

′ ′ It follows from (8.15), (8.16) and the bounds for CN1,N2,N1,N 2,N1,N2 t, τ in (8.17), (8.18), k,l ( ) (8.19) that 2N1 2N2 II 2 Q k kt k,2 t ≲I1 +I2 +I3. (8.23) k~0 S S ⟨ ⟩ S ( )S = By the bound (8.23), in order to obtain the claim (8.10), it suﬃces to show t −1~3 Ij t ≲ ǫζ t , Ij τ dτ ≲ ǫν ζ t , j 1, 2, 3, (8.24) ( ) ( ) S0 ( ) ( ) = which will be achieved below.

∞ Lt bounds for I1 t . To bound I1, we start with (8.21) and the deﬁnition of C1 in (8.20). Then, using the Cauchy–Schwarz( ) inequality in τ, and then the Young’s convolution inequality for the sums, we bound

I1 t ( ) t 2 −3 −3 1~3 −3~2 1~2 l lτ ν t τ r τ fˆk−l τ G̃ dτ Q Q S − l Nmax ≤ k~0 l~0 0 S S ⟨ ⟩ ⟨ ( )⟩ ( )Y ( )Y = = t t 2 −6 −5~2 1~3 −3 1~2 −7~2 ˆ 2 1~2 l lτ ν t τ rl τ dτ τ fk−l τ G̃ dτ ≲ Q Q S − S Nmax k~0 l~0( 0 S S ⟨ ⟩ ⟨ ( )⟩ ( ) ) ( 0 ⟨ ⟩ Y ( )Y ) = = t 2 t −6 −5~2 1~3 −3 1~2 −7~2 ˆ 2 l lτ ν t τ rl τ dτ τ fk τ G̃ dτ . ≲ Q S − Q S ⟨ ⟩ Nmax l~0( 0 S S ⟨ ⟩ ⟨ ( )⟩ ( ) ) k 0 Y ( )Y = (8.25) 2 By (8.14) (allowing ⟨τ⟩ growth), t t t −7~2 ˆ 2 −2~3 −7~2 2 −3~2 τ fk τ ̃ dτ ν τ f τ (0) dτ ǫ τ dτ ǫ. (8.26) Q S ⟨ ⟩ GN ≲ S ⟨ ⟩ Ẽ ≲ S ⟨ ⟩ ≲ k 0 Y ( )Y max 0 Y ( )Y Nmax 0 Thus, substituting this into (8.25), we obtain t 2 −6 −5~2 1~3 −3 1~2 I1(t) ≲ ǫ Q S l lτ ν t − τ rl τ dτ . (8.27) l~0( 0 S S ⟨ ⟩ ⟨ ( )⟩ ( ) ) = To proceed, a direct computation using (8.5) shows t 2 t 2 −6 −5~2 1~3 −3 1~2 −6 −5~2 1~2 Q S l lτ ν t − τ rl τ dτ ≲ Q S l lτ rl τ dτ l~0( 0 S S ⟨ ⟩ ⟨ ( )⟩ ( ) ) l~0( 0 S S ⟨ ⟩ ( ) ) = = (8.28) −7~2 2 ≲ Q l sup sup rl′ τ ≲ ζ t . (l~0 S S ) ( l′ ~0 0 τ t ( )) ( ) = = ≤ ≤ ∞ Plugging this into (8.27) proves the Lt estimates in (8.24).

1 1 Lt bounds for I1 t . In view of (8.27), to prove the Lt bound for I1 t , it suﬃces to understand ( ) ( ) t s 2 −6 −5~2 1~3 −3 1~2 S Q S l lτ ν s − τ rl τ dτ ds. 0 l~0( 0 S S ⟨ ⟩ ⟨ ( )⟩ ( ) ) = s −5~2 1~3 −3 −5~2 We split the τ-integration: when τ max 2 , 1 , we have lτ ν s − τ ≲ lτ ≲ − − − − ≥ − − lτ 5~4 ls 5~4 lτ 5~4 s 5~4; while when {τ max} s , 1 ,⟨ we⟩ have⟨ lτ( 5~2 )⟩ν1~3 s ⟨ τ⟩ 3 ⟨ ⟩ ⟨ ⟩ ≲ ⟨ ⟩ ⟨ ⟩ ≤ { 2 } ⟨ ⟩ ⟨ ( − )⟩ ≲ THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 50

lτ −5~2 ν1~3s −3. Hence, ⟨ ⟩ ⟨ ⟩ t s 2 −6 −5~2 1~3 −3 1~2 S Q S l lτ ν s − τ rl τ dτ ds 0 l~0( 0 S S ⟨ ⟩ ⟨ ( )⟩ ( ) ) = t s 2 −6 −5~4 1~2 −5~4 ≲ S Q S l lτ rl τ dτ s ds 0 l~0( 0 S S ⟨ ⟩ ( ) ) ⟨ ⟩ = t s 2 −6 −5~2 1~2 1~3 −3 + S Q S l lτ rl τ dτ ν s ds 0 l~0( 0 S S ⟨ ⟩ ( ) ) ⟨ ⟩ = t t −7~2 2 −5~4 1~3 −3 −1~3 ≲ Q l sup sup rl′ τ S ⟨s⟩ ds + S ⟨ν s⟩ ds ≲ ν ζ(t). (l~0 S S ) ( l′ ~0 0 τ t ( )) 0 0 = = ≤ ≤ Combining this with (8.27) yields the desired conclusion in (8.24).

∞ −2 −2 Lt bounds for I2(t). This is similar to I1, except that we use ⟨k − l⟩ instead of l for summability. More precisely, we argue as in (8.27) except for distributing the ℓ1 and ℓS2S sums diﬀerently in the application of Young’s convolution inequality, to obtain I2 t ( ) t 2 −1 −2 −3 1~3 −3~2 1~2 l k l lτ ν t τ r τ fˆk−l τ G̃ dτ Q Q S − − l Nmax ≤ k~0 l~0 0 S S ⟨ ⟩ ⟨ ⟩ ⟨ ( )⟩ ( )Y ( )Y = = t −2 −5~2 1~3 −3 1~2 ≲ Q Q S l lτ ν t − τ rl τ dτ k~0 l~0( 0 S S ⟨ ⟩ ⟨ ( )⟩ ( ) ) = = t 2 −7~2 −4 ˆ 2 1~2 × τ k − l fk−l τ G̃ dτ (S0 ⟨ ⟩ ⟨ ⟩ Y ( )Y Nmax ) t t 2 −2 −5~2 1~3 −3 −2 −7~2 ˆ 2 1~2 l lτ ν t τ rl τ dτ k τ fk τ G̃ dτ ≲ Q S − Q⟨ ⟩ (S ⟨ ⟩ Nmax l~0 0 S S ⟨ ⟩ ⟨ ( )⟩ ( ) k 0 Y ( )Y ) = t t −2 −5~2 1~3 −3 −7~2 ˆ 2 l lτ ν t τ rl τ dτ τ fk τ G̃ dτ , ≲ Q S − Q S Nmax l~0 0 S S ⟨ ⟩ ⟨ ( )⟩ ( ) ( k 0 ⟨ ⟩ Y ( )Y ) = (8.29) where in the very end, we used the Cauchy–Schwarz inequality for the sum in k. Using (8.26), we thus obtain t −2 −5~2 1~3 −3 I2 t ≲ ǫ Q S l lτ ν t − τ rl τ dτ, (8.30) ( ) l~0 0 S S ⟨ ⟩ ⟨ ( )⟩ ( ) = Finally, we use (8.5) to bound t t −2 −5~2 1~3 −3 −5~2 Q S l lτ ν t − τ rl τ dτ ≲ S τ dτ sup Q rl τ ≲ ζ t . (8.31) l~0 0 S S ⟨ ⟩ ⟨ ( )⟩ ( ) ( 0 ⟨ ⟩ )(0 τ t l~0 ( )) ( ) = ≤ ≤ = Plugging this into (8.30) gives the desired conclusion in (8.24).

1 L bounds for 2 t . We bound the integral in (8.30) using Fubini’s theorem: t I ( ) t s −2 −5~2 1~3 −3 Q S S l lτ ν s − τ rl τ dτ ds l~0 0 0 S S ⟨ ⟩ ⟨ ( )⟩ ( ) = t t t −5~2 1~3 −3 −1~3 −5~2 ≲ Q S τ S ν s − τ ds rl τ dτ ≲ ν Q S τ rl τ dτ. l~0 0 ⟨ ⟩ ( τ ⟨ ( )⟩ ) ( ) l~0 0 ⟨ ⟩ ( ) = = THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 51

~~Then, using H¨older’s inequality and (8.5), we obtain~~

t t −1~3 −5~2 −1~3 −5~2 −1~3 ν Q S τ rl τ dτ ≲ ν S τ dτ sup Q rl τ ≲ ν ζ t . l~0 0 ⟨ ⟩ ( ) ( 0 ⟨ ⟩ )(0 τ t l~0 ( )) ( ) = ≤ ≤ = Combining these two estimates with (8.30) yields the desired bound in (8.24).

∞ ∞ Lt bounds for I3 t . Next, we bound I3 in Lt . Starting with (8.20), (8.22) and then ˆ 2 ( ) using fk−l τ G̃ ≲ ǫ (by (8.14)), we have Y ( )Y Nmax−2 t 2 −1 −4 −2 1~2 3 t l k l kt lτ r τ fˆk−l τ G̃ dτ I ≲ Q Q S − − l Nmax−2 ( ) k~0 l~0 0 S S ⟨ ⟩ ⟨ ⟩ ( )Y ( )Y = = t 2 ′ −1 −4 −2 ≲ ǫ sup sup rl′ τ Q Q l k − l S kt − lτ dτ (8.32) ( l′ ~0 τ ′ [0,t] ( )) k~0 l~0 S S ⟨ ⟩ 0 ⟨ ⟩ = ∈ = = 2 ′ −2 −4 ≲ ǫ sup sup rl′ τ Q Q l k − l . ( l′ ~0 τ ′ [0,t] ( )) k~0 l~0 S S ⟨ ⟩ = ∈ = = Hence, Young’s convolution inequality gives

~~2 2 −2 −4 −4 −4 Q Q l k − l ≲ Q l Q⟨k⟩ ≲ 1. k~0 l~0 S S ⟨ ⟩ l~0 S S k = = = Plugging this back into (8.32) and using (8.5) yields~~

~~I3(t) ≲ ǫζ(t).~~

~~1 1 ˆ 2 Lt bounds for I3(t). To bound the Lt norm for I3(t), we ﬁrst use fk−l τ G̃ ≲ ǫ (by Y ( )Y Nmax−2 (8.14)) to obtain~~

t 3 s ds S0 I ( ) t s 2 −1 −4 −2 1~2 l k l ks lτ r τ fˆk−l τ G̃ dτ ds ≲ Q S Q S − − l Nmax−2 (8.33) k~0 0 l~0 0 S S ⟨ ⟩ ⟨ ⟩ ( )Y ( )Y = = t s 2 ǫ l −1 k l −4 ks lτ −2r1~2 τ τ s. ≲ Q S Q S − − l d d k~0 0 l~0 0 S S ⟨ ⟩ ⟨ ⟩ ( ) = = By Schur’s test,

t t −1 −4 −2 S I3 s ds ≲ ǫ sup sup S Q l k − l ks − lτ dτ 0 ( ) k~0 s 0 l~0 S S ⟨ ⟩ ⟨ ⟩ = = t t (8.34) −1 −4 −2 ′ ′ × sup sup S Q l k − l ks − lτ ds Q S rl′ τ dτ . l~0 τ 0 k~0 S S ⟨ ⟩ ⟨ ⟩ l′ ~0 0 ( ) = = = Each of the integrals can be easily checked to be bounded, so that by (8.5) we have

~~t t −1~3 S I3 s ds ≲ ǫ Q S rl τ dτ ≲ ǫν ζ t . 0 ( ) l~0 0 ( ) ( ) = THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 52~~

8.3. Nonlinear interaction II. In this section, under the bootstrap assumption (8.1) on f, we bound t IIIk t ν Sk t τ Γ f, f k τ µ dv dτ. S S 3 − √ ( ) = 0 R ( )[( Æ( )) ( )] We will prove that

t 2N1 2N2 III 2 1~3 2N1 2N2 III 2 2 2~3 Q k kt k t + ν S Q k kτ k τ dτ ≲ ǫ ν . (8.35) k~0 S S ⟨ ⟩ S ( )S 0 k~0 S S ⟨ ⟩ S ( )S = = To prove (8.35), we use (6.12) in Proposition 6.4 for the semigroup S t τ) with η kτ. k − = Thus, for any N1, N2 such that N1 N2 N , ( + ≤ max

Sk t τ Γ f, f k τ µ dv S 3 − √ U R ( )[( Æ( )) ( )] U −N2 1~3 −3~2 ≲ kt ν t − τ Γ f, f k τ E(10,0)′ (8.36) ⟨ ⟩ ⟨ ( )⟩ Y( Æ( )) ( )Y Landau,k,kτ,N2 −N1 −N2 1~3 −3~2 SβS~3 10 α β ω k kt ν t τ ν v ∂ ∂ Y Γ f, f τ 2 ≲ − Q x v k Lv S S ⟨ ⟩ ⟨ ( )⟩ SαS N1, SβS 1, SωS N2 Y⟨ ⟩ [ ( ( ))]̂( )Y = ≤ ≤ in which we used Y Γ τ Y Γ τ , recalling the vector ﬁeld Y t . 0,kτ k Å k ∇x + ∇v By Lemma 4.10, ̂ ( ) = ( ) ( ) =

2SβS~3 10 α β ω 2 ν v ∂ ∂ Y Γ f, f τ 2 Q Q x v k Lv k~0 SαS N1, SβS 1, SωS N2 Y⟨ ⟩ [ ( ( ))]̂( )Y = = ≤ ≤ 2 10 α′ β′ ω′ 10 α′′ β′′ ω′′ v ∂x ∂v Y f L2 v ∂x ∂v Y f L2 ≲ Q v v L2 Sα′S+Sα′′S N1, Sω′S+Sω′′S N2 [Y⟨ ⟩ Y Y⟨ ⟩ Y [ x (8.37) 1 =Sβ′S+Sβ′′S 2 ≤ ≤ ≤ 2 2~3 10 α′ β′ ω′ 10 α′′ β′′ ω′′ ν v ∂x ∂v Y f L2 v ∂x ∂v Y f L2 . + Q v v L2 Sα′S+Sα′′S N1, Sω′S+Sω′′S N2 [Y⟨ ⟩ Y Y⟨ ⟩ Y [ x Sβ=′S+Sβ′′S 3 ≤ = The two terms in (8.37) are to be controlled in appropriate E and D norms. We point out three important observations: ̃ ̃ ● Note that the v -weights in (8.37) are signiﬁcantly lower than what is encoded in the energy and⟨ dissipation⟩ norms. One term could have N 3 derivatives (when, say, α′ β′ ω′ 0), but in that ● max + = = = case exactly three of the derivatives must be ∂v, so it can be controlled with the D(0) norm. ̃Nmax ● In each term of (8.37), at least one factor has at least one ∂v derivative. We put that factor in the D norm, and the other factor in the E so as not to incur a loss of ν−1~3. ̃ ̃ We only consider the second term in (8.37); the ﬁrst term is similar and slightly simpler. ′ ′′ ′ ′′ ′ ′′ ● Take α + α N1, ω + ω N2 and β + β 3. S S S S = ′ S′ S ′ S S ≤ ′′ ′′ S ′′S S S = ● After switching α , β ,ω with α , β ,ω if necessary, we assume without loss of generality α′ β(′ ω′ ) N (2 2 . We) can apply Sobolev embedding in x to the + + ≤ max + correspondingS S term,S S S notingS ⌊( that since)~ N⌋ max 9, we have Nmax 2 2 2 Nmax 2. ′ ′− + +′′ ′′−− ′ ′′ ≥ β ⌊( β ej )~ ⌋β ≤ β ej ● By the pigeonhole principle, β 1 or β 1, i.e. ∂v ∂vj ∂v or ∂v ∂vj ∂v . S S ≥ S S ≥ = = THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 53

We use H¨older’s inequality and then the Sobolev inequality in x to obtain 2 2~3 10 α′ β′ ω′ 10 α′′ β′′ ω′′ ν v ∂ ∂ Y f 2 v ∂ ∂ Y f 2 x v Lv x v Lv 2 [Y⟨ ⟩ Y Y⟨ ⟩ Y [Lx −4~3 2Sβ′S~3 10 α′′′ α′ β′ ω′ 2 2Sβ′′S~3 10 α′′ β̃′′ ω′′ 2 ν ν v ∂ ∂ ∂ Y f 2 ν v ∂ ∂ Y f ≲ Q x x v Lx,v × Q x v ∆x,v ( ′′′ ′′ ′′ Sα S 2 Y⟨ ⟩ Y )β̃ β −ej Y⟨ ⟩ Y ≤ = ′− 2Sβ′S~3 10 α′′′ α′ β ej ω′ 2 2Sβ′′S~3 10 α′′ β′′ ω′′ 2 ν v ∂ ∂ ∂v Y f ν v ∂ ∂ Y f 2 + Q x x ∆x,v x v Lx,v (8.38) Sα′′′S 2 Y⟨ ⟩ Y Y⟨ ⟩ Y ≤ ′ ′ β̃ β −ej = −4~3 2 2 2 2 ν f 0 f 0 f 0 f 0 ≲ Ẽ( ) D̃( ) + D̃( ) Ẽ( ) (Y Y Nmax−2 Y Y Nmax Y Y Nmax−2 Y Y Nmax ) −2~3 2 −4~3 2 ≲ ǫν f D̃(0) + ǫν f D̃(0) , Y Y Nmax Y Y Nmax−2 where in the very last line we used the bootstrap assumption (8.1). The ﬁrst term in (8.37) can be bounded similarly so that we have

2SβS~3 10 α β ω 2 ν v ∂ ∂ Y Γ f, f τ 2 Q Q x v k Lv k~0 SαS N1, SβS 1, SωS N2 Y⟨ ⟩ [ ( ( ))]̂( )Y = = ≤ ≤ (8.39) −2~3 2 −4~3 2 ǫν f 0 ǫν f 0 . ≲ D̃( ) + D̃( ) Y Y Nmax Y Y Nmax−2 We now square (8.36), multiply it by k 2N1 kt 2N2 , sum over k, and plug in (8.39). Using the Cauchy–Schwarz inequality for the τS integral,S ⟨ ⟩ we bound 2N1 2N2 III 2 Q k kt k t k~0S S ⟨ ⟩ S ( )S = t 2 2 2N1 2N2 ν k kt Sk t τ Γ f, f k τ µ dv dτ Q S SR3 − √ ≤ k~0 S S ⟨ ⟩ 0 ( )[( Æ( )) ( )] = t t t (8.40) 4~3 2 2~3 2 1~3 −3 ǫν f τ 0 dτ ǫν f τ 0 dτ ν t τ dτ ≲ S D̃( ) + S D̃( ) S − 0 Y ( )Y Nmax 0 Y ( )Y Nmax−2 0 ⟨ ( )⟩ t t 2 1~3 2 2~3 ǫν f τ 0 dτ ǫν f τ 0 dτ ǫν , ≲ S D̃( ) + S D̃( ) ≲ 0 Y ( )Y Nmax 0 Y ( )Y Nmax−2 where we have also used the bootstrap assumption (8.1) at the end. This gives the desired L∞ bounds in (8.35). The L2 estimates in (8.35) also follow from essentially the same computation as (8.40), after also using Fubini’s theorem, namely, t 2N1 2N2 III 2 S Q k ks k s ds 0 k~0 S S ⟨ ⟩ S ( )S = t s 4~3 1~3 −3~2 2 ǫ ν ν s τ f τ (0) dτ ds ≲ S0 S0 − D̃ ⟨ ( )⟩ Y ( )Y Nmax (8.41) t s s′ 2~3 1~3 −3~2 2 1~3 ′ −3~2 ν ν s τ f τ 0 dτ ds sup ν s τ dτ + S S − D̃( ) S − 0 0 Nmax−2 0 s′ t 0 ⟨ ( )⟩ Y ( )Y ≤ ≤ ⟨ ( )⟩ t t 2~3 2 2 2 1~3 ǫν f τ 0 dτ ǫ f τ 0 dτ ǫ ν . ≲ S D̃( ) + S D̃( ) ≲ 0 Y ( )Y Nmax 0 Y ( )Y Nmax−2

~~This ends the proof of Proposition 8.2, and so that of Theorem 8.1. THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 54~~

~~9. Nonlinear energy estimates In this section, we derive energy estimates for the full nonlinear Vlasov–Poisson–Landau equation (2.2a)–(2.2b) under the bootstrap assumptions (8.1) and (8.2).~~

~~The main result of this section is the following.~~

Theorem 9.1. Consider data as in Theorem 3.1. Suppose there exists TB 0 such that the T3 R3 > solution f to (2.2a)–(2.2b) remains smooth in 0, TB × × and satisfies the bootstrap assumptions (8.1) and (8.2). [ ) Then, for ϑ 0, 2 , 0 t TB, and 0 N Nmax, the following energy estimate holds: ∈ { } ≤ < ≤ ≤ t − − + f τ 2 ν1~3 f τ 2 τ ǫ2ν2~3 ν 1~3, t max{0,N Nmax 2}, sup Ẽ(ϑ) + D̃(ϑ) d ≲ min (9.1) 0 τ t N S0 N ≤ ≤ Y ( )Y Y ( )Y { ⟨ ⟩} where ⋅ Ẽ(ϑ) and ⋅ D̃(ϑ) are the global energy and dissipation norms defined as in (5.17). Y Y N Y Y N The main step in the proof of Theorem 9.1 is the following estimates for the inhomogeneous terms : R̃α,β,ω Proposition 9.2. Fix ϑ 0, 2 . Under the assumptions of Theorem 9.1, for α + β + ω N, ∈ { } S S S S S S ≤ 1~2 −2 2 1~2 2~3 2 1~3 ǫ t f t ǫ ν f t ν f t (ϑ) f t (ϑ) f t (ϑ) Rα,β,ω ≲ Ẽ(ϑ) + D̃(ϑ) + Ẽ D̃ D̃ ̃ ⟨ ⟩ Y ( )Y N Y ( )Y N Y ( )Y N Y ( )Y N Y ( )Y Nmax−2 α ω (9.2) (ϑ) (ϑ) 2 min f t D̃ , f t Ẽ ∂x Y ρ~0 t Lx , + N N Q = {Y ( )Y Y ( )Y } SαS+SωS N Y ( )Y ≤ where are as defined in Proposition 5.11. R̃α,β,ω We now show that Proposition 9.2 implies Theorem 9.1.

~~d f 2 Proof of Theorem 9.1 assuming Proposition 9.2. Computing dt Ẽ(ϑ) and using the main Y Y N energy estimates in Proposition 5.11, we obtain~~

d −1 −1 e1+⟨t⟩ f 2 θν1~3e1+⟨t⟩ f 2 Ẽ(ϑ) + D̃(ϑ) dt( Y Y N ) Y Y N 1+⟨t⟩−1 d 2 1~3 2 −2 1+⟨t⟩−1 2 e f θν f t e f ϑ Q ̃0 + ̃0 − Ẽ( ) (9.3) dt ,Nα,β ,N,Nω ,Nα,β ,N,Nω N = Nα,β+Nω N ( Y YE Y YD ) ⟨ ⟩ Y Y ≤ −2 2 2 2 1 t f (ϑ) C ν a, b, c ∂tφ L∞ φ ,∞ f (ϑ) Rα,β,ω , − Ẽ + + x + Wx Ẽ + Q ≤ ⟨ ⟩ Y Y N ( S( )S (Y Y Y Y )Y Y N SαS+SβS+SωS N ̃ ) ≤ 1 noting 1 e1+⟨t⟩− e2. Since f satisﬁes the nonlinear Vlasov–Poisson–Landau system, the conservation≤ law and≤ (3.1) imply

f µ dv dx v f µ dv dx v 2f µ dv dx 0, (9.4) 3 3 √ 3 3 j √ 3 3 √ UT ×R = UT ×R = UT ×R S S = i.e. a, b, c 2 0; and the bootstrap assumption (8.2) implies that the second term in (9.3) is = − boundedS( )Sǫ1~2ν1~3 t 2 f 2 . Plugging in also the estimates for R from Proposition 9.2, ≲ Ẽ(ϑ) α,β,ω ⟨ ⟩ Y Y N ̃ THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 55 we obtain

d −1 e1+⟨t⟩ f 2 θν1~3 f 2 t −2 f 2 Ẽ(ϑ) + D̃(ϑ) + Ẽ(ϑ) dt( Y Y N ) Y Y N ⟨ ⟩ Y Y N 1~2 −2 2 1~2 2~3 2 1~3 ≲ ǫ t f t Ẽ(ϑ) + ǫ ν f t D̃(ϑ) + ν f t Ẽ(ϑ) f t D̃(ϑ) f t D̃(ϑ) (9.5) ⟨ ⟩ Y ( )Y N Y ( )Y N Y ( )Y N Y ( )Y N Y ( )Y Nmax−2 α ω (ϑ) (ϑ) 2 min f t D̃ , f t Ẽ ∂x Y ρ~0 t Lx . + N N Q = {Y ( )Y Y ( )Y } SαS+SωS YN ( )Y ≤ for ǫ suﬃciently small, the ﬁrst two terms on the right hand side of (9.5) can be absorbed by the last two terms on the left hand side, i.e.

d −1 e1+⟨t⟩ f 2 θν1~3 f 2 t −2 f 2 Ẽ(ϑ) + D̃(ϑ) + Ẽ(ϑ) dt( Y Y N ) Y Y N ⟨ ⟩ Y Y N 1~3 α ω ν f t (ϑ) f t (ϑ) f t f t (ϑ) , f t (ϑ) ∂ Y ρ t 2 . Ẽ D̃ D̃N 2 min D̃ Ẽ x ~0 Lx ≲ N N max− + N N Q = Y ( )Y Y ( )Y Y ( )Y {Y ( )Y Y ( )Y Sα}S+SωS YN ( )Y ≤ (9.6) Define now t t f t f t 2 ν1~3 f τ 2 dτ τ −2 f τ 2 dτ. FN ∶ Ẽ(ϑ) + S D̃(ϑ) + S Ẽ(ϑ) [ ]( ) = Y ( )Y N 0 Y ( )Y N 0 ⟨ ⟩ Y ( )Y N Thus FN f t can be bounded in terms of the t-integral of the right hand side of (9.6). This will in turn[ ]( be) controlled below for different values of N. The case N Nmax − 2. Consider first the case N Nmax − 2. Using (8.1), we have 1~2 ≤1~3 ≤ f Ẽ(ϑ) ≲ ǫ ν . Thus, Y Y Nmax−2 t 1~3 ν f τ Ẽ(ϑ) f τ D̃(ϑ) f τ D̃(ϑ) dτ S 2 0 Y ( )Y N Y ( )Y N Y ( )Y Nmax− t (9.7) ǫν2~3 f τ 2 dτ ǫν1~3 f t . ≲ S D̃(ϑ) ≲ FN 0 Y ( )Y Nmax−2 [ ]( )

On the other hand, since ρ~0 is v-independent and has vanishing x-mean by definition, Poincaré’s inequality implies that= for N N 2, ≤ max − α ω −2 2 α′ α ω −2 α ω ∂ Y ρ t 2 t t ∂ ∂ Y ρ t 2 t ∂ Y ρ t 2 . Q x ~0 Lx ≲ Q Q x x ~0 Lx ≲ Q x ~0 Lx = ′ = = SαS+SωS YN ( )Y ⟨ ⟩ Sα S 2 SαS+SωS N Y⟨ ⟩ ( )Y ⟨ Sα⟩S+SωS NYmax ( )Y ≤ = ≤ ≤ Thus, using (8.3) and Young’s inequality, we obtain, for any η 0, > t α ω (ϑ) (ϑ) 2 min f τ D̃ , f τ Ẽ ∂x Y ρ~0 τ Lx dτ S N N Q = 0 {Y ( )Y Y ( )Y Sα}S+SωS YN ( )Y ≤ t ∞ −2 1~3 −1 2 2~3 dτ (9.8) f τ (ϑ) τ ǫν dτ η f t η ǫ ν ≲ S Ẽ ≲ FN + S 2 0 Y ( )Y N ⟨ ⟩ [ ]( ) 0 τ −1 2 2~3 ⟨ ⟩ ≲ ηFN f t + η ǫ ν . [ ]( ) Plugging (9.7) and (9.8) into (9.6), and bounding the initial data term by (3.2), we thus obtain 1~3 −1 2 2~3 FN f t ≲ ǫν + η FN f t + η ǫ ν . [ ]( ) ( ) [ ]( ) Choosing ǫ0, ν0 and η sufficiently small, we can absorb the first term on the right to the left, giving the desired bound for FN t . ( ) THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 56

The case Nmax − 1 N Nmax. We consider the case N Nmax; the case N Nmax − 1 is similar. Note that we≤ need≤ to prove two estimates: one allowing= for a loss in ν−=2~3, and the other allowing for a growth in t 2. As above, we will bound the⟨ time-integral⟩ of the terms in (9.6), now for N N . First, = max t 1~3 ν f τ Ẽ(ϑ) f τ D̃(ϑ) f τ D̃(ϑ) dτ S 2 0 Y ( )Y Nmax Y ( )Y Nmax Y ( )Y Nmax− t t 1~3 2 1~2 1~3 2 1~2 (9.9) sup f τ (ϑ) ν f τ dτ ν f τ dτ ≲ Ẽ S D̃(ϑ) S D̃(ϑ) 0 s t Nmax 0 Nmax 0 Nmax−2 ( ≤ < Y ( )Y )( Y ( )Y ) ( Y ( )Y ) 1~2 1~3 − ≲FNmax f t FNmax 2 f t ≲ ǫν FNmax f t , [ ]( ) [ ] ( ) [ ]( ) − where in the final estimate, we used the bound for FNmax 2 f t derived above. [ ](∞) We have two ways for bounding the other term. Using the Lt bound in (8.3), we have t α ω min f τ (ϑ) , f τ (ϑ) ∂ Y ρ τ 2 dτ D̃ Ẽ Q x ~0 Lx S0 Nmax Nmax = {Y ( )Y Y ( )Y }SαS+SωS NYmax ( )Y ≤ t t 1~3 −1 2 2~3 2 (9.10) f τ (ϑ) ǫν dτ η f t η ǫ ν dτ ≲ S Ẽ ≲ FNmax + S 0 Y ( )Y Nmax [ ]( ) ( 0 ) −1 2 2~3 2 ≲ ηFNmax f t + η ǫ ν t . [ ]( ) ⟨ ⟩ 2 and, using instead the Lt bound in (8.3), we obtain t α ω f τ (ϑ) , f τ (ϑ) ∂ Y ρ τ 2 τ min D̃ Ẽ Q x ~0 Lx d S0 Nmax Nmax = {Y ( )Y Y ( )Y }SαS+SωS NYmax ( )Y ≤ t t 1~3 2 −1 −1~3 α ω 2 (9.11) ην f τ (ϑ) dτ η ν ∂ Y ρ 0 τ 2 dτ ≲ D̃ + Q x ~ Lx S0 N S0 = Y ( )Y max SαS+SωS NmaxY ( )Y ≤ −1 2 ≲ ηFNmax f t + η ǫ . [ ]( ) Combining (9.9)–(9.11), integrating (9.6), and controlling initial data by (3.2), we thus obtain 1~3 −1 2 2~3 2 FNmax f t ≲ ǫν + η FNmax f t + η ǫ min ν t , 1 . [ ]( ) ( ) [ ]( ) { ⟨ ⟩ } We can thus conclude as before by choosing ǫ0, ν0 and η small. The remainder of this section is thus devoted to the proof of Proposition 9.2, after some preliminary bounds on the electric field in the next subsection. 9.1. Bounds on the electric field. In this section, we give estimates on the electric field. Lemma 9.3. Let 0 N N . Then ≤ ≤ max α ω α ω α ω ∂x Y φ t L2 ∂x Y E t L2 ∂x Y ρ~0 t L2 (9.12) Q x + Q x ≲ Q = x SαS+SωS N+2 Y ( )Y SαS+SωS N+1 Y ( )Y SαS+SωS N Y ( )Y ≤ ≤ ≤ and α ω α ω α ω ∂x Y φ t L∞ ∂x Y E t L∞ ∂x Y ρ~0 t L2 . (9.13) Q x + Q x ≲ Q = x SαS+SωS N Y ( )Y SαS+SωS N−1 Y ( )Y SαS+SωS N Y ( )Y ≤ ≤ ≤ −2 Proof. The estimate (9.12) follows from the Poisson equation φˆk k ρˆk and the definition ˆ ˆ Z3 = S S Ek −ikφk for each Fourier mode k 0 . The bound (9.13) then follows from Sobolev embedding.= ∈ { } THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 57

~~Lemma 9.4. The electric potential φ satisﬁes −2 α ω ∂ φ ∞ t ∂ Y f 2 . (9.14) t Lx ≲ Q x Lx,v Y Y ⟨ ⟩ SαS+SωS 3 Y Y ≤~~

Proof. Since 3 ∂ φ dx 0, we apply Poincar´e’s and Sobolev’s inequalities to obtain ∫T t = −2 α ω −2 α+α′ ω ∂ φ ∞ t ∂ Y ∂ φ ∞ t ∂ Y ∂ φ 2 . t Lx ≲ Q x t Lx ≲ Q Q x t Lx Y Y SαS+SωS 2⟨ ⟩ Y Y ⟨ ⟩ Sα′S 2 SαS+SωS 2 Y Y ≤ ≤ ≤ To control the ﬁnal L2 norm by the right hand side of (9.14), we use the elliptic equation for ∂tφ:

∆∂tφ ∂tρ x vf µ dv, − = = −∇ ⋅ SR3 √ where the last identity is the conservation of mass. 9.2. Estimates on Q,ℓ . In this subsection, we give claimed bounds on Q,ℓ that appear Rα,β,ω R̃α,β,ω in Proposition 5.11 in term of energy and dissipation norms, under the assumptions of Q,ℓ Theorem 9.1. (The bounds for Zα,β,ω will be derived later in Section 9.3.) Precisely, we shall bound 2SβS~3 Q,ℓ−2Sα′S 2(SβS+Sβ′S)~3 Q,ℓ−2Sβ′S Q ν Rα+α′,β,ω + Q ν Rα+,β+β′,ω Sα′S 1 Sβ′S 2 ≤ ≤ for α + β + ω Nmax. We recall from Lemma 5.5 and (5.3) that S S S S S S = Q,ℓ 2(q+1)φ 2 α β ω α β ω α β ω e w ∂ ∂ Y f ∂ ∂ Y Ej∂v µ E v E v, ∂ ∂ Y f Rα,β,ω U 3 3 x v x v j √ + ⋅ ∇ − ⋅ x v = T ×R [ ] [ ] α β ω + ν∂x ∂v Y Γ f, f dv dx ( ) Q,ℓ,1 Q,ℓ,2 Q,ℓ,3 =∶ Rα,β,ω + Rα,β,ω + Rα,β,ω Q,ℓ,j in which Rα,β,ω correspond to the integral involving each term in the bracket. The claimed estimates on Q,ℓ in Proposition 9.2 are thus a combination of Lemmas 9.5–9.7 below R̃α,β,ω giving bounds on each of these integral terms. 1~2 1~3 −2 1~2 φ ∞ ǫ ν t ǫ Before we proceed, let us remark that since the Lx ≲ ≲ by (8.2), we can replace any factors of e(q+1)φ by 1 (and vice versa)Y Y without changing⟨ ⟩ the bounds.

Lemma 9.5. For α + β + ω N, we have S S ′ S S ′ S S ≤ 2(SβS+Sβ′S)~3 Q,ℓ−2Sα S−2Sβ S,1 α′′ ω′′ (ϑ) (ϑ) 2 ν α+α′,β+β′,ω min f t Ẽ , f t D̃ ∂x Y ρ~0 Lx R ≲ N N Q = {Y ( )Y Y ( )Y } Sα′′S+Sω′′S N Y Y ≤ ǫ1~2 t −2 f 2 + Ẽ(ϑ) ⟨ ⟩ Y Y N when either (1) α′ 1 and β′ 0, or (2) α′ 0, β′ 2. S S ≤ = = S S ≤ Proof. Let us consider only the case α′ 1 and β′ 0. The other case is similar after noting D(ϑ) S S ≤ = that the N norm by deﬁnition controls the corresponding term with more ∂v derivative and that ̃ρ is independent of v. We compute α′ α β ω α′ α ω′ β+ω′′ ∂x ∂x ∂v Y Ej∂vj √µ Q ∂x ∂x Y Ej∂vj ∂v √µ. [ ] = ω′+ω′′ ω = + ′′ ∂ ∂β ω µ v Notice that vj v √ decays rapidly in . THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 58

~~E(ϑ) Bounding with the N norm. Using the fact that E gains one derivative over ρ~0, the above computations and̃ the Cauchy–Schwarz inequality implies =~~

2SβS~3 Q,ℓ−2Sα′S,1 2SβS~3 2(q+1)φ 2 α+α′ β ω α+α′ β ω ν + ′ ν e w ∂ ∂ Y f∂ ∂ Y Ej∂v µ dv dx Rα α ,β,ω U 3 3 x v x v j √ ≤ U T ×R [ ] U 2SβS~3 α+α′ β ω α′′+α′ ω′′ ν ∂ ∂ Y f 2 ∂ Y E 2 ≲ x v Lx,v Q Q x j Lx Y Y Sα′S 1 Sα′′S+Sω′′S N Y Y ≤ ≤ α′′ ω′′ (ϑ) 2 f Ẽ ∂x Y ρ~0 Lx . ≲ N Q = Y Y Sα′′S+Sω′′S N Y Y ≤ D(ϑ) ′ D(ϑ) Bounding with the N norm. When α 1, we can thus use the N norm to control α+α′ β ω ̃ S S = ̃ ∂x ∂v Y f so that

2SβS~3 Q,ℓ−2,1 2SβS~3 2(q+1)φ 2 α+α′ β ω α+α′ β ω ν + ′ ν e w ∂ ∂ Y f∂ ∂ Y Ej∂v µ dv dx Rα α ,β,ω 3 3 x v x v j √ ≤ U UT ×R [ ] U 2SβS~3 α+α′ β ω α′′+α′ ω′′ ν ∂ ∂ Y f 2 ∂ Y E 2 ≲ x v Lx,v Q Q x j Lx Y Y Sα′S 1 Sα′′S+Sω′′S N Y Y ≤ ≤ α′′ ω′′ (ϑ) 2 f D̃ ∂x Y ρ~0 Lx , ≲ N Q = Y Y Sα′′S+Sω′′S N Y Y ≤ D(ϑ) where we have used the definition of N to bound f, and used (9.12) to bound E. ′ ̃ D(ϑ) When α 0, note that directly bound the term with the N norm would cause a loss of ν−1~3. Instead,= we integrate by parts in x: recalling E ̃φ, we get = −∇x 2SβS~3 Q,ℓ,1 2SβS~3 2(q+1)φ 2 α β ω α β ω ν ν e w ∂x ∂ ∂ Y f∂ ∂ Y φ∂v µ dv dx Rα,β,ω U 3 3 j x v x v j √ = T ×R [ ] 2SβS~3 2(q+1)φ 2 α β ω α β ω 2 q 1 ν ∂x φe w ∂ ∂ Y f∂ ∂ Y φ∂v µ dv dx. + + U 3 3 j x v x v j √ ( ) T ×R [ ] − v 2 ∞ S S Using (9.12), the bootstrap assumption (8.2) on ∂xφ Lx and the fact that µ e decays − = rapidly, the second integral is clearly bounded byYǫν1~3Y t 2 f t 2 . As for the first integral N term, we use the rapid decay in v , Hölder’s inequality⟨ ⟩ andY (9.12)YE) to bound ⟨ ⟩ 2SβS~3 2(q+1)φ 2 α β ω α β ω ν e w ∂x ∂ ∂ Y f∂ ∂ Y φ∂v µ dv dx U 3 3 j x v x v j √ U T ×R [ ] U 2SβS~3 α β ω α ω′ ν ∂ ∂ ∂ Y f 2 ∂ Y φ 2 ≲ xj x v Lx,v Q x Lx Y Y Sω′S SωS Y Y ≤ SβS~3 α′′ ω′′ ν f (ϑ) ∂ Y ρ 2 , ≲ D̃ Q x Lx Y Y N Sα′′S+Sω′S N Y Y ≤ giving the lemma.

Lemma 9.6. For α + β + ω N, we have S′ S ′S S S S ≤ 2(SβS+Sβ′S)~3 Q,ℓ−2Sα S−2Sβ S,2 α′′ ω′′ (ϑ) (ϑ) 2 ν α+α′,β+β′,ω ǫ min f t Ẽ , f t D̃ ∂x Y ρ~0 Lx R ≲ N N Q = {Y ( )Y Y ( )Y } Sα′′S+Sω′′S N Y Y ≤ (9.15) ǫ1~2 t −2 f 2 + Ẽ(ϑ) ⟨ ⟩ Y Y N when either (1) α′ 1 and β′ 0, or (2) α′ 0, β′ 2. ≤ = = ≤ S S S S ′ Proof. Take α + β + ω N. To avoid notational confusion, we consider only the case α 0 and β′ 0; theS S otherS S S casesS ≤ are almost identical upon using the higher derivative control= of = α,β,ω and α,β,ω and the fact that E gains one ∂x derivative over ρ~0 (see (9.12)). Ẽ D̃ = THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 59

Let us start by estimating the integral involving E ⋅ ∇v. By definition, we compute α β ω α′′′ ω′′′ α′′ β ω′′ Ej∂vj , ∂x ∂v Y f ≲ Q ∂x Y Ej ∂vj ∂x ∂v Y f . (9.16) S[ ] S Sα′′′S+Sα′′S SαS S( ) S Sω′′′S+Sω′′S=SωS Sα′′′S+Sω′′′=S 1 ≥ ′′′ ′′′ ∞ Consider first the case when α + ω 4, for which the L bounds on the electric field in the bootstrap assumption (8.2S ) canS S beS used.≤ We have

2SβS~3 2(q+1)φ 2 α β ω α′′′ ω′′′ α′′ β ω′′ ν e w ∂ ∂ Y f ∂ Y Ej ∂v ∂ ∂ Y f dv dx U 3 3 x v x j x v U T ×R ( ) U 2SβS~3 α β ω α′′′ ω′′′ α′′ β ω′′ ν ∂ ∂ Y f 2 ∂ Y E ∞ ∂ ∂ ∂ Y f 2 (9.17) ≲ x v Lx,v(ℓα,β,ω,ϑ) x Lx vj x v Lx,v(ℓα,β,ω,ϑ) 1~2 Y1~3 −2 Y−1~3 2 Y 1~2 −2 Y 2 Y Y ≲ ǫ ν t ν f Ẽ(ϑ) ≲ ǫ t f Ẽ(ϑ) , ( ⟨ ⟩ )( Y Y N ) ⟨ ⟩ Y Y N ′′ ′′ upon recalling α + ω + 1 α + ω . S S S S ≤ S S S S′′′ ′′′ Next, we consider the case when α + ω 5, which we will bound by the ﬁrst term in S′′ S S′′ S ≥ ′′ ′′ (9.15). In this case, we must have α + ω α + ω − 5 (and in particular α + β + ω S S S S ≤ S S S S S S S S S S ≤ Nmax − 5), and so upon using Sobolev embedding in x and the bootstrap assumption (8.1), we have SβS~3 α′′ β ω′′ ∞ 2 ν ∂vj ∂x ∂v Y f Lx Lv(ℓα,β,ω,ϑ) Y 3 ′′′ Y′′ ′′ −1 3 1 2 SβS~ α α β ω 2 2 ~ ~ ν ∂x ∂vj ∂x ∂v Y f L L (ℓ ,ϑ) ν f ̃(ϑ) ǫ . ≲ Q x v α,β,ω ≲ E 2 ≲ Sα′′′S 2 Y Y Y Y Nmax− ≤ Therefore, using H¨older’s inequality and Lemma 9.3, we bound

2SβS~3 2(q+1)φ 2 α β ω α′′′ ω′′′ α′′ β ω′′ ν e w ∂x ∂v Y f ∂x Y Ej ∂vj ∂x ∂v Y f dv dx U UT3×R3 ( ) U 2SβS~3 α β ω α′′′ ω′′′ α′′ β ω′′ ν ∂ ∂ Y f 2 ∂ Y E 2 ∂ ∂ ∂ Y f ∞ 2 ≲ x v Lx,v(ℓα,β,ω,ϑ) x j Lx vj x v Lx Lv(ℓα,β,ω,ϑ) (9.18) Y α′′′ ω′′′Y 1~2 Y1~2 Y Y α′′′ ω′′′ Y (ϑ) 2 (ϑ) 2 f Ẽ ∂x Y Ej Lx ǫ ǫ f Ẽ ∂x Y ρ~0 Lx . ≲ N ≲ N Q = Y Y Y Y Y Y Sα′′′S SαS Y Y Sω′′′S≤SωS ≤ We also need a bound with E(ϑ) above replaced by D(ϑ). Noticing that a direct estimate ̃N ̃N with D(ϑ) causes a loss of ν−1~3, we integrate by parts in ∂ . (We remark that when α′ 0 ̃N vj = ′ D(ϑ) ~ or β 0, such an integration by parts is unnecessary, by deﬁnition of the N norm.) After integration=~ by parts, we argue as above with Sobolev embedding, noting̃ also that since ′′ ′′ α + ω α + ω − 5, we have additional weights in v . In other words, we bound S S S S ≤ S S S S ⟨ ⟩ 2SβS~3 2(q+1)φ 2 α β ω α′′′ ω′′′ α′′ β ω′′ ν e w ∂ ∂ Y f ∂ Y Ej ∂v ∂ ∂ Y f dv dx U 3 3 x v x j x v U T ×R ( ) U 2SβS~3 2(q+1)φ 2 α β ω α′′′ ω′′′ α′′ β ω′′ ν e w ∂v ∂ ∂ Y f ∂ Y Ej ∂ ∂ Y f dv dx ≲ U 3 3 j x v x x v U T ×R ( ) U 2SβS~3 2(q+1)φ 2 α β ω α′′′ ω′′′ α′′ β ω′′ (9.19) ν e ∂v w ∂ ∂ Y f ∂ Y Ej ∂ ∂ Y f dv dx + U 3 3 j x v x x v U T ×R ( ) ( ) U 1~2 α′′′ ω′′′ (ϑ) 2 ǫ f D̃ ∂x Y ρ~0 Lx . ≲ N Q = Y Y Sα′′′S SαS Y Y Sω′′′S≤SωS ≤ Combining (9.17), (9.18) and (9.19), we have thus proven the desired estimate corresponding to the commutator term in (9.16). THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 60

Similarly, we now treat the integral involving E ⋅ v. We compute α β ω α′′′ ω′′′ α′′ β′ ω′′ α′′′ ω′′′ α′′ β ω′′ Ejvj, ∂x ∂v Y f ≲ Q ∂x Y Ej ∂x ∂v Y f + Q vj ∂x Y Ej ∂x ∂v Y f . S[ ] SαS ′′′S+Sα′′S(S SαS ) SαS′′′S+Sα′′SS Sα(S ) S Sω′′′S+Sω′′S=SωS Sω′′′S+Sω′′S=SωS Sβ′S SβS−=1 Sα′′′S+Sω′′′=S 1 Sα′′′S+=Sω′′′S 1 ≥ ≥ α β ω The first term is similar to previous terms in Ej∂vj , ∂x ∂v Y f, and is in fact better because it has two fewer ∂v derivatives. The second[ term experiences] a linear growth of v . ′′ ′′ S−4S This growth however causes no loss of v-weight, since α + ω α + ω − 1, gaining v in the v-weight. The lemma follows. S S S S ≤ S S S S ⟨ ⟩ Lemma 9.7. For α β ω N, we have + + ≤ 2(SβS+Sβ′S)~3 Q,ℓS−2Sα′S−S2SβS ′S,3S S 1~3 2 1~3 ν + ′ + ′ ν f (ϑ) f (ϑ) ν f Ẽ f (ϑ) f D̃ Rα α ,β β ,ω ≲ Ẽ D̃ + N D̃ Nmax−2 Y Y Nmax−2 Y Y N Y Y Y Y N Y Y when either (1) α′ 1 and β′ 0, or (2) α′ 0, β′ 2. S S ≤ = = S S ≤ Proof. Using Lemma 4.9 with ℓ ℓ , we bound = α,β,ω e2(q+1)φw2∂α∂βY ωf∂α∂βY ωΓ f, f dv dx U 3 3 x v x v U T ×R ( ) U α β ω α′ β′ ω′ α′′ β′′ ω′′ ∂ ∂ Y f ∂ ∂ Y f 2 ∂ ∂ Y f ≲ Q x v ∆x,v(ℓα,β,ω,ϑ) x v Lx,v x v ∆x,v(ℓα,β,ω,ϑ) ′ + ′′ Sα S Sα S SαSY Y Y α′ β′ ω′Y Y α′′ β′′ ω′′Y Sβ′S+Sβ′′S≤SβS ∂ ∂ Y f ∂ ∂ Y f 2 + x v ∆x,v x v Lv(ℓα,β,ω,ϑ) Sω′S+Sω′′S≤SωS ′ Y Y Y Y ≤ α′ β ω′ noting the norms involving ∂x ∂v Y f can have any weight in v. Therefore, by definition, we have νν2SβS~3 Q,ℓ,3 νν2SβS~3 e2(q+1)φw2∂α∂βY ωf∂α∂βY ωΓ f, f dv dx Rα,β,ω U 3 3 x v x v = U T ×R ( ) U 1~3 2 1~3 ν f (ϑ) f (ϑ) ν f Ẽ f (ϑ) f D̃ . ≲ Ẽ D̃ + N D̃ Nmax−2 Y Y Nmax−2 Y Y N Y Y Y Y N Y Y Q,ℓ−2Sα′S−2Sβ′S,3 By definition of the energy and dissipation norms, the same bounds hold for Rα+α′,β+β′,ω , upon assigning the respective v-weight and ν-scaling. Q,ℓ Q,ℓ−2 9.3. Estimates on Zα,β,ω. Finally, in this section, we give bounds on Zα,β,ω , defined as in Lemma 5.7, that appear in (5.22), noting the v-weight function is indexed at ℓα,β,ω − 2. Q,ℓ−2 Recalling the definition of Zα,β,ω from Lemma 5.7, we write Q,ℓ−2 Q,ℓ−2,1 Q,ℓ−2,2 Zα,β,ω = Zα,β,ω +Zα,β,ω Q,ℓ−2,1 where Zα,β,ω is defined by Q,ℓ−2,1 2(q+1)φ 2 −4 α β ω α β ω 2 e w v ∂x ∂ ∂ Y f ∂v ∂ ∂ Y Ej∂v µ dv dx Zα,β,ω U 3 3 j x v j x v j √ = T ×R ⟨ ⟩ ( ) [ ] 2(q+1)φ 2 −4 α β ω α β ω e w v ∂x ∂ ∂ Y f E v, ∂v ∂ ∂ Y f dv dx + U 3 3 j x v ⋅ ∇ j x v T ×R ⟨ ⟩ ( )[ ] 2(q+1)φ 2 −4 α β ω α β ω e w v ∂x ∂ ∂ Y f E v, ∂v ∂ ∂ Y f dv dx − U 3 3 j x v ⋅ j x v T ×R ⟨ ⟩ ( )[ ] 2(q+1)φ 2 −4 α β ω α β ω ν e w v ∂x ∂ ∂ Y f ∂v ∂ ∂ Y Γ f, f dv dx + U 3 3 j x v j x v T ×R ⟨ ⟩ ( ) ( ) Q,ℓ−2,2 and Zα,β,ω is defined in a symmetric way, switching ∂xj and ∂vj in each of the integrals above. Now observe that all the integral terms are estimated similarly, if not identically, as THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 61

already done for the similar integral terms in Lemma 9.5 (the ﬁrst term above), Lemma 9.6 (the second and third), and Lemma 9.7 (the last), respectively. This completes the proof of Q,ℓ−2 the claimed bounds on Zα,β,ω , and hence the proof of Theorem 9.1.

10. Global existence of solutions Theorem 10.1. Consider data as in Theorem 3.1. Then the unique smooth solution arising from the given initial data is global in time. Moreover, the estimates (8.3) and (9.1) hold for TB replaced by ∞. Proof. Using a standard local existence result, we can carry out a bootstrap argument. Suppose there exists TB 0 such that the solution f to (2.2a)–(2.2b) remains smooth in T3 R3 > 0, TB × × and satisﬁes the bootstrap assumptions (8.1) and (8.2). It suﬃces to show that[ in) fact (8.1) and (8.2) hold with ǫ replaced by Cǫ2, for some constant C 0 independent of ǫ and ν. > ● The improvement for (8.1) follows from Theorem 9.1. ● The improvement for (8.2) is an immediate consequence of Lemmas 9.3, 9.4 and the bounds obtained in Theorem 9.1. This closes the bootstrap argument and implies that the solution is global. Finally, since we have closed the bootstrap, the bounds (8.3) and (9.1) follow from Theorems 8.1 and 9.1.

11. Nonlinear density estimates: stretched exponential decay In the next two section, we will turn to the proof of the stretched exponential decay. Sim- ilarly as for the boundedness of the solution, the proof is split into two parts: the nonlinear density estimates are treated in this section, and the nonlinear energy decay estimates will be treated in Section 12. We ﬁrst point out a few key points for the density estimates, especially in contrast to the bounds proven in Section 8: (1) In order to prove the stretched exponential decay, we need to prove a density estimate also with a stretched exponential decay factor; see e t factors in Theorem 11.1. ( ) (2) We prove the estimate (11.2), which is of the same size for all p 2, ∞ . This is in ∈ [ ] 2 contrast with the boundedness estimates for ρ~0 in Section 8, where the Lt estimate has a weaker ν power (see (8.3)). = The main diﬀerence in the argument comes from the term IIk, where we crucially rely on the fact that we are at a lower order, and that both the boundedness of the higher order density estimates and the higher order energy estimates were already established in the previous sections (see (8.3) and (9.1)). (1) (2) (2) (3) We need a decomposition of ρ~0 ρ~0 + ρ~0 : the piece ρ~0 is better in terms of the = = = = 2 = 2 2~3 size, and (its ∂x derivatives) obeys an O ǫ ν instead of an O ǫ ν bound; the (1) 2 2~3 ( ) ( ) piece ρ~0 only obeys an O ǫ ν upper bound, but importantly one can also take 1 = ( ) Lt norm with the same upper bound. This decomposition is important for closing the energy decay estimates in Sec- tion 12. THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 62

~~1 3 1 3 2 3 We put forth another bootstrap argument. For e t eδ(ν ~ t) ~ , eδ(νt) ~ , introduce the following bootstrap assumption: ( ) ∈ { }~~

TB 2 1~3 2 2~3 t f t 2 ′ ν τ f τ 2 ′ τ ǫν , sup e Ẽ( ) + e D̃( ) d (11.1) 0 t T 1,1,0,0 S0 1,1,0,0 ≤ ≤ < B ( )Y ( )Y ( )Y ( )Y 2 q′SvS 2 where Ẽ(2)′ and D̃(2)′ are deﬁned as in (5.16) but with exponential weights e ⋅ 1,1,0,0 ⋅ 1,1,0,0 Y Y 2 Y Y qSvS instead of e 2 (cf. (5.19)).

~~The main result of this section is the following.~~

Theorem 11.1. Consider data as in Theorem 3.1. Suppose there exists TB 0 such that > 3 3 the solution f to (2.2a)–(2.2b) satisﬁes the bootstrap assumption (11.1) in 0, TB T R . 1 3 1 3 2 3 × × Then, for e t eδ(ν ~ t) ~ , eδ(νt) ~ , the following hold: [ ) ( ) ∈ { } For any p 2, , ρ~0 obeys the bound ● ∈ ∞ = [ ] 1~2 α 2 2 2~3 e t ∂x ρ~0 p 0 ; 2 ǫ ν . Q = Lt ([ ,TB ) Lx) ≲ (11.2) SαS 1 Y ( ) Y ≤ (1) (2) ● ρ~0 t, x admits a decomposition ρ~0 ρ~0 + ρ~0 such that = ( ) = = = = 1~2 α (1) 2 2 2~3 e t ∂x ρ 1 2 ǫ ν , (11.3) Q ~0 Lt ([0,TB );Lx) ≲ SαS 1 Y ( ) = Y ≤ 1~2 α (2) 2 2 e t ∂x ρ 2 2 ǫ ν. (11.4) Q ~0 Lt ([0,TB );Lx) ≲ SαS 1 Y ( ) = Y ≤

As in Section 8, we split the density contribution into the terms Ik, IIk and IIIk as in (8.7). Define (1) I II (2) III Nk t ∶ k + k t , Nk t ∶ k t , (11.5) 1~3 1~3 ( ) 2~=3 ( )( ) ( ) = ( ) and, for e τ eδ(ν τ) , eδ(ντ) and j 1, 2 , define ∈ ∈ ( ) { } (j) { } (j) Mk t ∶ e τ Nk t . (11.6) ( ) = ( ) ( ) (j) (j) ik⋅x The density decomposition asserted in Theorem 11.1 is then defined as ρ~0 ∑k~0 ρk e , where = = = t (j) (j) (j) ρk t ∶ Nk t + Gk t − s Nk s ds. (11.7) ( ) = ( ) S0 ( ) ( ) Similarly as done before, we introduce t 2 ζ t τ k 2 ρ(1) 2 τ τ k 2 ρ(1) 2 τ 1~2 τ e ∶ sup Q e k + S Q e k d ( ) = τ [0,t] k~0 ( )S S S S ( ) 0 [k~0 ( )S S S S ( )] ∈ = = t (11.8) ν−1~3 τ k 2 ρ(2) 2 τ ν−1~3 τ k 2 ρ(2) 2 τ τ. + sup Q e k + S Q e k d τ [0,t] k~0 ( )S S S S ( ) 0 k~0 ( )S S S S ( ) ∈ = = The following are the main estimates that will be used to prove Theorem 11.1:

1 3 1 3 2 3 Proposition 11.2. For e τ eδ(ν ~ τ) ~ , eδ(ντ) ~ , the following hold for all t 0, T : ∈ ∈ B ( ) { t } 2 [ ) k 2 (1) 2 τ k 2 (1) 2 τ 1~2 τ ǫ2ν2~3 ǫζ t sup Q Mk + S Q Mk d ≲ + e (11.9) 0 τ t k~0 S S S S ( ) 0 [k~0 S S S S ( )] ( ) ≤ ≤ = = THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 63 and t k 2 (2) 2 τ k 2 (2) 2 τ τ ǫ2ν. sup Q M + Q S Mk d ≲ (11.10) 0 τ t k~0 S S S S ( ) k~0 0 S S S S ( ) ≤ ≤ = = Proof of Theorem 11.1 assuming Proposition 11.2. First note that the bounds (11.9) and (j) 2 (j) 2 (11.10) for Mk τ imply the same estimates for e τ ρk τ using (11.7). For instance, S ( )S (j) (j) (j) ( )S ( )S using the deﬁnitions of Nk , Mk and ρk in (11.5)–(11.7), t t k 2 ρ(j) t 2 k 2 (j) t 2 t k 2 G t τ (j) τ τ 2. Q e k ≲ Q Mk + Q e S k − Nk d (11.11) k~0 ( )S S S ( )S k~0 S S S ( )S k~0 ( )S S [ 0 S ( )SS S( ) ] = = = Notice that by (7.3), choosing δ small depending on δ′′ so that −δ′′(ν1~3(t−τ))1~3 −δ′′(ν(t−τ))2~3 e t min e , e ≲ e τ , (11.12) ( ) { } ( ) we can bound the last term in (11.11) by t t t k 2 G t τ (j) τ τ 2 k −1 k t τ −2 k (j) τ τ 2 Q e S k − Nk d ≲ Q S − Mk d k~0 ( )S S [ 0 S ( )SS S( ) ] k~0[ 0 S S ⟨ ( )⟩ S SS S( ) ] = = t 2 k −2 k t τ −2 τ k′ 2 (j) 2 τ ′ ≲ Q S ⟨ ( − )⟩ d sup sup Mk′ k~0 S S 0 k′ ~0 τ ′ [0,t] S S S S ( ) = = ∈ −4 ′ 2 (j) 2 ′ ′ 2 (j) 2 ′ ≲ Q k sup sup k Mk′ τ ≲ sup sup k Mk′ τ . k 0 k′ ~0 τ ′ [0,t] k′ ~0 τ ′ [0,t] [ ~ S S ] = ∈ S S S S ( ) = ∈ S S S S ( ) = (11.13) Combining (11.11) and (11.13), and then using (11.9) and (11.10), we obtain, for j 1, 2 , ∈ { } 2 (1) 2 2 2~3 2 (2) 2 2 sup Q e τ k ρk τ ≲ ǫ ν + ǫζe t , sup Q e τ k ρk τ ≲ ǫ ν. (11.14) τ [0,t] k~0 ( )S S S ( )S ( ) τ [0,t] k~0 ( )S S S ( )S ∈ = ∈ = 1 2 (1) 2 2 (2) We can control the Lt ℓk norm of k ρk and the Lt ℓk norm of k ρk in a similar manner. Indeed, using (11.5)–(11.7) and thenS SS the boundsS for Gk in (7.3),S SS S t 2 τ k 2 ρ(1) τ 2 1~2 τ S Q e k d 0 [k~0 ( )S S S ( )S ] = t 2 t τ 2 k 2 (1) τ 2 1~2 τ τ k 2 G τ s (1) s s 2 1~2 τ ≲ S Q Mk d + S Q e S k − Nk d d 0 [k~0 S S S ( )S ] 0 [k~0 ( )S S ( 0 S ( )SS S( ) ) ] = = t 2 t τ 2 k 2 (1) τ 2 1~2 τ k −1 k τ s −2 k (j) s s 2 1~2 τ . ≲ S Q Mk d + S Q S − Mk d d 0 [k~0 S S S ( )S ] 0 [k~0[ 0 S S ⟨ ( )⟩ S SS S( ) ] ] = = 1 2 To control the ﬁnal term, we use Minkowski’s inequality to exchange the order of Ls and ℓk, 1 1 and then use Fubini’s theorem to exchange the order of Lt and Ls so as to obtain t τ 2 k −1 k τ s −2 k (1) s s 2 1~2 τ S Q S − Mk d d 0 [k~0[ 0 S S ⟨ ( )⟩ S SS S( ) ] ] = t τ 2 k −2 k τ s −4 k 2 (1) 2 s 1~2 s τ ≲ S S Q − Mk d d 0 0 [k~0 S S ⟨ ( )⟩ S S S S ( )] = t τ 2 τ s −2 k −2 k 2 (1) 2 s 1~2 s τ ≲ S S ⟨ − ⟩ Q Mk d d 0 0 [k~0 S S S S S S ( )] = t 2 t 2 k −2 k 2 (1) 2 s 1~2 s k 2 (1) 2 s 1~2 s . ≲ S Q Mk d ≲ S Q Mk d 0 [k~0 S S S S S S ( )] 0 [k~0 S S S S ( )] = = THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 64

Thus, using also (11.9), we have obtained t 2 τ k 2 ρ(1) τ 2 1~2 τ ǫ2ν2~3 ǫζ t . S Q e k d ≲ + e (11.15) 0 [k~0 ( )S S S ( )S ] ( ) = 2 2 1~2 (2) A similar but slightly simpler argument also allows us to bound the Lt ℓk norm of e τ k ρk using (11.10) so that we have ( )S SS S t τ k 2 ρ(2) τ 2 τ ǫ2ν. S Q e k d ≲ (11.16) 0 k~0 ( )S S S ( )S = Recalling the deﬁnition of ζe in (11.8) and using (11.14), (11.15) and (11.16), we obtain 2 2~3 ζe t ≲ ǫ ν + ǫζe t , ( ) ( ) which implies 2 2~3 ζe t ≲ ǫ ν . (11.17) ( ) At this point, using (11.17), we easily conclude Theorem 11.1: ● Plugging (11.17) back into (11.15) and (11.16) yields the estimates (11.3) and (11.4). Plugging (11.17) into (11.14) yields the p case of (11.2). ● = ∞ ● Finally, the p 2, ∞ cases of (11.2) can be obtained by interpolating between the p case and∈ the[ bounds) (11.3) and (11.4). = ∞ The remainder of this section will be devoted to the proof of Proposition 11.2.

11.1. Initial data contribution. By (8.8) with N1 N2 1, we have = = 2 I 2 2 2~3 −4 Q e t k k t ≲ ǫ ν t , k~0 ( )S S S ( )S ⟨ ⟩ = which obeys the bounds required in (11.9). 11.2. Nonlinear interaction I. Recall the decomposition in (8.9); as in Section 8.2, we only consider the term IIk,2, as the term IIk,1 is easier. II ∞ 1 For the term k,2, we prove below the Lt and Lt according to (11.9). ∞ ′ ′ Proving the L bound. Arguing as in (8.15), with N1, N2 1, 0 and N1, N2 5, 2 , and taking into the extra stretched exponential decay( given by) = (6.14( )) to obtain( ) = ( ) t II −1 −5 −2 k k,2 t ≲ Q S l k k − l kt − lτ S SS ( )S l~0 0 S S S S⟨ ⟩ ⟨ ⟩ = (11.18) −δ′(ν1~3(t−τ))1~3 −δ′(ν(t−τ))2~3 ˆ min e , e ρˆl τ fk−l τ G̃′ dτ × 2 { }S ( )SY ( )Y Nmax− where − 1 2 ˆ (SβS 1)~3 10 α β ω 4 4 q0SvS α β fk τ G̃′ ν v ∂x ∂v Y f τ L2 v e ∂x ∂v f τ L2 , (11.19) N ∶ Q k v + k v Y ( )Y Sα=S+SωS N Y⟨ ⟩ ( )̂( )Y Y⟨ ⟩ ( )̂( )Y 1 SβS ≤2 ≤ ≤ 1 2 noting the last additional term (compared with (8.13)) with the exponential weight e 4 q0SvS slower than what is encoded in the energy and dissipation norms (5.16). See also Remark 6.3. In particular, we note that ˆ 2 −2~3 2 −2~3 2 fk t ̃′ ν f t (2)′ ν f t (2) ǫ (11.20) Q G 2 ≲ Ẽ ≲ Ẽ ≲ k Y ( )Y Nmax− Y ( )Y Nmax−2 Y ( )Y Nmax−2 THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 65

and 2 ˆ 2 −2~3 2 e t k fk t ̃′ ν e t f t (2)′ ǫ, (11.21) Q G0 ≲ Ẽ ≲ k~0 ( )S S Y ( )Y ( )Y ( )Y 1,1,0,0 = where we used, respectively, the energy bounds (9.1) and the bootstrap assumption (11.1). Next, use (11.12) with δ′′ replaced by δ′, and notice that k l −1 k l −1 1. We obtain S SS S ⟨ − ⟩ ≲ t 1~2 −2 1~2 −4 ˆ e t k IIk,2 t kt lτ e τ ρˆl τ k l fk−l τ G̃ dτ. (11.22) ≲ Q S − − Nmax−2 ( )S SS ( )S l~0 0 ⟨ ⟩ ( )S ( )S⟨ ⟩ Y ( )Y = Using (11.20), the Cauchy–Schwarz inequality in τ, and the Young’s convolution inequality for the sums, we obtain 2 II 2 Q e t k k,2 t k~0 ( )S S S ( )S = t t 2 2 1~2 −8 −4 ˆ 2 1~2 e τ ρˆl τ dτ k l kt lτ fk−l τ G̃ dτ ≲ Q Q S S − − Nmax−2 k~0 l~0( 0 ( )S ( )S ) ( 0 ⟨ ⟩ ⟨ ⟩ Y ( )Y ) = = t t 2 (11.23) 2 1~2 −8 −4 1~2 ≲ ǫ Q Q S e τ ρˆl τ dτ S k − l kt − lτ dτ k~0 l~0( 0 ( )S ( )S ) ( 0 ⟨ ⟩ ⟨ ⟩ ) = = t t 2 −4 2 2 ≲ ǫ Q S e τ ρˆl τ dτ Q k ≲ ǫ Q S e τ ρˆl τ dτ . (l~0 0 ( )S ( )S )[ k ⟨ ⟩ ] (l~0 0 ( )S ( )S ) = = It remains to check that t 2 2 Q S e τ l ρˆl τ dτ ≲ ζe t . (11.24) l~0 0 ( )S S S ( )S ( ) = t 2 (2) 2 Indeed, the bound for l~0 0 e τ l ρˆl τ dτ is immediate from the deﬁnition (11.8), ∑ = ∫ t 2 (1() )S S2 S ( )S 1 while that for ∑l~0 0 e τ l ρˆl τ dτ follows from interpolating between the Lt and the ∞ = ∫ ( )S S S ( )S Lt bounds in (11.8).

~~1 1 Proving the L bound. To obtain the Lt bound, we need to estimate~~

t s 2 1~2 1~2 −1 e t k l ρˆl τ Sk s τ vf k−l τ √µ dv dτ ds. (11.25) S Q Q S SR3 − ∇Ä 0 k~0 l~0 0 ( )S SS S S S( )U ( )[ ( )] U = = To estimate (11.25), we split the τ-integral into lτ ks 2 and lτ ks 2. In the latter integral, we further split the sums into the l k andS theS < S l S~k parts.S S ≥ S S~ First, consider the case lτ ks 2. We estimate=~ the integrand= as in (11.22), i.e. S S < S S~ 1~2 −1 e t k l ρˆl τ Sk s τ vf − τ √µ dv S 3 − ∇Ä k l ( )S SS S S S( )U R ( )[ ( )] U (11.26) −2 1~2 −4 ˆ ks lτ e τ ρˆl τ k l fk−l τ G̃′ ≲ − − 2 ⟨ ⟩ ( )S ( )S⟨ ⟩ Y ( )Y Nmax− ˆ −2 with fk−l τ G̃′ deﬁned as in (11.19). Since we imposed lτ ks 2, we have ks lτ 2 − ≲ Y ( )Y Nmax− S S < S S~ ⟨ ⟩ ks −2 ks −5~4 lτ −3~4 s −5~4 τ −3~4. It thus suﬃces to bound ⟨ ⟩ ≲ ⟨ ⟩ ⟨ ⟩ ≲ ⟨ ⟩ ⟨ ⟩ t s 2 −3~4 −4 2 1~2 −5~4 τ e τ ρˆl τ k l fˆk−l τ G̃′ dτ s ds . S Q Q S − Nmax−2 0 [k~0(l~0 0 ⟨ ⟩ ( )S ( )S⟨ ⟩ Y ( )Y ) ] ⟨ ⟩ = = THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 66

1~2 −5~4 Note that fˆk−l τ G̃′ ǫ by (11.20). Then, integrating the s factor, and using 2 ≲ Y ( )Y Nmax− ⟨ ⟩ the Cauchy–Schwarz inequality in τ and the Young’s convolution inequality for the sums,

t s 2 −3~4 1~2 −4 2 1~2 −5~4 τ e τ ρˆl τ k l fˆk−l τ G̃′ dτ s ds S Q Q S − Nmax−2 0 [k~0(l~0 0 ⟨ ⟩ ( )S ( )S⟨ ⟩ Y ( )Y ) ] ⟨ ⟩ = = t −3~4 1~2 −4 2 ≲ ǫ Q Q S τ e τ ρˆl τ k − l dτ k~0(l~0 0 ⟨ ⟩ ( )S ( )S⟨ ⟩ ) = = t t 2 (11.27) 2 1~2 −3~2 −8 1~2 ≲ ǫ Q Q S e τ ρˆl τ dτ S τ k − l dτ k~0 l~0( 0 ( )S ( )S ) ( 0 ⟨ ⟩ ⟨ ⟩ ) = = t 2 t 2 −4 2 ≲ ǫ Q S e τ ρˆl τ dτ Q k ≲ ǫ Q S e τ ρˆl τ dτ ≲ ǫζe t , l~0 0 ( )S ( )S k ⟨ ⟩ l~0 0 ( )S ( )S ( ) = = where at the end we used (11.24). Next, we turn to the case lτ ks 2 and k l. Here, we bound the integrand differently: S S ≥ S S~ =~ ′ ′ starting with (8.15) but taking instead N1, N2 4, 3 and N , N 1, 0 , we have ( ) = ( ) ( 1 2) = ( ) 1~2 −1 e t k l ρˆl τ Sk s τ vf − τ √µ dv 3 − ∇Ä k l ( )S SS S S S( )U SR ( )[ ( )] U (11.28) −5 −1 −3 1~2 4 3 l k k l lτ e τ l lτ ρˆl τ k l fˆk−l τ G̃′ , ≲ S S S S⟨ − ⟩ ⟨ ⟩ ( )(S S ⟨ ⟩ S ( )S)(S − SY ( )Y 0 ) −1 −1 where fˆk−l τ G̃′ is defined as in (11.19) with N 0. Observe that k l k l 1, and 0 − ≲ Y ( )Y −3 = −3~2 −3~2 S SS−3S~2 ⟨ −3~2⟩ that since lτ ks 2, it also holds that lτ ≲ ks lτ ≲ s τ . Hence, it suffices to boundS S ≥ S theS~ following term: ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩

t s 2 −4 −3~2 −3~2 1~2 4 3 2 1~2 l s τ e τ l lτ ρˆl τ k l fˆk−l τ G̃′ dτ ds S Q Q S − 0 0 [k~0(l~0,k 0 S S ⟨ ⟩ ⟨ ⟩ ( )(S S ⟨ ⟩ S ( )S)S SY ( )Y ) ] = = t t −4 −3~2 1~2 4 3 ˆ 2 −3~2 2 l τ e τ l lτ ρˆl τ k l fk−l τ G̃′ dτ s ds ≲ Q Q S − 0 S k~0(l~0,k 0 S S ⟨ ⟩ ( )(S S ⟨ ⟩ S ( )S)S SY ( )Y ) ( 0 ⟨ ⟩ ) = = t t 2 −8 −3~2 8 6 2 1~2 −3~2 2 ˆ 2 1~2 l τ l lτ ρˆl τ dτ τ e τ k l fk−l τ ̃′ dτ ≲ Q Q S S − G0 k~0 l~0,k( 0 S S ⟨ ⟩ (S S ⟨ ⟩ S ( )S ) ) ( 0 ⟨ ⟩ ( )S S Y ( )Y ) = = t 2 −4 ′ 4 ′ 3 ′ −3~2 2 ˆ 2 1~2 l sup sup l l τ ρˆl′ τ τ e τ k l fk−l τ G̃′ dτ ≲ Q Q S − 0 k~0 l~0,k S S ( l′ ~0 τ ′ [0,t] S S ⟨ ⟩ S ( )S)( 0 ⟨ ⟩ ( )S S Y ( )Y ) = = = ∈ 2 t 1~3 −4 −3~2 2 ˆ 2 ǫν l τ e τ k fk τ ̃′ dτ ≲ Q Q S G0 l~0 S S k~0 0 ⟨ ⟩ ( )S S Y ( )Y = = t 1~3 ′ 2 ˆ ′ 2 −3~2 ǫν sup e τ k fk τ G̃′ τ dτ ≲ Q 0 S (τ ′ [0,t] k~0 ( )S S Y ( )Y ) 0 ⟨ ⟩ ∈ = 1~3 ′ 2 ˆ ′ 2 ǫν sup e τ k fk τ ′ ǫ, ≲ Q G̃0 ≲ τ ′ [0,t] k~0 ( )S S Y ( )Y ∈ = where we used the Cauchy–Schwarz and the Young’s convolution inequalities, respectively, for ′ 4 ′ 3 ′ 1~3 the τ-integral and for the sums, as well as bounded supl′ ~0 supτ ′ [0,t] l l τ ρˆl′ τ ≲ ǫν = ∈ S S ⟨ ⟩ S ( )S using (8.3). Finally, we used (11.21) in the very last inequality.

The case where lτ ks 2 with k l has to be treated diﬀerently, since in this case fˆk−l corresponds to theS zerothS ≥ S modeS~ and= does not experience enhanced dissipation. We bound THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 67

the integrand using (11.26). Considering only k l, it thus suffices to bound = t s 1~2 2 −1 −2 1~2 2 k k s τ e τ k ρˆk τ fˆ0 τ G̃′ dτ ds . S Q S − Nmax−2 0 k~0( 0 S S ⟨ ( )⟩ ( )S SS ( )SY ( )Y ) = We use Minkowski’s inequality so that the ℓ2 sum in k is taken first, and then use Fubini’s theorem to integrate out the s τ −2 factor. More precisely, ⟨ − ⟩ t s 1~2 2 −1 −2 1~2 2 k k s τ e τ k ρˆk τ fˆ0 τ G̃′ dτ ds S Q S − Nmax−2 0 k~0( 0 S S ⟨ ( )⟩ ( )S SS ( )SY ( )Y ) = t s 1~2 2 −2 −4 2 2 ˆ 2 k k s τ e τ k ρˆk τ f0 τ G̃′ dτ ds ≲ S S Q − 2 0 0 k~0 S S ⟨ ( )⟩ ( )S S S ( )S Y ( )Y Nmax− = (11.29) t 1~2 t 2 −2 2 2 ˆ 2 −2 k e τ k ρˆk τ f0 τ G̃′ s τ ds dτ ≲ S Q 2 S − 0 k~0 S S ( )S S S ( )S Y ( )Y Nmax− ( τ ⟨ ⟩ ) = t 2 2 2 1~2 e τ k ρˆk τ fˆ0 τ G̃′ dτ . ≲ S Q Nmax−2 0 [k~0 ( )S S S ( )S ] Y ( )Y = (1) (2) To proceed, we splitρ ˆk ρˆk + ρˆk as in (11.7), so that by using Hölder’s inequality, (11.20), and (11.8), we have = t 2 2 2 1~2 ˆ e τ k ρˆk τ f0 τ G̃′ dτ S Q Nmax−2 0 [k~0 ( )S S S ( )S ] Y ( )Y = t 2 2 2 (1) 2 1~2 ˆ ′ e τ k ρˆk τ dτ sup f0 τ G̃′ (11.30) ≲ S Q Nmax−2 0 [k~0 ( )S S S ( )S ] τ ′ [0,t] Y ( )Y = ∈ t t 2 (2) 2 ˆ 2 e τ k ρˆ τ dτ f0 τ G̃′ dτ ǫζe t . + S Q k S 2 ≲ 0 k~0 ( )S S S ( )S 0 Y ( )Y Nmax− ( ) = Combining all the above cases, we have thus proven the bound (11.9) for the term (11.25).

11.3. Nonlinear interaction II. Finally, we prove the bounds for IIIk t corresponding to those required in (11.10). ( ) We argue as in (8.36) with N1 1, N2 0, but also take into account the stretched exponential decay given by (6.13) to= obtain =

k Sk t τ Γ f, f k τ √µ dv S SU SR3 ( − )[( Æ( )) ( )] U 1 3 1 3 2 3 1 2 −δ′(ν ~ (t−τ)) ~ −δ′(ν(t−τ)) ~ SβS~3 2 q0SvS α β min e , e ν v e 4 ∂ ∂ Γ f, f τ 2 . ≲ Q x v k Lv { } SαS 1, SβS 1 Y⟨ ⟩ [ ( ( ))]̂( )Y = ≤ To control the Γ f, f term, we argue as in (8.38), (8.39), with the help of Lemma 4.10, ( ) except for noticing that, importantly, there is exactly one factor with a ∂x derivative. As in (8.38), (8.39), we still put a factor with at least one ∂v derivative in the D̃ norm, and another 2 factor in the norm. We then put the factor with the ∂x derivative in Lx, and the other Ẽ ∞ factor will be bounded in Lx together with Sobolev embedding. The factor with exactly one ′ ′ ∂ E(2) D(2) x derivative can then by put into either that ̃1,1,0,0 or the ̃1,1,0,0 norm. Hence, we have 1 2 2SβS~3 2 q0SvS α β 2 ν v e 4 ∂ ∂ Γ f, f τ 2 Q Q x v k Lv k~0 SαS 1, SβS 1 Y⟨ ⟩ [ ( ( ))]̂( )Y = = ≤ (11.31) −4~3 2 2 2 2 ν f τ 2 ′ f τ 2 ′ f τ 2 ′ f τ 2 ′ . ≲ Ẽ( ) D̃( ) + D̃( ) Ẽ( ) (Y ( )Y Nmax−2 Y ( )Y 1,1,0,0 Y ( )Y Nmax−2 Y ( )Y 1,1,0,0 ) THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 68

1 3 1 3 2 3 Therefore, taking e t eδ(ν ~ t) ~ , eδ(νt) ~ , noting ( ) ∈ { } ′ 1~3 1~3 ′ 2~3 ′ 1~3 1~3 ′ 2~3 e t min e−δ (ν (t−τ)) , e−δ (ν(t−τ)) e τ min e−(δ ~2)(ν (t−τ)) , e−(δ ~2)(ν(t−τ)) , ( ) { } ≲ ( ) { } and using the Cauchy–Schwarz inequality in τ, we obtain t 2 2 2 2 2 e t k IIIk t ν e t k Sk t τ Γ f, f k τ √µ dv dτ Q Q S SR3 − Æ k~0 ( )S S S ( )S = k~0 ( )S S U 0 ( )[( ( )) ( )] U = = t 2 2 2 (δ′~2)(ν1~3(t−τ))1~3 ν e t k e Sk t τ Γ f, f k τ √µ dv dτ ≲ Q S SR3 − Æ k~0 ( )S S 0 U ( )[( ( )) ( )] U = t ′ 1~3 1~3 e−(δ ~2)(ν (t−τ)) dτ × S0 t 2 5~3 2 (δ′~2)(ν1~3 (t−τ))1~3 ν e t k e Sk t τ Γ f, f k τ √µ dv dτ ≲ Q S SR3 − Æ k~0 ( )S S 0 U ( )[( ( )) ( )] U = t 1~3 ′ 2 −(δ′ ~2)(ν1~3 (t−τ))1~3 2 ν sup f τ 2 ′ e τ e f τ 2 ′ dτ ≲ Ẽ( ) S D̃( ) τ ′ [0,t] Y ( )Y Nmax−2 0 ( ) Y ( )Y 1,1,0,0 ∈ t 1~3 ′ ′ 2 −(δ′~2)(ν1~3 (t−τ))1~3 2 ν sup e τ f τ 2 ′ e f τ 2 ′ dτ. + Ẽ( ) S D̃( ) τ ′ [0,t] ( )Y ( )Y 1,1,0,0 0 Y ( )Y Nmax−2 ∈ By the energy bound (9.1) and then the bootstrap assumption (11.1), this implies t 2 2 2 2~3 ′ ′ 2 e t k III t ǫν e τ f τ 2 ′ dτ ǫν sup e τ f τ 2 ′ Q k ≲ S D̃( ) + Ẽ( ) k~0 ( )S S S ( )S 0 ( )Y ( )Y 1,1,0,0 τ ′ [0,t] ( )Y ( )Y 1,1,0,0 = ∈ 2 4~3 ≲ ǫ ν . 2 An identical argument, using additionally Fubini’s theorem, gives the desired Lt bound: t 2 III 2 S Q e s k k s ds 0 k~0 ( )S S S ( )S = t t −(δ′~2)(ν1~3 (s−τ))1~3 2 2 ǫν e ds e τ f τ 2 ′ f τ 2 ′ dτ ≲ S S D̃( ) + D̃( ) 0 ( τ )[ ( )Y ( )Y 1,1,0,0 Y ( )Y Nmax−2 ] t 2~3 2 2 ǫν e τ f τ 2 ′ f τ 2 ′ dτ ≲ S D̃( ) + D̃( ) 0 [ ( )Y ( )Y 1,1,0,0 Y ( )Y Nmax−2 ] 2 ≲ ǫ ν. This ends the proof of Proposition 11.2, and thus of Theorem 11.1.

12. Nonlinear energy decay In this section, we establish the nonlinear energy decay estimates for the full nonlinear Vlasov–Poisson–Landau equation (2.2a)–(2.2b). Throughout this section, we shall use primed energy and dissipation norms ⋅ Ẽ(2)′ and ⋅ D̃(2)′ , which are deﬁned as in (5.16) Y Y ∗,∗,0,0 Y Y ∗,∗,0,0 q′SvS2 ′ 1 with the primed exponential weights e for q 2 q (cf. (5.19)). The main result of this section is the following.= Theorem 12.1. Consider data as in Theorem 3.1. Then, the following hold: (1) The energy of f decays with the following stretched exponential rate: ∞ δ(ντ)2~3 2 1~3 δ(ντ)2~3 2 2 2~3 e f τ 2 ′ ν e f τ 2 ′ τ ǫ ν . sup Ẽ( ) + D̃( ) d ≲ 0 τ ∞ Y ( )Y 0,0,0,0 S0 Y ( )Y 0,0,0,0 ≤ < THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 69

(2) The energy of ∇xf decays with the following enhanced stretched exponential rate: ∞ 2 1~3 2 2~3 τ f τ 2 ′ ν τ f τ 2 ′ τ ǫν , sup e Ẽ( ) + e D̃( ) d 0 τ ∞ ( )Y ( )Y 1,1,0,0 S0 ( )Y ( )Y 1,1,0,0 ≤ ≤ < 1 3 1 3 2 3 for e t eδ(ν ~ t) ~ , eδ(νt) ~ . ( ) ∈ { } 12.1. Preliminary energy estimates. The ﬁrst step in the proof of Theorem 12.1 is the following energy estimates for the lowest order energies.

~~Proposition 12.2. The following energy estimates hold:~~

d 1+⟨t⟩−1 1~3 2 −2 2 e f t (2)′ θν f t 2 ′ t f t 2 ′ Ẽ + D̃( ) + Ẽ( ) dt( Y ( )Y 0,0,0,0 ) Y ( )Y 0,0,0,0 ⟨ ⟩ Y ( )Y 0,0,0,0 1~3 ν f t (2)′ f t (2)′ f t ̃ (12.1) ≲ Ẽ D̃ DN −2 Y ( )Y 0,0,0,0 Y ( )Y 0,0,0,0 Y ( )Y max

~~min f t (2)′ , f t (2)′ ρ t 2 , + Ẽ D̃ ~0 Lx {Y ( )Y 0,0,0,0 Y ( )Y 0,0,0,0 }Y = ( )Y and~~

d 1+⟨t⟩−1 1~3 2 −2 2 e f t (2)′ θν f t 2 ′ t f t 2 ′ Ẽ + D̃( ) + Ẽ( ) dt( Y ( )Y 1,1,0,0 ) Y ( )Y 1,1,0,0 ⟨ ⟩ Y ( )Y 1,1,0,0 1~3 ν f t (2)′ f t (2)′ f t ̃ ≲ Ẽ D̃ DN −2 (12.2) Y ( )Y 1,1,0,0 Y ( )Y 1,1,0,0 Y ( )Y max α (2)′ (2)′ 2 min f t Ẽ , f t D̃ ∂x ρ~0 t Lx . + 1,1,0,0 1,1,0,0 Q = {Y ( )Y Y ( )Y } SαS 1 Y ( )Y ≤ Proof. As in the proof of Theorem 9.1 in Section 9, the key step is to bound the inhomogeneous terms R̃α,β,ω. Here, we take R̃α,β,ω to be as in the ϑ 2 case in Proposition 5.11, qSvSϑ q′SvSϑ ′ 1 = except that the e weights are replaced by e with q 2 q. In view of the proof of Proposition 9.2, with N 0 and N= 1, we ﬁrst note the following = = bounds on the inhomogeneous terms R̃α,0,0: 1~2 −2 2 1~2 2~3 2 1~3 ǫ t f t 2 ′ ǫ ν f t 2 ′ ν f t (2)′ f t (2)′ f t (2)′ R0,0,0 ≲ Ẽ( ) + D̃( ) + Ẽ D̃ D̃ ̃ ⟨ ⟩ Y ( )Y 0,0,0,0 Y ( )Y 0,0,0,0 Y ( )Y 0,0,0,0 Y ( )Y 0,0,0,0 Y ( )Y Nmax−2

~~min f t (2)′ , f t (2)′ ρ t 2 , + Ẽ D̃ ~0 Lx {Y ( )Y 0,0,0,0 Y ( )Y 0,0,0,0 }Y = ( )Y and~~

1~2 −2 2 1~2 2~3 2 ǫ t f t 2 ′ ǫ ν f t 2 ′ Q R̃α,0,0 ≲ Ẽ( ) + D̃( ) SαS 1 ⟨ ⟩ Y ( )Y 1,1,0,0 Y ( )Y 1,1,0,0 = 1~3 + ν f t Ẽ(2)′ f t D̃(2)′ f t D̃(2)′ Y ( )Y 1,1,0,0 Y ( )Y 1,1,0,0 Y ( )Y Nmax−2 α (2)′ (2)′ 2 min f t Ẽ , f t D̃ ∂x ρ~0 t Lx . + 1,1,0,0 1,1,0,0 Q = {Y ( )Y Y ( )Y } SαS 1 Y ( )Y ≤ The proposition thus follows in a similar manner as in deriving (9.6).

12.2. Decay estimates. We now give the proof of Theorem 12.1. We shall only prove the enhanced decay rate, part (2) in the theorem; the other part is similar, if not simpler. THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 70

Applying Lemma A.1. We proceed by a bootstrap argument. Assume that there is TB 0 such that the bootstrap assumption (11.1) holds. In particular, we can use the bounds derived> in Theorem 11.1. We will use the Strain–Guo type estimate in Lemma A.1. For the remainder of the proof, 2 3 1 3 2 3 ﬁx either e t eδ(νt) ~ or e t eδ(ν ~ t) ~ . Deﬁne g and h so that ( ) = ( ) = 2 1+⟨t⟩−1 2 2 1+⟨t⟩−1 2 g t, v dv e f t 2 ′ , h t, v dv e f t 2 ′ , (12.3) 3 Ẽ( ) 3 D̃( ) SR ( ) = Y ( )Y 1,1,0,0 SR ( ) = Y ( )Y 1,1,0,0

~~1 noting that the factor e1+⟨t⟩− is harmless. Note also that the primed exponential weights 2 eq′SvS are used. Speciﬁcally,~~

2 4M−4Sα′S α+α′ 2 1~3 4M−4 α α g t, v A0 v ∂x f dx ν v x∂x f, v∂x f dx ∶ Q Q ST3 + ST3 ∇ ∇ ( ) = SαS 1 Sα′S 1 ⟨ ⟩ S S ⟨ ⟩ ⟨ ⟩ = ≤ ′ − ′ ′ + −1 1 2 2Sβ S~3 4M 4Sβ S α β 2 1 ⟨t⟩ 2 q0SvS ν v ∂x ∂v f dx e e , + Q ST3 Sβ′S 1,2 ⟨ ⟩ S S = 1 2 noting the exponential weight e 2 q0SvS inserted above. A similar deﬁnition is introduced for h2 t, v to satisfy (12.3). (By deﬁnition,) we note that

−1 2 1~3 2 1~3 −4 2 1~3 2 ν v g t, v dv ν f 2 ′ , ν v g t, v dv ν f 2 ′ , 3 ≲ D̃( ) 3 ≲ D̃( ) SR ⟨ ⟩ ( ) Y Y 1,1,0,0 SR ⟨ ⟩ ( ) Y Y 1,1,0,0 where Poincaré’s inequality was used in obtaining the second inequality, upon noting that α ∂x f has zero x-mean with α 1. S S = E(2)′ Therefore, after taking θ smaller if necessary, Proposition 12.2 and the definitions of 1,1,0,0 ′ ̃ D(2) θν1~3 θν1~3 and ̃1,1,0,0 imply that the differential inequality (A.2) holds with c , b , m 4, i.e. = = = d g2 t, v dv θν1~3 v −4g2 t, v dv θν1~3 h2 t, v dv F t , (12.4) dt SR3 ( ) + SR3 ⟨ ⟩ ( ) + SR3 ( ) ≲ ( ) and with c θν, b θν1~3, m 1, i.e. = = = d g2 t, v dv θν v −1g2 t, v dv θν1~3 h2 t, v dv F t , (12.5) dt SR3 ( ) + SR3 ⟨ ⟩ ( ) + SR3 ( ) ≲ ( ) where F t is given by ( ) 1~3 F t ∶ ν f t Ẽ(2)′ f t D̃(2)′ f t D̃(2)′ ( ) = Y ( )Y 1,1,0,0 Y ( )Y 1,1,0,0 Y ( )Y Nmax−2 α (12.6) (2)′ (2)′ 2 min f t Ẽ , f t D̃ ∂x ρ~0 t Lx . + 1,1,0,0 1,1,0,0 Q = {Y ( )Y Y ( )Y } SαS 1 Y ( )Y ≤ 2 3 1 3 1 3 Let e τ eδ(νt) ~ , eδ(ν ~ t) ~ . We will prove below that for any η 0, the following holds uniformly( ) ∈ for{ all T 0, T : } > ∈ ( B) T e t F t dt S0 ( ) ( ) T (12.7) −1 2 2~3 1~3 2 1~3 2 η ǫ ν ǫν η t f t 2 ′ ν τ f τ 2 ′ τ . ≲ + + sup e Ẽ( ) + e D̃( ) d ( ) 0 t T ( )Y ( )Y 1,1,0,0 S0 ( )Y ( )Y 1,1,0,0 ≤ < THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 71

Note also that in view of the primed energy and dissipation norms, the boundedness of the corresponding unprimed norms yields the boundedness of exponential moments for g (as is needed by (A.1) in Lemma A.1). Namely, using Theorem 9.1, we have

1 2 2 q0SvS 2 2 2 2~3 e g t, v dv f t 2 ǫ ν . 3 ≲ Ẽ( ) ≲ SR ( ) Y ( )Y 1,1,0,0 Therefore, we can apply Lemma A.1, using (12.7) and recalling (12.3), to deduce T 2 1~3 2 t f t 2 ′ ν τ f τ 2 ′ τ sup e Ẽ( ) + e D̃( ) d 0 t T ( )Y ( )Y 1,1,0,0 S0 ( )Y ( )Y 1,1,0,0 ≤ < T (12.8) −1 2 2~3 1~3 2 1~3 2 η ǫ ν ǫν η t f t 2 ′ ν τ f τ 2 ′ τ . ≲ + + sup e Ẽ( ) + e D̃( ) d ( ) 0 t T ( )Y ( )Y 1,1,0,0 S0 ( )Y ( )Y 1,1,0,0 ≤ < Taking ǫ0, ν0 and η sufficiently small, we can absorb the final term to the LHS, which yields T 2 1~3 2 2 2~3 t f t 2 ′ ν τ f τ 2 ′ τ ǫ ν , sup e Ẽ( ) + e D̃( ) d ≲ (12.9) 0 t T ( )Y ( )Y 1,1,0,0 S0 ( )Y ( )Y 1,1,0,0 ≤ < after fixing η 0. This then improves the bootstrap assumption (11.1). In particular, this > closes the bootstrap argument, and show that (12.9) holds for all t 0, ∞ , which implies the desired estimate in Theorem 12.1. ∈ ( ) It thus remains to prove the claim (12.7), under the bootstrap assumption (11.1). Controlling F t . To prove (12.7), we control each of the two terms in (12.6). First, by Hölder’s inequality,( ) T 1~3 ν e t f t Ẽ(2)′ f t D̃(2)′ f t D̃(2)′ dt S0 ( )Y ( )Y 1,1,0,0 Y ( )Y 1,1,0,0 Y ( )Y Nmax−2 T T 1~2 1~3 2 1~2 1~3 2 1~2 ′ sup e τ f τ ̃(2) ν e t f t (2)′ dt ν f t (2)′ dt ≲ E1 1 0 0 S D̃ S D̃ (0 τ T ( )Y ( )Y , , , )( 0 ( )Y ( )Y 1,1,0,0 ) ( 0 Y ( )Y Nmax−2 ) ≤ < T 1~3 2 1~3 2 ǫν t f t 2 ′ ν τ f τ 2 ′ τ , ≲ sup e Ẽ( ) + e D̃( ) d 0 t T ( )Y ( )Y 1,1,0,0 S0 ( )Y ( )Y 1,1,0,0 ≤ < (12.10)

1~3 T 2 where we have used the estimate established in (9.1) for ν f t ′ dt. ∫0 D̃(2) Y ( )Y Nmax−2 (1) (2) For the other term, we decompose ρ~0 ρ~0 + ρ~0 according to Theorem 11.1 so that = = = = α (2)′ (2)′ 2 e t min f t Ẽ , f t D̃ ∂x ρ~0 t Lx 1,1,0,0 1,1,0,0 Q = ( ) {Y ( )Y Y ( )Y } SαS 1 Y ( )Y = (12.11) α (1) α (2) e t f t (2)′ ∂x ρ t L2 e t f t (2)′ ∂x ρ t L2 ≲ Ẽ1 1 0 0 Q ~0 x + D̃1 1 0 0 Q ~0 x ( )Y ( )Y , , , SαS 1 Y = ( )Y ( )Y ( )Y , , , SαS 1 Y = ( )Y = = Thus, using (11.3) and (11.4) respectively, as well as H¨older’s and Young’s inequality, we have T α (1) e t f t ̃(2)′ ∂x ρ t L2 dt S E1 1 0 0 Q ~0 x 0 ( )Y ( )Y , , , SαS 1 Y = ( )Y = T 1 2 2 −1 1 2 α (1) 2 ~ ~ 2 (12.12) η sup e t f t ̃(2)′ η e t ∂x ρ t L dt ≲ E1 1 0 0 + S Q ~0 x (0 t T ( )Y ( )Y , , , ) [ 0 ( ) SαS 1 Y = ( )Y ] ≤ < = 2 −1 2 2~3 η t f t 2 ′ η ǫ ν , ≲ sup e Ẽ( ) + 0 t T ( )Y ( )Y 1,1,0,0 ≤ < THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 72

as well as T α (2) e t f t ̃(2)′ ∂x ρ t L2 dt S D1 1 0 0 Q ~0 x 0 ( )Y ( )Y , , , SαS 1 Y = ( )Y = T T 1~3 2 −1 −1~3 α (2) 2 ην e t f t (2)′ dt η ν e t ∂x ρ t 2 dt (12.13) ≲ S D̃ + S Q ~0 Lx 0 ( )Y ( )Y 1,1,0,0 0 ( ) SαS 1 Y = ( )Y = T 2 −1 2 2~3 η t f t 2 ′ t η ǫ ν . ≲ e D̃( ) d + S0 ( )Y ( )Y 1,1,0,0 Combining (12.10)–(12.13), and recalling (12.6), we have thus obtained (12.7). This ends the proof of Theorem 12.1.

13. Putting everything together The main theorem, Theorem 3.1, now follows straightforwardly. Indeed, ● Global existence of smooth solutions follows from Theorem 10.1. ● The estimates (3.3a) and (3.3b) follows from (9.1). ● The bounds (3.4) and (3.5) follow from interpolating Theorem 12.1 with (3.3b). ● Finally, for the uniform Landau damping statement (3.6), we bound, using Parseval’s theorem, interpolation, (8.3) and (11.2):

−Nmax+1 α ω ρˆk t k t 1 ∂x Y ρ~0 t L2 ≲ + Q = x S S( ) ⟨ ( )⟩ SαS+SωS Nmax−1 Y ( )Y ≤ −Nmax+1 1~Nmax α ω (Nmax−1)~Nmax k t 1 ρ~0 2 ∂x Y ρ~0 t L2 ≲ + = Lx Q = x ⟨ ( )⟩ Y Y (SαS+SωS Nmax Y ( )Y ) ≤ − + − 1~3 1~3 − 2~3 ǫν1~3 k t 1 Nmax 1 min e δ(ν t) , e δ(νt) , ≲ ⟨ ( + )⟩ { } after taking δ smaller. This completes the proof of Theorem 3.1.

Appendix A. Strain–Guo type lemmas R3 R Lemma A.1. Let T 0, ∞ and g ∶ 0, T × → be a smooth function. Suppose there exist C 0, c 0, b 0∈, (m 0], q 0, 2[ and) p 0, q such that the following holds: > > > ≥ ∈ ( ) ∈ ( 2 ) (1) There is a uniform bound of Gaussian moments:

2 sup eqSvS g2 t, v dv C. (A.1) 3 t [0,T ) SR ( ) ≤ ∈ (2) The following diﬀerential inequality holds for all t 0, : ∈ [ ∞) d g2 t, v dv c v −mg2 t, v dv b h2 t, v dv F t , (A.2) dt SR3 ( ) + SR3 ⟨ ⟩ ( ) + SR3 ( ) ≤ ( ) for some function h 0, T R3 R, and some function F 0, T R satisfying ∶ [ ) × → ∶ [ ) → T 2 ep(ct) 2+m F t dt C. (A.3) S0 ( ) ≤ Then, there exists C 0 (depending only on q and m) such that q,m > 2 T 2 p(ct) 2+m 2 p(ct) 2+m 2 sup e g t, v dv b e h t, v dv dt Cq,mC. (A.4) 3 + 3 t [0,T ) SR ( ) S0 SR ( ) ≤ ∈ THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 73

~~Proof. We compute using (A.2) that~~

2 2 d 2 2 ep(ct) +m g2 t, v dv bep(ct) +m h2 t, v dv dt( SR3 ( ) ) + SR3 ( ) 2 (A.5) 2 2+m 2 p(ct) 2+m 2pc 2 −m 2 p(ct) 2+m e m g dv c v g dv e F t . 3 − 3 + ≤ ( 2 m t 2+m SR SR ⟨ ⟩ ) ( ) ( + ) 1 1 We control the ﬁrst term in (A.5). Splitting into v ct 2+m and v ct 2+m , we bound −m 2 ⟨ ⟩ ≤ ( ) ⟨ ⟩ ≥ ( ) the low velocity by 3 v g t, x, v dv dx and the high velocity using (A.1): ∫R ⟨ ⟩ ( ) 2 2 2 2 2pc +m 2 c +m 2 m g t, v dv m 1 1 g t, v dv 3 + t 2+m SR ( ) ≤ t 2+m (S{vS⟨v⟩ (ct) 2+m } S{vS⟨v⟩ (ct) 2+m }) ( ) 2 + m ≤ ≥ ( ) 2 2 2+m 2 −m 2 c q −q(ct) 2+m qSvS 2 (A.6) c v g dv dx m e e e g dv 3 + 3 ≤ SR ⟨ ⟩ t 2+m SR 2 2+m 2 −m 2 c q −q(ct) 2+m c v g dv dx m Ce e . 3 + ≤ SR ⟨ ⟩ t 2+m q −m 2 We plug (A.6) into (A.5) and use p . Note that the 3 v g t, v dv terms cancel. ≤ 2 ∫R ⟨ ⟩ ( ) 2 2 d 2 2 ep(ct) +m g2 t, v dv bep(ct) +m h2 t, v dv dt( SR3 ( ) ) + SR3 ( ) 2 (A.7) 2 2 2 2 c +m − q 2 p(ct) +m q 2 (ct) +m e F t + m Ce e . ≤ ( ) t 2+m 2 Integrating, using (A.1) to bound the initial term R3 g 0, v dv, and using (A.3) to bound 2 ∫ ( ) the L1 norm of ep(ct) 2+m F t , we have t ( ) 2 T 2 sup ep(ct) 2+m g2 t, v dv b ep(ct) 2+m h2 t, v dv dt 3 + 3 t [0,T ) SR ( ) S0 SR ( ) ∈ 2 (A.8) ∞ 2 2 c +m − q 2 q 2 (ct) +m 2C + C m e e dt. ≤ S0 t 2+m To bound the integral in (A.8), split the integration domain into 0, c−1 and c−1, so that [ ] [ ∞) 2 ∞ −1 ∞ 2+m 2 2 c 2 c − q t 2+m dt − q t 2+m ′ q 2 (c ) q 2+m 2 (c ) m e e dt e c m e d ct C (A.9) + 1 q,m S0 t 2+m ≤ ( S0 t 2+m Sc− ( )) ≤ ′ for some Cq,m. Plugging (A.9) back into (A.8) yields the conclusion. R3 R Lemma A.2. Let g ∶ 0, ∞ × → be a smooth function. Suppose there exist C 0 and c 0 such that [ ) > > (1) There is a uniform bound of the 4m-th moments:

sup v 4mg2 t, v dv C. (A.10) 3 t [0,∞) SR ⟨ ⟩ ( ) ≤ ∈ (2) The following diﬀerential inequality holds for all t 0, : ∈ [ ∞) d g2 t, v dv c v −mg2 t, v dv 0. (A.11) dt SR3 ( ) + SR3 ⟨ ⟩ ( ) ≤ THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 74

Then 35π g2 t, v dv 1 C ct −3. (A.12) SR3 ( ) ≤ ( 2 + ) ⟨ ⟩ Proof. We compute using (A.11) that d 3c2t ct 3 g2 t, v dv ct 3 g2 t, v dv c v −mg2 t, v dv . (A.13) dt(⟨ ⟩ SR3 ( ) ) ≤ ⟨ ⟩ ( ct 2 SR3 ( ) − SR3 ⟨ ⟩ ( ) ) ⟨ ⟩ m 1 m 1 To bound the first term in (A.13), we split into v 3 ct and v 3 ct , then bound −m 2 ⟨ ⟩ ≤ ⟨ ⟩ ⟨ ⟩ ≥ ⟨ ⟩ the low velocity by 3 v g t, v dv and the high velocity using (A.10): ∫R ⟨ ⟩ ( ) 3c2t 3c2t g2 t, v dv g2 t, v dv 2 3 2 1 + 1 ct SR ( ) = ct (S{vS⟨v⟩m ⟨ct⟩} S{vS⟨v⟩m ⟨ct⟩}) ( ) ⟨ ⟩ ⟨ ⟩ ≤ 3 ≥ 3 (A.14) 35c2t 35Cc2t c v −mg2 dv v 4mg2 dv c v −mg2 dv . ≤ SR3 ⟨ ⟩ + ct 6 SR3 ⟨ ⟩ ≤ SR3 ⟨ ⟩ + ct 6 ⟨ ⟩ ⟨ ⟩ −m 2 We plug (A.14) into (A.13), noting that the 3 v g t, v dv terms cancel. So ∫R ⟨ ⟩ ( ) d 35Cc2t 35Cc ct 3 g2 t, v dv . (A.15) dt(⟨ ⟩ SR3 ( ) ) ≤ ct 3 ≤ ct 2 ⟨ ⟩ ⟨ ⟩ Integrating, and using (A.10) for the t 0 term yield the conclusion. = References [1] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, and T. Yang. The Boltzmann equation without angular cutoff in the whole space: II, Global existence for hard potential. Anal. Appl. (Singap.), 9(2):113–134, 2011. [2] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, and T. Yang. Global existence and full regularity of the Boltzmann equation without angular cutoff. Comm. Math. Phys., 304(2):513–581, 2011. [3] R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu, and T. Yang. The Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential. J. Funct. Anal., 262(3):915–1010, 2012. [4] C. Bardos and P. Degond. Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data. Ann. Inst. H. PoincaréAnal. Non Linéaire, 2(2):101–118, 1985. [5] C. Bardos, P. Degond, and F. Golse. A priori estimates and existence results for the Vlasov and Boltzmann equations. In Nonlinear systems of partial differential equations in applied mathematics, Part 2 (Santa Fe, N.M., 1984), volume 23 of Lectures in Appl. Math., pages 189–207. Amer. Math. Soc., Providence, RI, 1986. [6] Margaret Beck and C. Eugene Wayne. Metastability and rapid convergence to quasi-stationary bar states for the two-dimensional Navier-Stokes equations. Proc. Roy. Soc. Edinburgh Sect. A, 143(5):905– 927, 2013. [7] Jacob Bedrossian. Suppression of plasma echoes and Landau damping in Sobolev spaces by weak collisions in a Vlasov-Fokker-Planck equation. Ann. PDE, 3(2):Paper No. 19, 66, 2017. [8] Jacob Bedrossian. Nonlinear echoes and Landau damping with insufficient regularity. Tunis. J. Math., 3(1):121–205, 2021. [9] Jacob Bedrossian and Michele Coti Zelati. Enhanced dissipation, hypoellipticity, and anomalous small noise inviscid limits in shear flows. Arch. Ration. Mech. Anal., 224(3):1161–1204, 2017. [10] Jacob Bedrossian, Pierre Germain, and Nader Masmoudi. Dynamics near the subcritical transition of the 3d Couette flow II: Above threshold case. arXiv:1506.03721, preprint, 2015. [11] Jacob Bedrossian, Pierre Germain, and Nader Masmoudi. On the stability threshold for the 3D Couette flow in Sobolev regularity. Ann. of Math. (2), 185(2):541–608, 2017. [12] Jacob Bedrossian, Pierre Germain, and Nader Masmoudi. Stability of the Couette flow at high Reynolds numbers in two dimensions and three dimensions. Bull. Amer. Math. Soc. (N.S.), 56(3):373–414, 2019. [13] Jacob Bedrossian, Pierre Germain, and Nader Masmoudi. Dynamics near the subcritical transition of the 3D Couette flow I: Below threshold case. Mem. Amer. Math. Soc., 266(1294):v+158, 2020. THE VLASOV–POISSON–LANDAU SYSTEM IN THE WEAKLY COLLISIONAL REGIME 75

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~~(Sanchit Chaturvedi) Department of Mathematics, Stanford University, 450 Jane Stanford Way, Bldg 380, Stanford, CA 94305, USA Email address: [email protected]~~

~~(Jonathan Luk) Department of Mathematics, Stanford University, 450 Jane Stanford Way, Bldg 380, Stanford, CA 94305, USA Email address: [email protected]~~

~~(Toan T. Nguyen) Penn State University, Department of Mathematics, State College, PA 16803, USA Email address: [email protected]~~