Linearized Wave-Damping Structure of Vlasov-Poisson in R3

Home , Landau damping

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References 25

1 Introduction

1.1 The problem at hand One of the fundamental equations in the kinetic theory of plasmas is the Vlasov-Poisson equations for an inﬁnitely extended plasma (see e.g. [12, 7]),

∂ f + v f + E(t,x) f = 0, t ·∇x ·∇v  E(t,x)= xW x ρ(t,x), −∇ ∗  (1.1)  ρ(t,x)= f(t, x, v)dv n0,  R3 − Z f(t = 0, x, v)= fin(x, v),   for the time-dependent probability density function f(t, x, v) 0 of the electrons in the phase space ≥ (x, v) R3 R3, n is the number density of the constant ion background, and where W is the ∈ × 0 kernel of the Coulomb interaction1. q2 W (x)= 4πǫ m x 0 e | | with q the electron charge, me the electron mass, and ǫ0 the vacuum permittivity. We will consider (1.1) linearized about the homogeneous Maxwellian background with ﬁxed temperature T

2 0 n0 − me|v| f (v) := e 2T . (1.2) (2πT )3/2

In 1930s and 1940s, Vlasov [39, 38] suggested to neglect collisions and derive the so-called Vlasov- Poisson equation for long-range interactions, independently from Jeans’ derivation [23] in stellar dynamics. Motivated by the mathematical understanding of the plasma oscillations previously theorized in particular by Langmuir, he studied the linearized approximation of (1.1) (see (1.5) below) formally searching for eigenmodes in the form of planar waves e−iωt+ikxF (v), given some velocity distribution F , and computed various dispersion relations (see for instance equation (50) in [39]). He asserted, not quite correctly but almost correctly as we shall see and clarify in this paper, that in the long-wave limit k 1, where k is the Fourier variable in space, one has the | | ≪ dispersion relation

2 2 2 3T 2 4 q n0 ω = ωp + k + k as k 0, where ωp := (1.3) me | | O | | → ǫ0me is the cold plasma frequency. The relation ω2 = ω2 + 3T k 2 is called the Bohm-Gross dispersion p me | | relation2, and arises from the following compressible Euler-Poisson for the spatial density n(t,x) 0 ≥ 1Note that we are slightly abusing notation as E in (1.1) actually denotes the acceleration of the electrons due to the electric field, not the electric field itself. This convention is taken for the remainder of the paper. 2Technically, the relation seems to have first appeared in Vlasov’s earlier work [39] before appearing in [6], however, it is likely that access to Vlasov’s work was difficult at that time. See also e.g. pg 260 of [12].

2 and macroscopic velocity u(t,x) R3 ∈ ∂ n + (nu) = 0 t ∇x ·  m n [∂ u + (u )u]= qnE(t,x) 3T n e t ·∇x − ∇x   ǫ E(t,x)= qn 0∇x ·  when linearized around a constant density state n0 > 0, u0 = 0: ∂ n + n u = 0 t 0∇x ·  m n ∂ u = qn E(t,x) 3T n (1.4) e 0 t 0 − ∇x   ǫ E(t,x)= qn. 0∇x ·  This model is sometimes referred to as a warm plasma model in the physics literature see e.g. [Chapter 16; [12]] for more detail. Vlasov’s prediction was shown to be incomplete by Landau in 1946 [25], who showed that in fact the linearized electric field decays to zero as t , a phenomenon →∞ now known as Landau damping, and in particular that non-trivial planar waves never satisfy the dynamics. The damping arises due to mixing/filamentation in phase space, not unlike a scalar quantity being stirred in a fluid (as first pointed out in [37]). Landau damping was later observed in experiments [26, 27] and is considered one of the most important properties of collisionless plasmas [32, 12, 35]. On T3 R3, the linearized electric field decays rapidly provided the initial data is regular, × specifically, its decay in time is comparable to the decay in Fourier variable associated with v of the initial data [28, 4]. Analogous estimates are also true for the density on R3 R3 if one uses a model × with Debye shielding (this arises when studying the ion distribution function), i.e. where W is the fundamental solution to the elliptic problem ∆W + αW = δ for some constant α> 0 (see [5, 19] − and the references therein). However, such rapid decay estimates on the linearized electric field are false in R3 R3 for Vlasov-Poisson. Landau himself predicted an extremely slow Landau damping × of long waves approximately solving the dispersion relation (1.3) (see also e.g. [12, 35] for modern exposition in the physics literature), something that we will make much more precise below. This lack of significant Landau damping was proved rigorously by Glassey and Schaefer [10, 11], though a precise characterization of the dynamics was still lacking. In this work, we precise the linearized dynamics and prove that the linearized electric field can be split into two contributions: one contribution at long spatial waves k 0 that is a Klein- ∼ Gordon-like propagation matching (1.3) to leading order with very weak Landau damping with rate “ ( k ∞)” and another contribution, which is properly Landau damping and decays at a rate faster O | | than kinetic free transport for long-waves k 0, i.e. faster than Vlasov-Poisson equation linearized 0 ∼ around f = 0, n0 = 0. In particular, our work shows that the hydrodynamic description (linearized Euler-Poisson) in fact is the leading order description of the electric field at long waves, despite the lack of collisional effects. As such, we remark that Vlasov was essentially correct to leading order in his prediction of the long wave dynamics of (1.5). For long waves, we show that the distribution function decomposes (to leading order) into two pieces: a term that factorizes as E˜(t,x) f 0(v) ·∇v (where f 0 is the Maxwellian background) where E˜ solves a Klein-Gordon-type equation (including p an additional tiny damping) and a separate contribution that scatters to free transport in Lx,v for p> 6. Hence, the hydrodynamic behavior arises from a large-scale collective motion of the plasma that is insensitive to the filamentation in phase space normally associated with Landau damping. Understanding the relationship between Landau damping (or other kinetic effects) and the observed large-scale hydrodynamic behavior in collisionless plasmas has been an area of research in the physics literature for some time (see e.g. [17, 16, 33, 20] and the references therein). Our work

3 provides a precise description of the hydrodynamic behavior and its leading order corrections due to Landau damping and other kinetic effects, for the simple linearized problem; studies of more physical settings (e.g. external and/or self-consistent magnetic fields, inclusion of ions, inhomogeneous backgrounds etc) and/or the inclusion of nonlinear effects is an interesting direction of further research. Our work is only linear, but we remark that much progress has been made at the nonlinear level in recent years in other settings. After the earlier work of [8] (see also [21]), the major breakthrough came in [28], when Landau damping for the nonlinear problem was shown on T3 R3 for all × sufficiently small and smooth initial perturbation of Landau-stable stationary solutions; the actual smoothness required being Gevrey or analytic. We also refer to [4, 13] for simplified proofs and [3] for a study of the problem with collisions. The work [2] shows that the results therein do not hold in finite regularity (see also [14]). Our previous work [5] studied the nonlinear problem with shielding on R3 R3; see also [19, 29] for an alternative approach and some refinements. Other works, old × and new, have studied the nonlinear Cauchy problem near the vacuum state, that is without the presence of a non-zero spatially homogeneous background equilibrium f 0, see e.g. [1, 22]. As can be seen from our results below, the dynamics are significantly different in this case.

1.2 Main results We linearize the Vlasov-Poisson equation (1.1) on R3 R3 around an infinitely-extended, homoge- x × v neous background f 0(v) 0. This models a spatially localized disturbance of an infinite plasma in ≥ which collisions can be neglected, a fundamental problem in the kinetic theory of plasmas [12, 7, 35]. For a localized disturbance h := f 0 f, the linearized Vlasov-Poisson equations for the electron − distribution function are given by ∂ h + v h + E(t,x) f 0 = 0, t ·∇x ·∇v  E(t,x)= xW x ρ(t,x), −∇ ∗  (1.5)  ρ(t,x)= h(t, x, v)dv,  R3 Z h(t = 0, x, v)= h (x, v),  in   where we take the charge neutrality assumption R3×R3 hin(x, v)dxdv = 0. Define the standard plasma constants (number density, plasma frequency and temperature): R 2 0 2 q n0 me 2 0 n0 := f (0), ωp := , T := v f (v)dv, ǫ m 3n R3 | | 0 e 0 Z and as stated above, we assumec the Maxwellian distribution background (1.2). Our main results are asymptotic decomposition of the electric field and the distribution function; we give slightly simplified statements for readability and refer to the main body of the paper for k+0,p more detailed expansions. See Subsection 1.3 for the notation x , x,y , , and Ww . h i h i h∇i We remark that another preprint proving similar results with somewhat different methods [18] has recently been completed as well. These works have been completed totally independently.

Theorem 1. Suppose R3×R3 hindxdv = 0 and let h and E solve (1.5). There is δ0 > 0 and a decomposition of the electric ﬁeld E = E + E between a ‘Klein-Gordon’ and ‘Landau damped’ R KG LD parts, with E supported in spatial frequencies k <ν and: KG | | 0 E satisﬁes the following Landau-damping-type decay estimates for any σ N: • LD ∈ σ 1 3 x,t x ELD(t) L2 . hin σ+ +0,1 kh∇ ∇ i k x t 5/2 k kW 2 h i 0 4 σ 1 x,t x ELD(t) L∞ . hin σ+3+0,1 . (1.6) kh∇ ∇ i k x t 4 k kW0 h i E further decomposes as E = E(1) + E(2) with the pointwise-in-time estimates: • KG KG KG KG (1) E (t) . Ein 2 + vhin 2 + hin 0,1 KG 2 L Lx,v W4 Lx k k k k k k (2) E (t) . hin 0,1 KG 2 W5 Lx k k (2) −3/2 E (t) . t hin 3/2,1 KG ∞ W Lx h i k k 5 1 1 (1) −3 2 − p . 0,1 EKG(t) p p t x hin W ( 2 p< ) Lx h i kh i k 4 ∀ ≤ ∞ (1) −3/2 xE (t) . t hin 0,1 . KG ∞ W4 ∇ Lx h i k k

E(1) solves a weakly damped Klein-Gordon type equation in the following sense: there are • KG bounded, smooth functions λ, Ω, k B(0, δ ) such that (for errors independent of t), ∈ 0 [(1) −λ(k)t −λ(k)t ik sin (Ω(k)t) E (t, k)= Ein(k)e cos (Ω(k)t)+ e k ηhin(k, 0) KG k 2 ·∇ Ω(k) | | b+ ( k 2)e−λ(k)t+iΩ(k)t + ( k 2)e−λ(k)t−iΩ(k)ct, O | | O | | ∞ 2 2 3T 2 4 where λ(k) > 0, λ(k)= ( k ) , Ω (k)= ωp + k + k as k 0. O | | me | | O | | → Furthermore, here holds for all 2

(1) (2) which further emphasizes that for small k, EKG is much larger than EKG. (2) Remark 1. The EKG electric ﬁeld is essentially a Klein-Gordon-type evolution subjected to a Landau damping external forcing. See Section 2 for more details.

Remark 2. For frequencies bounded away from zero, i.e. for any δ > 0, P≥δE, Landau damps at a polynomial rate t −σ provided the initial data is W σ,1, hence the extremely slow Landau damping h i of EKG manifests only at k = 0. As expected, for frequencies bounded from zero, the electric field −hktis (both ELD and EKG) damps exponentially fast if the initial data is analytic (and analogously e for Gevrey initial data). 1 Remark 3. As the electric field is not shielded, it is too restrictive to assume that Ein L , 1 ∈ regardless of how well localized the initial hin is. If xhin L and R3 ×R3 hindxdv = 0, then ∈ X v −3 p Ein(x) x as x (hence, Ein L for p > 1 but not p = 1) butR one cannot obtain faster ≈ | | → ∞ ∈ α decay at infinity unless one has higher zero-moment conditions, such as R3 R3 x hindxdv = 0 and x× v the latter are not propagated by the semi-group. R Remark 4. In the case of the kinetic free transport, the electric field decays only as EF T (t) ∞ . k kLx t −2, which is a full power of t slower than (1.6). For frequencies bounded away from zero, both the h i free transport electric field EF T and our ELD are Landau-damped exponentially fast for analytic data (and polynomial for Sobolev data); the difference in decay rates comes from the contribution of long waves.

5 Remark 5. There is a clear distinction between Landau damping and dispersive decay above: for the Landau damping contributions, each derivative buys one power of time (a structure that can be guessed from free transport), whereas for the Klein-Gordon-like contributions, there is no such behavior.

Remark 6. We have not endeavored to get the sharpest dependence on the initial data that could be possible. This could be important for nonlinear extensions.

We deduce a similar decomposition at the level of the distribution function:

Theorem 2. The solution to (1.5) decomposes as h = h + h with h = E˜ (t,x) f 0(v) KG LD KG KG ·∇v for some eﬀective electric ﬁeld E˜KG, with the following estimates:

for all m 0, r [1, ] and all p [2, ), (E˜KG is essentially a phase-shifted version of • (1) ≥ ∈ ∞ ∈ ∞ EKG and satisﬁes the same estimates)

m v hKG(t) L2 Lr .m hin 0,1 (1.7) kh i k x v k kW4 1 1 m −3 2 − p v hKG(t) Lp Lr .p,m t x hin 0,1 kh i k x v h i kh i kW4 m −3/2 v xhKG(t) L∞Lr .m t hin 0,1 ; kh i ∇ k x v h i k kW4

p hLD scatters to free transport in all Lx,v, p > 6 (provided one has enough initial regularity). • σ m In particular, if v hin is integrable for σ, m N large enough, then for all p> 6, there h∇ip h i ∈ exists an h Lx,v such that ∞ ∈ m t→+∞ v hLD(t,x vt, v) h∞(x, v) p 0. h i − − Lx,v −−−−→ h i

Remark 7. Using the Strichartz estimates [9] for the transport equation one obtains

hLD q p r . hin σ,1 k kLt LxLv k kWm for all (p, q, r, a) satisfying

1 1 1 1 1 1 2 1 1 1 1 1 1 < 1, 1 + , = n , = + , a> 6. r − n p ≤ r ≤ ≤ p r q r − p a 2 r p Theorem 1 shows that the leading-order term in the asymptotics of the distribution h factorises between a function of (t,x) and a fixed function of v, which is reminiscent of a hydrodynamical limit even though the equations are collisionless. One can actually push this intuition further: a long-wave re-scaling of the electric field converges weakly (in negative Sobolev spaces) to the electric field solving the linearized Euler-Poisson system (1.4). See Subsection 2.5 for a proof.

3 Theorem 3. Consider an initial data hin(x, v) = ǫ 0(ǫx, v) such that 0 has zero average and 5+0,1 H H W , and denote H0 ∈ 5 2 q n0 −1 E0 = x( ∆x) 0( , v) dv. ǫ m ∇ − R3 H · 0 e Z

6 Let be the unique solution in the natural energy space L∞H1 and ∂ L∞L2 to the E E ∈ t x tE ∈ t x following Klein-Gordon equation,

2 2 3T ∂t + ωp ∆ = 0 E − me E  (0,x)= E (x)  E 0 (1.8)   ∂t (0,x)= n0 x vhindv .  E − ∇ ·  Z  Then, for all s (5/2, 7/2), 0<ǫ 1, for all 0

Moreover, given such initial data with appropriate scaling, both the electric fields in the linearized Vlasov-Poisson and the linearized Euler-Poisson system are asymptotic to the Bohm-Gross dispersion relation at large scale: E(t,x) = E + E and (t,x) = + with the following weak L2 + − E E+ E− x limit on t [ T, T ] for any T fixed finite ∈ − t weak 3T 1 ∓iωp 2 t x 1 ± i∆t e ǫ E± , e me E0 ǫ2 ǫ2 ǫ −−−→L2 2 x t weak 3T 1 ∓iωp 2 t x 1 ± i∆t e ǫ ± , e me 0. ǫ2 E ǫ2 ǫ −−−→L2 2 E x 1.3 Notation Denote x = (1+ x 2)1/2 and the Fourier multiplier defined by h i | | h∇i [ f(ξ)= ξ f(ξ); h∇i h i we define other Fourier multipliers in the analogous way.b We define P≤N to be the Littlewood-Paley low-pass filter in x, in particular, define χ C∞(B(0, 2)) with χ(x)=1 for x 1, and define ∈ c | |≤ k P\f(k)= χ f(k). ≤N N Denote the weighted Sobolev spaces (note that the weightb index refers to the variable v)

1/p 1/p p p f Lp := f(x, v) dxdv , ρ Lp := ρ(x) dx k k x,v R3 R3 | | k k x R3 | | Z × Z m σ f σ,p := v x,v f p . k kWm kh i h∇ i kL σ+0,p For inequalities involving norms Wm we use the notation +0 if we mean that the estimate holds for σ + δ with δ (0, 1) with a constant that depends on δ. For p [1, ] we denote p′ = p the ∈ ∈ ∞ p−1 Hölder conjugate. Let f : [0, ) C satisfy e−µtf(t) L1 for some µ R. Then for all complex ∞ → ∈ ∈ numbers z µ, we can define the Fourier-Laplace transform via the (absolutely convergent) ℜ ≥ integral 1 ∞ fˆ(z) := e−ztf(t)dt. (1.9) 2π Z0 7 This transform is inverted by integrating the f˜ along a so-called ‘Bromwich contour’ via the inverse Fourier-Laplace transform: let γ>µ and define:

γ+i∞ fˇ(t) := eztf(z)dz. Zγ−i∞ 2 Decomposition of the electric ﬁeld

2.1 Volterra equation The most important property of the linearized Vlasov equations is that one can reduce the problem to a Volterra equation for the density separately for each spatial frequency. We now recall how this is done. Writing a Duhamel representation along free transport gives

t h(t, x, v)= h (x vt, v)+ ( W ρ)(t,x v(t s)) f 0(v)ds. in − ∇x ∗x − − ·∇v Z0 2 −1 Taking the Fourier transform in x and integrating in v gets (with w0 := ωpn0 )

t ρˆ(t, k)= h (k,kt) w (t τ)f 0(k(t τ))ˆρ(τ, k)dτ. (2.1) in − 0 − − Z0 Taking the Fourier-Laplace transformc (1.9) in time for cz suﬃciently large gives ℜ ρ˜(z, k)= H(z, k)+ (z, k)˜ρ(z, k), (2.2) L where H(z, k) is the Fourier-Laplace transform of t h (k,kt) and the dispersion function is 7→ in +∞ +∞ w0 − z s 0 −zt c |k| 0 ˆ (z, k) := w0 tf (kt)e dt = 2 e s f ks ds, (2.3) L − 0 − k 0 Z | | Z c c with the standard notation kˆ = k/ k . This change of variable allows for the function s sf 0(ksˆ ) to | | 7→ have regularity bounds independently of the size of k. Our assumption (1.2) implies that z (z, k) 7→ L is an entire function. Moreover, we see that away from z = 0, (z, k) is also a smooth functionc of L k (even at k = 0 as we shall see in the expansions below).

2.2 Asymptotic expansions and lower bounds on the dispersion function Solving (2.2) for ρ works, formally at least, except where (z, k) gets close to one. In the case of L not-so-small spatial frequencies k, the treatment is similar to that of the periodic domain x T3, ∈ and so we merely sketch the argument here; see [28, 4] for more details.

Lemma 2.1 (Resolvent estimates for non-small frequencies). There exists a λ > 0 such that for any ν > 0, κ> 0 (depending on ν ) such that 0 ∃ 0

k >ν0, inf 1 (z, k) > κ. (2.4) ∀ | | ℜz>−λ|k| | −L |

Furthermore, the following estimate holds uniformly on the critical vertical line of this region 1 k >ν0, ω R, (λ k + iω, k) .λ . (2.5) ∀ | | ∈ |L | | | 1+ k 2 + ω2 | | 8 Proof. The estimate (2.4) is proved in e.g. [28, 4]. To see (2.5), we use integration by parts and the analyticity of f 0 to get ∞ w0 λs−i ω s (λ k + iω, k) = e |k| sf 0 ksˆ ds |L | | | k 2 Z0 | | ∞ c w0 1 λs− i ω s 2 |k| 0 ˆ = 2 2 ∂s e sf ks ds k 0 ω | | Z λ i |k| − ∞ c w0 0 ˆ 2 0 ˆ 2 2 2 n0 + 2 ∂sf ks + s ∂s f ks ds ≤ (λ k + ω ) | | 0 | | Z 1 c c .λ , 1+ k 2 + ω2 | | which completes the proof.

Next we turn to the low frequency estimates, which are more challenging, and contain the long-wave dispersive structure. For δ, δ′ > 0, deﬁne the following region in the complex plane:

ℜz = −(1 − δ)|ℑz|

Λδ,δ′

ℜz = −(1 − δ)|ℑz|

Figure 1: The region Λδ,δ′

We will next show that (z, k) stays uniformly away from one in the region Λ ′ z iω <ǫ L δ,δ \{| ± p| } where ωp is the cold plasma frequency. The proof relies on two representations: (1) an expansion obtained by the stationary phase method (i.e. successive integrations by parts in time) meaningful z for large values of |k| , (2) an approximation argument using the explicit formula obtained from the Plemelj formula at the imaginary line z = 0, that provides estimates near this line. The ﬁrst ℜ representation is given by the following lemma. Lemma 2.2 (Asymptotic expansion of ). Given δ′ > 0 suﬃciently small depending only on f 0, L 2 2 4 ωp 3T k k z ′ z Λδ,δ , (z, k)= 2 1+ | 2| + | |4 , as | | . (2.6) ∀ ∈ L − z " mez O z !# k →∞ | | | | 9 Note that this expansion only contains information for frequencies k z . | | ≪ | | ˆ0 j 0 Proof. Since s sf (ksˆ ) is odd, observe that ∂s (sf (ksˆ )) = 0 for all even j. Therefore, 7→ |s=0 integrating by parts repeatedly in s in (2.3) gives c ∞ w0 w0 − z s (z, k)= f 0(0) e |k| ∂2 sf 0 ksˆ ds L − z2 − z2 s Z0 2 2 ∞ w0 3w0 k w0 k − z s c0 ˆ ˆ 2c0 |k| 4 0 ˆ = 2 f (0) 4| | k k : f (0) |4 | e ∂s sf ks ds − z − z ⊗ ∇ − z 0 2h i 2Z w0c 3w0 k c w0 k c =: f 0(0) | | kˆ kˆ : 2f 0(0) | | ζ(z, k). − z2 − z4 ⊗ ∇ − z4 h i Since w f 0(0) = w n c= ω2 and 3n T := m kˆ kˆc: 2f 0(0) gives the leading order terms in (2.6). 0 0 0 p 0 e ⊗ ∇ Note that we have c ∞ c − z s |k| 4 0 ˆ ζ(z, k) := e ∂s sf ks ds. 0 Z It remains to show that for z Λ ′ , there holds c ∈ δ,δ k 2 ζ(z, k) . | | . | | z 2 | | First consider the region z δ′ k : integrating by parts two more times, we obtain ℜ ≥− | | 2 ∞ 2 ′ k δ′s 6 0 ˆ k z δ k , ζ(z, k) . | |2 1+ e ∂s sf ks ds .δ′ | |2 , ∀ℜ ≥− | | | | z 0 z | | Z | | c where the last line followed by the analyticity of f 0. Turn next to the region z < δ′ k with z > (1 δ) z . Observe then arg z2 [ π +β, 3π β] ℜ − | | ℜ − − |ℑ | ∈ 2 2 − for a small β > 0 depending on δ. Write

∞ 0 − z s − z s |k| 4 0 ˆ |k| 4 0 ˆ ζ(z, k)= e ∂s sf (ks) ds e ∂s sf (ks) ds =: ζ1(z, k)+ ζ2(z, k). −∞ − −∞ Z Z On the one hand, ζ is boundedc as in the region z δ′ kcdue to the now advantageous sign of 2 ℜ ≥− | | the exponent. On the other hand, ζ1 is a true Fourier-Laplace transform, and due to (1.2),

∞ 4 ∞ − z s z − z s |k| 4 0 ˆ |k| 0 ˆ ζ1(z, k)= e ∂s sf (ks) ds = 4 e f (ks)ds −∞ k −∞ Z | | Z 2 4 ∞ z 2 T 4 mez n0 z c − s−s n0 z c 2 = e |k| 2me ds = e 2T |k| . 3/2 k 4 3/2 k 4 me | | Z−∞ me | | Due to arg z2 [ π + β, 3π β], it holds z2 . z 2 . z 2 and this term vanishes (to infinite ∈ 2 2 − ℜ δ −|ℑ | δ −| | order) in terms of |z| . This completes the proof. |k| →∞ Lemma 2.2 suffices to estimate the resolvent in much of the areas of interest. The next lemma estimates the resolvent for low frequencies k in the half-plane z δ′ k (assuming δ′ to be small ℜ ≥− | | enough) and away from the cold plasma frequencies iω . Given ǫ> 0, define the following region: ± p ′ H ′ := z = λ + iω C : λ> δ k and z iω ǫ) . ǫ,δ ∈ − | | | ± p|≥ 10 |z − iωp| <ǫ

ℜz = −δ′|k|

Hǫ,δ′

|z + iωp| <ǫ

Figure 2: The region Hǫ,δ′

′ Lemma 2.3 (Low frequency resolvent estimates). Given ε, δ > 0, there are ν0,κ> 0 such that

k <ν , z H ′ , 1 (z, k) κ. ∀ | | 0 ∀ ∈ ǫ,δ | −L |≥ Proof. Let R> 0 be ﬁxed large depending on f 0 but independent of k. Case 1: z > R k . In this region the estimate follows from (2.6) taking R> 0 suﬃciently large. | | | | Case 2: z R k . In this region the asymptotic expansion (2.6) is no longer useful and we use | | ≤ | | the Plemelj formula instead. Writing z = λ + iω, it is classical that for λ = 0, one has (see e.g. [31, 28] for explanations),

0 ′ w0 (fk ) (r) w0π 0 ′ ω (iω, k)= ω dr + i (fk ) , L k 2 R r k 2 k | | Z − |k| | | | | where, for any k = 0, the partial hyperplane average is deﬁned as 6 R 0 0 r , fk (r) := f (v∗)dv∗. ∀ ∈ k r+k Z |k| ⊥ Moreover, observe that for any z C such that z R k and z ( δ′ k , 0], ∈ | |≤ | | ℜ ∈ − | | ∞ w ′ 1 0 δ s 0 ˆ ∂z (z, k) . 3 se f ks ds . 3 (2.7) | L | k 0 k | | Z | | c where we have used the analyticity of f 0 and taken δ′ small enough. Subcase 2.1: z R k and c k ω R k . Given any c> 0, we deduce from decay, smoothness, | |≤ | | | | ≤ | |≤ | | radial symmetry, and monotonicity of f 0 that

w0π 0 ′ ω 1 inf 2 (fk ) &c,R 2 , c< |ω|

11 and therefore, using (2.7), (z, k) & 1 holds uniformly for all z ( δ′ k , 0). |ℑL | ℜ ∈ − | | (f 0)′(r) Subcase 2.2: z R k and ω < c k . Observe that k is integrable since (f 0)′(0) = 0 and f 0 | |≤ | | | | | | r k smooth, and if we denote

0 ′ (fk ) (r) dr = c0 = 0, R r 6 Z we deduce that if c |c0| is small enough then ≤ 2

0 ′ 0 ′ (fk ) (r) ω (fk ) (r) ω c0 ω dr c0 = dr . | | | | R r − k R ω k ≤ 2 Z |k| Z r r |k| − | | − | |

and thus 1 1 (iω, k) & . | −L | k 2 | | Using again (2.7), for δ′ small enough we deduce that for z ( δ′ k , 0], ℜ ∈ − | | 1 (z, k) & 1. | −L | These diﬀerent cases above together prove that 1/ 1 is bounded from above on the strip | − L| z ( δ′ k , 0] and outside B(0,R k ) z > 0 , and since there are no poles within the remaining ℜ ∈ − | | | | ∩{ℜ } region B(0,R k ) z > 0 , the function is holomorphic in this region and the upper bound is | | ∩ {ℜ } also valid there by the maximum principle, which completes the proof.

2.3 Construction of the branches of poles From Lemma 2.2, we have a pole at k = 0 at the cold plasma frequency: ( iω , 0) = 1. It follows | | L ± p from Rouch´e’s theorem that if k if small enough, exactly two poles persist in respective neighbor- | | hoods of iω : Given ǫ> 0, the two functions F (z) := 1 (z, 0) and G(z) := (z, k) (z, 0) are ± p −L L −L holomorphic on the set z iω lǫ, and Lemma 2.2 implies 1 (z, 0) & ǫ and (z, 0) (z, k) . | ∓ p| | −L | |L −L | k 2 on z iω = ǫ. Therefore, F (z) = 1 (z, 0) and F (z) G(z) = 1 (z, k) have the same | | | ∓ p| −L − −L number of poles in z iω <ǫ provided that k is suﬃciently small relatively to ǫ. | ∓ p| | |

iωp 3T 4 2 p+(k)= i ω + |k| + O (|k| ) + Error(k) p meωp Error(k)= O (|k|∞) < 0

3T 2 4 p−(k)= −i ω + |k| + O (|k| ) + Error(k) p meωp −iωp

Figure 3: The branches of poles k p (k) 7→ ±

12 However knowing just the approximate location of the poles is not enough to deduce dispersive estimates. We next use the implicit function theorem to construct the branches of solutions p±(k). Lemma 2.4. There are ǫ,ν > 0 such that for all k <ν , there are unique p (k) C solution to 0 | | 0 ± ∈ (p (k), k) = 1 in z iω <ǫ and k p (k) =: λ(k) iΩ(k) are smooth (but not analytic) L ± {| ∓ p| } 7→ ± − ± and satisfy λ(k) > 0 and p (k) iω with the following expansions as k 0: ± ∼k→0 ± p →

2 2 3T 2 4 Ω(k) = ωp + k + k (2.8) me | | O | | 3T Ω(k)= i k + k 3 (2.9) ∇ meωp O | | 3T 2Ω(k)= i Id + k 2 (2.10) ∇ meωp O | | jλ(k) . k N for any j, N N . (2.11) ∇ j,N | | ∈ Remark 8. This expansion of p (k ) = Ω(k) provides the rigorous justiﬁcation for the Bohm- ℑ ± ± Gross dispersion relation in kinetic theory. Regarding the real part p (k) = λ(k), physicists ℜ ± − assert that (see e.g. [12, pp.419]),

3/2 4 2 me ωp meωp λ(k) √π 3 exp 2 , ≈ k T 3/2 −4 k T ! | | | | however at this time we lack a mathematically rigorous explanation for this exact prediction.

0 Proof. Since f is real, p+ = p− and it is enough to build the branch near +iωp. By the implicit function theorem applied to the function of (z, k) C Rd, the result follows by verifying L ∈ × ∂ (iω , 0) =c 0, since is smooth in (z, k) and analytic in z in this neighborhood. Roughly zL p 6 L speaking we want to take derivatives of the expansion (2.6). From (2.3) and integrating by parts as in the proof of Lemma 2.2,

∞ ∞ w0 − z s w0 − z s |k| 2 0 ˆ |k| 2 2 0 ˆ ∂z (z, k)= 3 e s f ks ds = 2 e ∂s s f ks ds L k 0 z k 0 | | Z ∞ | | Z 2w0n0 w0 − z s c |k| 3 2 0 ˆ c = 3 + 3 e ∂s s f ks ds z z 0 Z ∞ 2w0n0 2w0 k w0 k − z s = + | | kˆ f 0 ksˆ c+ | | e |k| ∂4 s2f 0 ksˆ ds z3 z4 ·∇ z4 s Z0 2w n w k h i = 0 0 + 0| |ζ′(z, k) c c z3 z4

′ 0 with ζ (z, k) that is uniformly bounded for z iωp <ǫ and k <ν0 (using the analyticity of f ). 2 | − | | | 2ωp 2i Hence ∂z (iωp, 0) = 3 = = 0. The implicit function theorem then implies the existence of a L (iωp) ωp 6 unique smooth solution k p (k) to the equation (p (k), k) = 1 in a neighborhood of iω . 7→ + L + p To get more precise information on the behavior of the poles near k = 0, we need to compute derivatives in k as well of . Expansion (2.8) immediately follows from Lemmas 2.2: L 2 2 2 2 3Tωp 2 4 2 3T 2 4 p+ = p+ (p+, k)= ωp 2 k + k = ωp + k + k . L − mep+ | | O | | me | | O | |

13 Next, observe that

( )(p , k) p = ∇kL + , (2.12) ∇ + − (∂ )(p , k) zL +

2p = (∂ )(p , k))−1 ( 2 )(p , k) + ( ∂ )(p , k) p + (∂2 )(p , k) p 2 (2.13). ∇ + − zL + ∇kL + ∇k zL + ·∇ + z L + |∇ +| By the same integration by parts method used in Lemma 2.2, we obtain

∞ w0 − z s 2w0k |k| 2 0 ˆ ˆ ˆ 2 0 3 k (z, k)= 3 e s f (ks)ds = 4 k k : f (0) + k . ∇ L − k 0 ∇ z ⊗ ∇ O | | | | Z h i c c 2 0 T 2 −1 Using f (0) = 3n0 Id and w0 = ω n , this yields ∇ me p 0

c 6T 3 k (p+(k), k)= 2 k + k , ∇ L meωp O | | which implies (2.9). The proof of (2.10) is similar, using now (2.13) instead of (2.12); the lengthier calculations are omitted for the sake of brevity. In these calculations a clear pattern emerges: in all derivatives m (z, k) (respectively m∂ (z, k)), for m N, the leading order as k 0 is an even ∇k L ∇k zL ∈ → (resp. odd) power of z−1, and thus at z = iω all derivatives m (iω , 0) (resp. m∂ (iω , 0)) ± p ∇k L p ∇k zL p are purely real (resp. purely imaginary). This proves all derivatives mp (0) are purely imaginary ∇ + and implies thus (2.11), i.e. p = λ vanishes to inﬁnite order at k = 0. Observe that since ℜ + − nevertheless p < 0 as k 0 (by Lemma 2.1), the function k p (k) diﬀers from its Taylor ℜ + ∼ 7→ + series at k = 0 and is therefore not analytic.

2.4 Spectral surgery and extraction of Klein-Gordon waves Through (2.2), the solution to the Volterra equation (2.1) is classically [15] given formally as:

t ρˆ(t, k)= h (k,kt)+ (t τ, k)h (k,kτ)dτ, (2.14) in R − in Z0 where the resolvent kernel is givenc by the inverse Laplacec transform R 1 γ+i∞ (z, k) (t, k)= ezt L dz, R 2iπ 1 (z, k) Zγ−i∞ −L for a suitable Bromwich contour such that z L(z,k) is holomorphic for z > γ 0. 7→ 1−L(z,k) ℜ − The calculations in the two previous Subsections 2.2-2.3, show that for k < ν0 suﬃciently L(·,k) | | ′ small, is holomorphic in the region H ′ represented in Figure 2 (the half-plane z δ k 1−L(·,k) ǫ,δ ℜ ≤− | | minus ǫ-discs around for the two poles), with one isolated pole p±(k) in each disc, depending on k as studied in the last Subsection. Therefore, by Cauchy’s Residue theorem,

′ ep+(k)t ep−(k)t 1 γ +i∞ (z, k) (t, k)= + + ezt L dz R ∂ (p (k), k) ∂ (p (k), k) 2iπ ′ 1 (z, k) − zL + − zL − ! Zγ −i∞ −L =: + (t, k)+ − (t, k) + (t, k), RKG RKG RRF T RKG(t,k) | {z } 14 ′ ′ for some γ ( δ , ǫ) so that the vertical line is to the left of the poles p (k) but still in H ′ where ∈ − − ± ǫ,δ /(1 ) is meromorphic. This decomposes the resolvent = + into a Klein-Gordon L −L R RKG RLD part and a remainder free transport part, which then yields a corresponding decomposition of the densityρ ˆ(t, k) through (2.14): t t ρˆ(t, k)= hin(k,kt)+ KG(t τ, k)hin(k,kτ)dτ + RF T (t τ, k)hin(k,kτ)dτ 0 R − 0 R − Z+ − Z =:cρ ˆF T (t, k) +ρ ˆKG(t, k) +ρ ˆKG(t,c k) +ρ ˆRF T (t, k). c ± We first prove a general expansion ofρ ˆKG(t, k) by successive integrations in time (producing additional powers of k). Terms in this expansion are either comparable to solutions to free transport (with additional Fourier multipliers) or to solutions to a Klein-Gordon-like evolution equation. Lemma 2.5 (Expansion of the Klein-Gordon density). For all k <ν and all ℓ N, we have | | 0 ∈ ℓ ℓ ρˆ± (t, k)= ep±(k)tA±(k) k⊗j : j h (k, 0) A±(k) k⊗j : j h (k,kt) KG j ∇η in − j ∇η in Xj=0 h i Xj=0 h i t c c ± ⊗(ℓ+1) ℓ+1 + KG(t τ, k)Aℓ+1(k) k : η hin(k,kτ) dτ (2.15) 0 R − ∇ Z h i where h (k, η) is the differential in the second Fourier variable,c and with the notation ∇η in J (k) 1 c A±(k) := ± and J (k) := . j −p (k)j+1 ± −∂ (p (k), k)) ± zL ± Remark 9. Note that the Fourier multipliers A± are smooth and bounded for k <ν . j | | 0 Proof. Integrating by parts in time gives t ρˆ± (t, k)= ± (t τ, k)h (k,kτ)dτ KG RKG − in Z0 t t J (k) p±(k)(t−τ) c ± −λ(k)(t−τ)±iΩ(k)(t−τ) = J±(k)e hin(k,kτ)dτ = ∂τ e hin(k,kτ)dτ 0 − 0 p±(k) Z Z J (k) J (k) t J (k) ± ±c p±(k)t ± p±(k)(t−τ) c = H(k,kt) e hin(k, 0) + e k ηhin(k,kτ) dτ, p±(k) − p±(k) 0 p±(k) ·∇ Z h i and iterating finitely many times yieldsc the result. c

− + Note that by symmetry J+(k)= J−(k) and Aj (k)= Aj (k), and the calculations of Lemma 2.4 give the expansion J (k)= ωp + O( k 2) which allows to expand the coefficients A±(k) in (2.15). ± ∓ 2i | | j Denoting p (k)= λ(k) iΩ(k) with λ(k) > 0 and Ω(k)= ω + ( k 2), it immediately implies: ± − ± p O | | Lemma 2.6 (Klein-Gordon coefficients at low frequencies). One has as k 0, → A+(k)+ A−(k)=1+ ( k 2) (2.16) 0 0 O | | A+(k)+ A−(k)= ( k 2) 1 1 O | | ep+(k)tA+(k)+ ep−(k)tA−(k)= e−λ(k)t cos [Ω(k)t]+ ( k 2)eiΩ(k)t + ( k 2)e−iΩ(k)t 0 0 O | | O | | h sin [Ω(k)t] i ep+(k)tA+(k)+ ep−(k)tA−(k)= e−λ(k)t + ( k 2)eiΩ(k)t + ( k 2)e−iΩ(k)t , 1 1 Ω(k) O | | O | | where the ( k 2) in the above represent infinitely differentiable, bounded functions of k which are O | | independent of time.

15 Remark 10. Note crucially that (2.16) cancels the leading order of the free transport evolution for long-waves as k 0, which leads to an improved decay of the Landau damping contribution of → the electric field for such long-waves. We are now able to precise our decomposition of the electric field. Definition 2.1. We define the variable precision decomposition,

ℓ ik ik E(1;ℓ)(t, k)= w 1 A+(k) A−(k) h (k,kt) w A±(k) k⊗j : j h (k,kt) LD 0 k 2 − 0 − 0 in − 0 k 2 j ∇η in | | | | j=1 X b ik t c c E(2) (t, k)= w (t τ, k)h (k,kτ)dτ, LD 0 k 2 RRF T − in | | Z0 ℓ b ik c E(1;ℓ)(t, k)= w ep±(k)tA±(k) k⊗j : j h (k, 0) KG 0 k 2 j ∇η in | | Xj=0 b t c (2;ℓ) ik ± ⊗(ℓ+1) ℓ+1 EKG (t, k)= w0 2 KG(t τ, k)Aℓ+1(k) k : η hin(k,kτ) dτ, k 0 R − ∇ | | Z h i andb accordingly deﬁne the particular decomposition we shall use inc the sequel (setting ℓ = 4)

(1) (1;4) (2) ELD := ELD , ELD := as above (1) (1;4) (2) (2;4) EKG := EKG , EKG := EKG , (1) (2) (1) (2) ELD := ELD + ELD, EKG := EKG + EKG. Next, we estimate the ‘remainder free transport part’ of the resolvent. The gain in powers of k present in 2.7 is critical to the high quality decay rate of the Landau damping electric ﬁeld.

Lemma 2.7 (Remainder free transport resolvent at low frequencies). There exists λ0 > 0 such that for all k <ν there holds, | | 0 k <ν , (t, k) . k 3 e−λ0|k|t. ∀ | | 0 |RRF T | | | Proof. We add and subtract by the expected leading order behavior as k 0 (using that the 2 −1 → integration path is away from z = 0), hence deﬁne for α := 3Tωpme , 2 2 ωp α k (z)= 2 2 + 2 | | 2 2 . Q z + ωp (z + ωp) The function z ezt (z, k) is holomorphic and decaying in the left half-plane z < γ′ k , and 7→ Q ℜ − | | hence we deduce, deforming the contour suitably, 1 (z, k) (t, k)= + + ezt L + (z, k) dz RRF T 2iπ 1 (z, k) Q ZΓ0 ZΓ+ ZΓ− −L =: 0 + + + − with RRF T RRF T RRF T Γ = z = λ + iω : λ = δ k ,ω ( R k ,R k ) 0 − | | ∈ − | | | | n o Γ = z = λ i(1 + δ)λ + i R (1 + δ)δ k : λ ( , δ k ] + − − | | ∈ −∞ − | | n o Γ = z = λ + i(1 + δ)λ i R (1 + δ)δ k : λ ( , δ k ] . − − − | | ∈ −∞ − | | n o 16 Γ+

R|k|

∂Λδ,δ′ Γ0

−δ|k|

−R|k|

Γ−

Figure 4: The contour of integration (note that it is slightly modiﬁed as compared to ∂Λδ,δ′ )

We separate cases as in Lemma 2.3. Consider ﬁrst z Γ with z < δ′ k . As in the proof of ∈ 0 |ℑ | | | Lemma 2.3, in this region, there holds the following expansion, valid for δ′ suﬃciently small,

2 ′ ωp z z Γ0 with z < δ k , ( z, k)= + |ℑ | , ∀ ∈ |ℑ | | | L ℑ − k 2 O k 3 | | | | and for all z . k , | | | | (z, k) . k −2 , ∂ (z, k) . k −3 . |L | | | | zL | | | Therefore, for δ and δ′ suﬃciently small, we have (z, k) k −2 for z Γ with z < δ′ k . Next L ≈ | | ∈ 0 |ℑ | | | consider the case z Γ with δ′ k z R k . By similar arguments as above and in Lemma ∈ 0 | | ≤ |ℑ | ≤ | | 2.3, we have 1 (z, k) & k −2 and (z, k) . k −2. Therefore, on Γ , the integrand ( k 2), | −L | | | |L | | | 0 O | | resulting in the estimate

z2 (z, k)+ ω2 α k 2 zt L p 3 −δ|k|t e 2 2 + 2 | | 2 2 dz . k e . Γ0 (1 (z, k))(z + ωp) (z + ωp) ! | | Z −L

This completes the estimates on Γ . 0 We next turn to Γ , Γ . We need only consider Γ ,Γ is analogous. We use now z k and + − + − | | ≫ | | the decomposition from Lemma 2.2:

ω2 3ω2T k 2 k 4 p p | | (z, k)= 2 4 + | 6| ζ(z, k) L − z − mez z with ζ(z, k) uniformly bounded on Γ+ and decaying at inﬁnity. Using this expansion we have (z, k) z2 (z, k)+ ω2 α k 2 L + (z, k)= L p + | | 1 (z, k) Q (1 )(z2 + ω2 (z2 ω2)(z2 + ω2) −L −L p − p p 17 αk2(z2 ω2 ζ(z, k)k4 = L− p + (z2 z2 )(z2 + ω2)2 z2(z2 z2 )(z2 + ω2) − L p − L p αk2(αk2 + ζ(z, k)k4z−2 ζ(z, k)k4 = + . (z2 z2 )(z2 + ω2)2 z2(z2 z2 )(z2 + ω2) − L p − L p Using the uniform boundedness of ζ(z, k) we integration then gives

∞ k 4 + . e−δ|k|t | | dx = k 3 e−δ|k|t. RRF T x2 | | ZR|k|

This completes the proof of Lemma 2.7.

We ﬁnally estimate the whole resolvent at frequencies bounded away from zero, which is simpler.

Lemma 2.8 (Non-small frequencies resolvent estimate). Given any ν0 > 0 there is λ1 > 0 such that ν 1 k 0 , (t, k) . e−λ1|k|t. ∀ | |≥ 2 R k | | Proof of Lemma 2.8. Choose λ > 0 as in Lemma 2.1 and deform the contour to get (there are no poles in z λ k when k ν /2) ℜ ≥− | | | |≥ 0 1 −λ|k|+i∞ (z, k) (t, k)= ezt L dz R 2iπ 1 (z, k) Z−λ|k|−i∞ −L and using the estimate of Lemma 2.1 gives ∞ −λ|k|t 1 1 −λ|k|t (t, k) . e 2 2 dω . e . |R | −∞ k + ω k Z | | | | | |

2.5 Another look at the long wave hydrodynamic behavior In this subsection we prove Theorem 3 on the basis of the decomposition of Deﬁnition 2.1 and the previous estimates of this section.

Proof of Theorem 3. We prove the first part of the theorem, as the second part is seen directly from the decomposition 2.1, together with some basic estimates that are similar or easier than what is required to prove the first part. For any field F , define 1 x F (t,x) := F t, , with Fourier transform F (t, k)= F (t,ǫk). ǫ ǫ3 ǫ ǫ Note that we have defined hin such that for all ǫ > 0, hin,ǫ(x, v)c = 0(x, vb). Define the initial 0 H macroscopic flux (recall that n0 is the total mass of f (v)) 1 1 jin(x) := vhindv with Fourier transform jin(k)= i ηhin(k, 0). n R3 n ∇ 0 Z 0 Denote, c c 1 J(k)= i η 0(k). n0 ∇ H b c 18 The rescaled electric field in the linearized Euler-Poisson system (1.8) satisfies Eǫ ik 2 k sinΩKG(ǫk)t ǫ ǫ(t, k)= 2 0(k) cos ΩKG(ǫk)t + ǫωp 2 k J(k) , E k H k · ΩKG(ǫk) | | | | where we defineb the exact Euler-Poissonc imaginary phase ΩKG(kb)

2 3T 2 ΩKG(k) := ωp + k . r m | | (1) Consider EKG first defined in the decomposition of Definition 2.1, since it is the contribution which is asymptotic to . Lemma 2.6 shows that Eǫ (1) −λ(ǫk)t ik −λ(ǫk)t 2 ik sin [Ω(ǫk)t] ǫE (t, k)= e 0(k) cos [Ω(ǫk)t]+ e ǫωp k J(k) KG,ǫ k 2 H k 2 · Ω(ǫk) | | | | 2 −λ(ǫk)t b + (ǫ ǫk )ec 0 0,1 . b O | | kH kW4 Note that the errors depend on time, but in a uniformly bounded way, and that they depend on v-moments of hin up to order 5. By the expansion of Ω in Lemma 2.4, we have

2 3T 2 4 2 3T 2 4 Ω(ǫk) ΩKG(ǫk) . ωp + ǫk + ( ǫk ) ωp + ǫk . ǫk | − | m | | O | | − m | | | | r r and thus

cos [Ω(ǫk)t] cos [Ω (ǫk)t] = ǫk 4 t , sin [Ω(ǫk)t] sin [Ω (ǫk)t] = ǫk 4 t , − KG O | | − KG O | | and any N > 0, 1 e−λ(ǫk)t = ǫk N t , and therefore we deduce, − O | | (1) 4 2 ǫ E ǫ . ǫk t + ǫ ǫk 0 0,1 . KG,ǫ − E | | | | kH kW4 and thus b b 1 2 4 8 2 2 (1) ǫ ǫk + ǫk t s− 3 −0 ǫ E (t) ǫ(t) . 0 0,1 | | | | dk . 0 0,1 ǫ 2 t KG,ǫ −s W 2s W −E H kH k 4 |ǫk|.1 k ! kH k 4 h i Z h i for s ( 3 , 7 ). Turn next to E(2) ; from its definition ∈ 2 2 KG 1/2 t 2 (2) b 1 10 5 s− 1 −0 ǫ E (t) . ǫ ǫk 0(k,kτ) dk dτ . 0 0,1 ǫ 2 , KG,ǫ −s 2 2s η W4 H 0 |ǫk|.1 ǫk k | | ∇ H ! kH k Z Z | | h i for s ( 3 , 19 ). This completes the treatment of the ‘Klein-Gordonc parts’ of the electric field. ∈ 2 2 We turn next to the Landau damping contributions, which in fact dominate the error. It is (1) convenient to subdivide the Landau damping field as in Definition 2.1. The contribution of ELD is straightforward, indeed, 1 2 4 2 2 (1) −2s 2 j [j ǫ 2 1 ǫ ELD,ǫ(t) . ǫ k ǫk ǫk η 0(k,kt) dk . 0 W , , H−s  R3 h i | | | | ∇ H  kH k 4 t j=0 Z X h i   5 (2) for s> 2 . Turn finally to ELD, which produces the dominant error. Then, 1/2 t 2 −1 2 (2) min( ǫk , ǫk ) −λ0|ǫk|(t−τ) 2 ǫ E (t) . ǫ | | | | e 0(k,kτ) dk dτ . ǫ 0 2,1 , LD,ǫ −s 2s W H 0 R3 k H ! kH k 0 Z Z h i 5 b for s> 2 . This completes the proof of Theorem 3.

19 3 Electric ﬁeld estimates

3.1 Landau damping estimates on the electric ﬁeld

In this section we provide estimates for ELD. We start with the optimal decay estimates for the density for the kinetic free transport (optimal in terms of time decay, not in the dependence on the initial data). Denote the spatial density of the solution to the free transport equation

H(t,x) := hin(x tv, v)dv with Fourier transform H(t, k)= hin(k,kt). R3 − Z Lemma 3.1. For all σ 0, b c ≥ σ x,t x H(t, ) 1 (3.1) kh∇ ∇ i · kLx . hin σ,1 k kW0 σ x,t x H(t, ) 2 kh∇ ∇ i · kLx −3/2 . t hin σ+3/2+0,1 h i k kW0 σ −3 x,t x H(t, ) L∞ . t hin σ+3+0,1 . (3.2) kh∇ ∇ i · k x h i k kW0 More generally, for all 1 p , j 0, ≤ ≤∞ ≥ 1 σ ⊗j ⊗j −j−3 1− p +3+ +0 1 x,t x x : v hin( tv, v)dv . t hin W σ j , . (3.3) h∇ ∇ i ∇ R3 ·− p h i k k j Z Lx

Proof. The inequality (3.1) is clear. To see (3.2) note that

j j j k −3−σ +3+ 1 +3+ 1 xH(t, ) L∞ . k H(t, k) dk . | |3+j+ dk hin W j , . t hin W j , . ∇ · x R3 | | R3 k,kt k k 0 h i k k 0 Z Z ! h i b The proof of (3.3) follows similarly (using also interpolation).

Next, we turn to estimates on the damped part of the electric ﬁeld. By Lemmas 2.6 and 3.1, (1) ELD satisﬁes the estimates claimed in Theorem 1. (2) Turn next to obtaining estimates on ELD. Lemma 3.2. There holds the following estimates

σ (2) 1 x,t x E (t) . hin σ+2+0,1 (3.4) LD 2 5/2 W0 h∇ ∇ i Lx t k k h i σ (2) 1 x,t x E (t) . hin σ+3+0,1 . (3.5) LD ∞ 4 W0 h∇ ∇ i Lx t k k h i

Proof. Consider ﬁrst the low spatial frequencies k . ν . Compute using Lemma 2.7, for any a> 0, | | 0 t σ (2) σ −1 x,t x E (t) x,t x x RRF T (t τ) x H(τ, ) dτ LD ∞ ∞ h∇ ∇ i Lx ≤ 0 h∇ ∇ i |∇ | − ∗ · Lx t Z σ −1 σ . k, (t τ)k k RF T (t τ, k) k,τk hin(k,τk) dkdτ 0 R3 h − i | | |R − | h i Z Zt 2 −3−a −3−a c+3+ 1 . k τk (t τ)k dk dτ hin W σ a, . | | R3 h i h − i k k 0 Z0 Z 20 We split the integral

t t 2 t k 2 τk −3−a (t τ)k −3−adk dτ = + k 2 τk −3−a (t τ)k −3−adk dτ, R3 | | h i h − i t R3 | | h i h − i Z0 Z Z0 Z 2 ! Z and change variables k′ = τk or k′ = (t τ)k in each one to obtain − t k 2 τk −3−a (t τ)k −3−adk dτ . t−4. R3 | | h i h − i Z0 Z Note that the k 2 in the numerator is crucial for obtaining this sharp rate. which concludes the | | proof of (3.5) at low frequencies. At frequencies bounded away from zero, the estimate is similar though more straightforward and with less loss of regularity. The L2 case (3.4) follows similarly and is omitted for brevity.

3.2 Dispersive estimates of the electric field We now consider the ‘Klein-Gordon part’ of the electric field in Definition 2.1. Consider first the following useful dispersive estimates: Lemma 3.3 (Dispersive estimates for weakly damped poles). We have 1 p 2 there holds for ≤ ≤ t 0 and f = f(x) Lp(R3 ), ≥ ∈ x 1 1 p±(∇)t −3 p − 2 e P≤ν0 f ′ . t f Lp . (3.6) Lp k k

p±(∇)t ∗ Remark 11. The linear propagator e is not a unitary operator, and the standard T T argument as in e.g. [24, 36] do not apply. Hence, it is not as trivial to obtain the homogeneous Strichartz estimates or the full range of expected inhomogeneous Strichartz estimates (although some inhomogeneous Strichartz estimates follow immediately from (3.6) and the O’Neil Young convolution inequality [30]). Proof. The case p = 2 is immediate. For the case p = 1, write

ep±(∇)tP f = K(t) f, ≤ν0 ∗x with the integral kernel

1 ix·k+itΩ(k)−λ(k)t K(t,x) := e aν0 (k)dk, (2π)d/2 R3 Z where a is a Schwartz class function compactly supported in a ball of radius 2ν corresponding ν0 ≤ 0 to the Littlewood-Paley projection. Hence the p = 1 case follows from the pointwise kernel estimate

−d/2 K(t) ∞ . t ( t 0). (3.7) k kL ∀ ≥ The intermediate exponents p (1, 2) are then obtained by the Riesz-Thorin interpolation theorem. ∈ Let us prove (3.7). Despite the complex phase, K is essentially a standard oscillatory integral, and we may easily adapt the standard arguments as in e.g. [Proposition 6, pg 344 [34]]. Let us ﬁrst explain the argument assuming that Ω(k)= k 2 (the case Ω(k)= ω2 + 3T k 2 is the same). In this p me | | x | | case, we make the change of variables y = k + 2t and we write (for some scale ǫ chosen below),

i|x|2/4t2 e it|y|2−λt y x K(t,x)= e χ aν0 y dy (2π)d/2 R3 ǫ − 2t Z 21 i|x|2/4t2 e it|y|2−λt y x + e 1 χ aν0 y dy (2π)d/2 R3 − ǫ − 2t Z h i =: KS(t,x)+ KNS(t,x), where χ C∞(B(0, 1)) with χ(z)=1 for z 1/2, and where we have split K into a ’stationary’ ∈ c | |≤ part and a ’non-stationary’ part. For the stationary part KS we simply bound the integrand and estimate the volume of integration:

K (t,x) . ǫd. (3.8) | S | ν0

For the non-stationary part KNS we integrate by parts using the non-vanishing of the phase. In particular observe that (note that 2 = x∗x for x Cn), |·| ∈ 2 2 t 2iy λ 2 eit|y| −λt = ( 2iy λ) eit|y| −λt. | −∇ | − −∇ ·∇y Therefore, if we deﬁne the diﬀerential operator 2iy λ Df := y − −∇ f , ∇ · 2iy λ 2 | −∇ | then by repeated integration by parts we have,

1 it|y|2−λt N y x KNS . N e D (1 χ( ))aν0 (y ) dy . | | t R3 − ǫ − 2t Z −2N We next show that the integrand is bounded by y ; for the proof in three dimensions, we only | | need N = 1, 2 though it holds for all N. The case N = 1 is easily checked. Indeed,

2iy λ 2iej ∂j λ −2 ∂j − −∇ = − − ∇ = ( y ), 2iy λ 2 − 2iy λ 2 O | | | −∇ | | −∇ | which is suﬃcient as the terms in which the derivative lands elsewhere are only better (estimates on nλ are provided by Lemma 2.4). For N = 2 we analogously have ∇ 2iy λ 2iej ∂j λ 2ieℓ ∂ℓ λ 2iej ∂j λ ∂ℓ − −∇ − − ∇ = − − ∇ − − ∇ 2iy λ 2 2iy λ 2 − 2iy λ 2 2iy λ 2 | −∇ | | −∇ | | −∇ | | −∇ | 2iy λ ∂ λ ( 2ie ∂ λ) (2iy λ) ( 2ie ∂ λ) + − −∇ ℓj∇ − ℓ − ℓ∇ −∇ · − j − j∇ 2iy λ 2 2iy λ 2 − 2iy λ 4 | −∇ | | −∇ | | −∇ | = ( y −4), O | | which is similarly suﬃcient (note the pattern that selects a particular dominant term whereas the more complicated error terms are smaller, hence the desired estimates hold for all N). Therefore, provided we choose N 2, ≥ 1 1 1 d−2N KNS . N 2N dy . N ǫ . | | t |y|≥ǫ y t Z | | Hence making the choice ǫ t−1/2 gives the result when combined with (3.8). The case using ∼ the true Ω(k) follows by Morse’s lemma due to (2.10) and the other estimates in Lemma 2.4 (see [Proposition 6, pg 344 [34]] for more details). This completes the main dispersive estimates (3.6).

22 (1) The estimates on EKG in Theorem 1 follow immediately from (3.6) and the decomposition 2.1 upon observing that for R3 ρindx = 0 we have for all 1

(2) Next, we will prove the pointwise-in-time decay estimates on EKG. Lemma 3.4. There holds for all 2 p , ≤ ≤∞ 1 1 (2) −3 2 − p E . t h 3/2,1 . KG p in W Lx h i k k 5

Proof. Deﬁne T as above:

⊗5 5 ⊗5 T (t,x)= v hin(x tv, v)dv, T (t, k)= i η hin(k, 0). R3 − ∇ Z By Minkowski’s inequality (and boundedness of the prefactob r multipliers onc Lp as they are Schwartz class functions),

t (2) p±(∇x)(t−τ) −1 5 EKG p . e P≤ν0 x x : T p dτ. Lx 0 |∇ | ∇ Lx +,− Z X On the one hand, by Bernstein’s inequality, for any p r′ > 2, it follows from (3.6) that we have ≥

p±(∇x)(t−τ) −1 ⊗5 p±(∇)(t−τ) −1 ⊗5 e x : T . e P≤ν0 x : T x p x r′ |∇ | ∇ Lx |∇ | ∇ Lx

1 −1 ⊗5 . P≤ν0 x T . 3 1 − 1 x r t τ ( r 2 ) |∇ | ∇ Lx

| − | On the the other hand, we similarly have

ep±(∇x)(t−τ) −1 ⊗5 : T . P −1 5 : T . x x p ≤ν0 x x 2 |∇ | ∇ Lx |∇ | ∇ Lx

Therefore, by Lemma 3.1, we have for r> 1,

t 1 1 1 3 (2) −3/2−3/2 −3( − 2 ) −3(1− )− E . h 3/2,1 min τ , t τ r τ r 2 dτ, KG p in W Lx k k 5 0 h i | − | h i +,− Z X which integrates to imply the stated result

4 Decomposition and scattering for the distribution function

In this section we prove Theorem 2. Denote the solution in the free transport moving frame

g(t, x, v) := h(t,x + vt, v), which satisﬁes ∂ g = E(t,x+tv) f 0(v). Therefore we have on the Fourier side (note h(t,k,η)= t − ·∇v g(t,k,η + kt)), b t [ b g(t,k,η)= h (k, η) E(τ, k) f 0(η kτ)dτ. in − · ∇v − Z0 b c b 23 (1) Consider the contribution EKG:

t (1) [ E (τ, k) f 0(η kτ)dτ KG · ∇v − Z0 ℓ b ik t [ = w ep±(k)τ A±(k) k⊗j : j h (k, 0) f 0(η kτ)dτ 0 k 2 j ∇η in · ∇v − +,− 0 | | Xj=0 X Z ℓ ± c ik Aj (k) [ = w ep±(k)t k⊗j : j h (k, 0) f 0(η kt) 0 k 2 p (k) ∇η in · ∇v − +,− ± | | Xj=0 X ℓ ± c ik Aj (k) [ w k⊗j : j h (k, 0) f 0(η) − 0 k 2 p (k) ∇η in · ∇v +,− ± | | Xj=0 X ℓ ± c ik t A (k) p±(k)τ j ⊗j j [0 w0 2 e k : ηhin(k, 0) η vf (η kτ)k dτ − k 0 p±(k) ∇ ·∇ ∇ − | | Xj=0 Z =: gKG + g2 + g3, c with hKG, h2, anddh3 definedb b analogously. Note that since hKG(t,k,η)= gKG(t,k,η + kt), [ h (t,k,η)= E˜ (t, k) df 0(η), KG KG · ∇v d where we define d 4 ± ik A (k) E˜ (t, k) := w ep±(k)t j k⊗j : j h (k, 0) . KG 0 k 2 p (k) ∇η in +,− ± | | Xj=0 X c (1) Arguing as for the EKG estimates in Theorem 1, hKG satisfies (1.7). Similarly, we define

t (2) [ h (k, η)= h (k, η)+ g (k, η)+ g (t,k,η) E[ + E (τ, k) f 0(η kτ)dτ.(4.1) LD in 2 3 − KG LD · ∇v − Z0 Thed term g is constantc in timeb and inbLp for all p 2 by the assumptionsd on the initial data. The 2 ≥ term g3 on the physical side is written in the form

t h (t, x, v)= E˜ (τ,x + τv) : v f 0(v) dτ, 3 ∇x 3 ⊗∇v Z0 ˜ (1) for a suitable E3. By straightforward variations of the arguments used to estimate EKG above, we have for any 6

t t ˜ 0 ˜ xE3(τ,x + τv) : v vf (v) p dτ . xE3(τ, ) p dτ ∇ ⊗∇ Lx,v ∇ · Lx Z0 Z0 t 1 1 −3 2 − p . hin 3+0,1 τ dτ, k kW5 h i Z0 p and hence h3 converges in Lx,v for all p> 6 as t . Due to the decay estimates in Lemma 3.3, (2) →∞ p the term in (4.1) involving EKG similarly converges in Lx,v for all p> 6.

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