DISCRETE AND CONTINUOUS Website: http://aimSciences.org DYNAMICAL SYSTEMS SERIES B Volume 10,Number4,November2008 pp. 801–822
THE STOCHASTIC PRIMITIVE EQUATIONS IN TWO SPACE DIMENSIONS WITH MULTIPLICATIVE NOISE
Nathan Glatt-Holtz and Mohammed Ziane Department of Mathematics University of Southern California Los Angeles, CA 90089, USA
(Communicated by Thomas Witelski)
Abstract. We study the two dimensional primitive equations in the presence of multiplicative stochastic forcing. We prove the existence and uniqueness of solutions in a fixed probability space. The proof is based onfinitedimen- sional approximations, anisotropic Sobolev estimates, andweakconvergence methods.
1. Introduction. The Primitive equations are a ubiquitous model in the study of geophysical fluid dynamics. They can be derived from the compressible Navier- Stokes equations by taking advantage of various properties common to geophysical flows. In particular, one uses the Boussinesq approximation,thatfluctuationsin the density of the fluid are much smaller than the mean density throughout. A scale analysis relying on the relative shallowness of the ocean at geophysical scales is employed to show that the pressure and the gravitational forces are the only relevant terms in the vertical momentum equation. This is referred to as the hydrostatic approximation. For further background and detailed physical derivations see [9]or [27], for example. The mathematical study of the Primitive equations was initiated in the early 1990’s with the work of Lions, Temam and Wang [21], [20], [19]. In these initial works the existence of global weak solutions and weak attractors were established and numerical schemes were developed. The case of local strong solutions was ad- dressed by Guill´en-Gonz´alez, Masmoudi, and Rodr´ıguez-Bellido [14]andbyHu, Temam and Ziane [15]. Also see the survey article of Temam and Ziane[30]. Re- cent breakthroughs have yielded global existence of strong solutions in three space dimensions. The case of Neumann boundary conditions was addressed by Cao and Titi [6]andlaterindependentlybyKobelkov[17]. In subsequent work of Kukavica and Ziane [18]adifferentproofwasdiscoveredwhichcoversthecaseofphysically relevant boundary conditions. For the two dimensional deterministic setting we mention theworkofPetcu, Temam, and Wirosoetisno in [28]andBresch,KazhikhovandLemoine[4]where both the cases of weak and strong solutions are considered. The 2-D primitive equations seem to be more difficult mathematically than the 2-DNavier-Stokes
2000 Mathematics Subject Classification. Primary: 76D03, 60H15; Secondary: 37L65. Key words and phrases. Primitive Equations, Geophysical Fluid Dynamics, Stochastic Partial Differential Equations, Galerkin Approximation, ExistenceandUniqueness. The second author is supported by NSF grant DMS-0505974.
801 802 NATHANGLATT-HOLTZANDMOHAMMEDZIANE equations. For instance, it is still an open problem whether weak solutions of the Primitive equations in the deterministic setting are unique. This is an easy exercise for the 2-D Navier Stokes equations. The addition of white noise driven terms to the basic governing equations for aphysicalsystemisnaturalforbothpracticalandtheoretical applications. For example, these stochastically forced terms can be used to account for numerical and empirical uncertainties and thus provide a means to studytherobustnessof abasicmodel.Specificallyinthecontextoffluids,complexphenomena related to turbulence may also be produced by stochastic perturbations. For instance, in the recent work of Mikulevicius and Rozovsky [25]suchtermsareshowntoarisefrom basic physical principals. Awidebodyofmathematicalliteratureexistsforthestochastic Navier-Stokes equations. This analytic program dates back to the early 1970’s with the work of Bensoussan and Temam [2]. For the study of well-posedness new difficulties related to compactness often arise due to the addition of a probabilistic parameter. For situations where continuous dependence on initial data remains open (for example in d =3whentheinitialdatamerelytakesvaluesinL2)ithasprovenfruitful to consider Martingale solutions. Here one constructs a probabilistic basis as part of the solution. For this context we refer the reader to the works of Cruzeiro [8], Capinski and Gatarek [7], Flandoli and Gatarek [13]andofMikuleviciusand Rozovskii [23]. On the other hand, when working in spaces where continuous dependence on the initial data can be expected, existence of solutions can sometimes be established on apreordainedprobabilityspace.Suchsolutionsareoftenreferred to as “strong” or “pathwise” solutions. In the two dimensional setting, Da Prato and Zabczyk [11]andlaterBreckner[3]aswellasMenaldiandSritharan[22]establishedthe existence of pathwise solutions where u L∞([0,T],L2), P a.s. On the other hand, Bensoussan and Frehse [1]haveestablishedlocalsolutionsin3-dfortheclass∈ − Cβ([0,T]; H2s)where3/4
2. Amodelforthestochasticprimitiveequationsintwospacedimen- sions. The two dimensional Primitive equations can be formally derived from the THE STOCHASTIC PRIMITIVE EQUATIONS IN TWO SPACE DIMENSIONS 803 full three dimensional system under the assumption of invariance with respect to the second horizontal variable y.Assuchwewillassumethattheinitialdataand the external forcing are independent of y.Byaddingasecondexternalforcingterm driven by a white noise, we arrive at the following non-linearstochasticevolution system: ∂ u ν∆u + u∂ u + w∂ u + ∂ p = f + g(u, t)W˙ (t)(1a) t − x z x ∂xu + ∂zw =0. (1b) In this formulation we have ignored the coupling with the temperature and salinity equations in order to focus our attention towards difficultiesarisingfromthenon- linear term (see [4]). This omission will be remedied in future work. The unknowns (u, w), p represent the field of the flow and the pressure respectively. The fluid fills adomain =[0,L] [0, h] (x, z). Note that p does not depend on the vertical variable z.M × − % We partition the boundary into the top Γ = z =0 ,thebottomΓ = z = h i { } b { − } and the sides Γs = x =0 x = L .Regardingtheboundaryconditions,we assume the Dirichlet{ condition}∪{u =0on} Γ,whileonΓ Γ we posit the free s i ∪ b boundary condition ∂zu =0,w =0.Wefurthersupposewithnolossofgenerality that: 0 0 0 fdz=0, gdz=0, udz =0. (2) !−h !−h !−h Due to (1b)wehavethat1: z w(x, z)= ∂ u(x, z˜)dz.˜ (3) − x !−h The stochastic term can be written in the expansion:
g(u, t)W˙ (t)= gk(u, t)β˙k(t). (4) "k The β˙k are the formal time derivatives of a collection of independent standard Brownian motions βk relative to some ambient, filtered, right continuous, probabil- ity space (Ω, , ( t)t≥0, P). The series converges in the appropriate function spaces and is subjectF toF uniform Lipschitz conditions in u.Allofthiswillbeformulated rigorously below.
3. Definitions. We will be working on the Hilbert spaces: 0 H = v L2( ): vdz=0 (5) ∈ M # !−h $ and 0 V = v H1( ): vdz=0,v =0onΓ . (6) ∈ M s # !−h $ These spaces are endowed with the L2 and H1 norms which we respectively denote by and .Weshallalsoneedtheintermediatespace: |·| '·' X = v H : ∂ v H (7) { ∈ z ∈ }
1In the geophysical literature w is often referred as a diagnostic variable as its value is com- pletely determined by u 804 NATHANGLATT-HOLTZANDMOHAMMEDZIANE with the norm u =(u 2 + ∂ u 2)1/2.TakeV $ to be the dual of V with the | |X | | | z | pairing notated by , .TheLerayoperatorPH is the orthogonal projection of L2( )ontoH.Theactionofthisoperatorisgivenexplicitlyby:(· ·) M 1 0 P v = v vdz. (8) H − h !−h It will also be useful to consider: 0 = v C∞( ¯ ): vdz=0,v =0onΓ ,∂ v =0onΓ Γ (9) V ∈ M s z b ∪ i # !−h $ which is a dense subset of H, X and V . We would now like to make (1)preciseasanequationinV $.Tothisend,we define a Stokes-type operator A as a bounded map from V to V $ via: v, Au =((v, u)). ( ) A can be extended to an unbounded operator from H to H according to Au = P ∆u with the domain: − H 0 D(A)= v H2( ): vdz=0,v =0onΓ ,∂ v =0onΓ Γ . (10) ∈ M s z b ∪ i # !−h $ By applying the theory of symmetric compact operators for A−1,onecanprove the existence of an orthonormal basis ek for H of eigenfunctions of A.Herethe associated eigenvalues λ form an unbounded,{ } increasing sequence. Define: { k} H =span e ,...,e n { 1 n} and take P to be the projection from H onto this space. Let Q = I P . n n − n Remark 1. For u D(A): ∈ 0 0 0 z=0 ∂xxudz = ∂xx udz =0, ∂zzudz = ∂zu]z=−h =0. !−h !−h !−h Thus, as in the case of periodic boundary conditions, we have P ∂ = ∂ − H xx − xx and PH ∂zz = ∂zz and therefore that PH ∆= ∆onD(A). The eigenvalue problem− for A reduces− to: − − ∆u = λu 0 udz =0 !−h u(0,z)=0=u(L, z),∂u(x, h)=0=∂ u(x, 0). z − z As such, the eigenfunctions and associated eigenvalues can be explicitly identified: 2 k πx k πz k2 k2 sin( 1 )cos( 2 ) , π2 1 + 2 . √hL L h L2 h2 # $k1,k2≥1 # % &$k1,k2≥1 This has the useful consequence that when j = l: + ∂ e ,e =0 (11) ( zz j l) and hence: P ( ∂ v)= ∂ v whenever v H . (12) n − zz − zz ∈ n THE STOCHASTIC PRIMITIVE EQUATIONS IN TWO SPACE DIMENSIONS 805
Next we address the nonlinear term. In accordance with (3)wetake: z (v)(x, z):= ∂ v(x, z˜)dzv˜ W − x ∈V !−h and let: B(u, v):=P (u∂ v + (u)∂ v). H x W z Below it will sometimes be convenient to denote B(u):=B(u, u). One would like to establish that B is a well defined and continuous mapping from V V V $ according to: × → B(u, v),φ = b(u, v, φ) ( ) where the associated trilinear form is given by:
b(u, v, φ):= (u∂ vφ (u)∂ vφ) d := b (u, v, φ) b (u, v, φ). x −W z M 1 − 2 !M This and more is contained in the following lemma: Lemma 3.1. (i) b is a continuous linear form on V V V and: × × b(u, v, φ) C u 1/2 u 1/2 v φ 1/2 φ 1/2 + ∂ u ∂ v φ 1/2 φ 1/2 (13) | |≤ | | ' ' ' '| | ' ' | x || z || | ' ' for any u, v, φ' V ( (ii) b satisfies the cancellation∈ property b(u, v, v)=0 (iii) b is also a continuous form on D(A) D(A) H (iv) For u D(A) we have the additional× cancellation× property: ∈ B(u),∂ u =0 ( zz ) (v) Moreover, for any '>0: 2 1/2 3/2 1/2 1/2 B(u),∂xxu C( ∂xu ∂xu + ∂xu ∂xu ∂zu ∂zu ) |( )|≤ | | ' ' | | ' ' | | ' ' (14) ' ∂ u 2 + C(')( ∂ u 4 + ∂ u 2 ∂ u 2 ∂ u 2) ≤ ' x ' | x | | x | | z | ' z ' (vi) If u, v V and ∂ v V then: ∈ z ∈ 1/2 1/2 B(u, v) ! C(( u + ∂ u ) ∂ v + u ∂ v + u ∂ v ∂ v )(15) | |V ≤ | | | z | | x | | |' z ' | || z | ' z ' Proof. Fix u, v, φ .Thefirsttermb admits the classical 2-D estimate: ∈V 1 b (u, v, φ) C u 1/2 u 1/2 v φ 1/2 φ 1/2. | 1 |≤ | | ' ' ' '| | ' ' The second term is estimated anisotropically: b (u, v, φ) | 2 | L z L sup ∂ udz¯ ∂ vφ dz dx ≤ x | z | !0 )z∈[−h,0] #!−h $ !0 * L 0 0 0 1/2 C ∂ u 2 dz ∂ v 2 dz φ 2 dz dx ≤ | x | · | z | · | | !0 %!−h !−h !−h & 0 1/2 L 0 0 1/2 C sup φ 2 dz ∂ u 2 dz ∂ v 2 dz dx ≤ | | | x | · | z | x∈[0,L] %!−h & !0 %!−h !−h & C ∂ u ∂ v φ 1/2 φ 1/2. ≤ | x || z || | ' ' 806 NATHANGLATT-HOLTZANDMOHAMMEDZIANE
For the final inequality we make use of the boundary conditions:
0 x 0 sup φ 2 dz =sup ∂ φ2 dz dx 2 φ φ . | | x ≤ | |' ' x∈[0,L] #!−h $ x∈[0,L] #!0 !−h $ To establish the cancellation property in (ii): 1 1 z b(u, v, v)= ∂ uv2 d ∂ udz˜ ∂ (v2) d −2 x M−2 x z M !M !M %!−h & 1 1 = ∂ uv2 d + ∂ uv2 d =0. −2 x M 2 x M !M !M Property (iii) is a direct application of H¨older’s Inequality and Sobolev embed- ding inequalities and is omitted. For (iv), noting that P ∂ = ∂ on D(A): − H zz − zz 1 B(u),∂ u = ( ∂ u(∂ u)2 u∂ u∂ u + ∂ u(∂ u)2) d =0. ( zz ) − x z − xz z 2 x z M !M The inequality given in (v) is addressed by estimating:
1 3 B(u),∂ u ∂ u 3 + ( (u)∂ u∂ u + ∂ (u)∂ u∂ u) d |( xx )|≤ 2| x |L (M) W zx x xW z x M +!M + + + 3 ∂ u 3 + + ∂ (u)∂ u∂ u d . + ≤| x |L (M) + | xW z x | M + !M For the first term above we use the Sobolev embedding H1/3 L3.Forthesecond term we have H1/2 L4 which justifies the estimate: ⊂ ⊂
∂ (u)∂ u∂ u d C ∂ u ∂ u 4 ∂ u 4 | xW z x | M≤ | xx || z |L | x |L !M C ∂ u 1/2 ∂ u 3/2 ∂ u 1/2 ∂ u 1/2. ≤ | x | ' x ' | z | ' z ' The second inequality is just an application of '-Young. For the final item (vi) fix φ V .Inthiscaseweestimateb anisotropically: ∈ 1 L 0 b (u, v, φ)= u∂ vφd 1 x M !0 !−h L 0 0 0 1/2 (16) C ( ∂ u + u )2dz ∂ v 2dz φ 2dz dx ≤ | z | | | · | x | · | | !0 %!−h !−h !−h & C( ∂ u + u ) ∂ v φ 1/2 φ 1/2. ≤ | z | | | | x || | ' ' For b2,byintegratingbypartsin,x we find: L 0 z b (u, v, φ)= ∂ udz¯ ∂ vφd 2 x z M !0 !−h %!−h & L 0 z = udz¯ ∂zxvφd − M (17) !0 !−h %!−h & L 0 z udz¯ ∂ v∂ φd − z x M !0 !−h %!−h & :=T1(u, v, φ)+T2(u, v, φ). THE STOCHASTIC PRIMITIVE EQUATIONS IN TWO SPACE DIMENSIONS 807
For T1 the anisotropic estimates yield: L 0 0 0 1/2 T (u, v, φ) C u 2dz ∂ v 2dz φ 2dz dx | 1 |≤ | | · | zx | · | | (18) !0 %!−h !−h !−h & u ∂ v φ 1/2 φ 1/2. ≤| |' z '| | ' ' ∞ The estimate for T2 is similar except that we make the Lx estimate on the middle term: L 0 0 0 1/2 T (u, v, φ) C u 2dz ∂ v 2dz ∂ φ 2dz dx | 2 |≤ | | · | z | · | x | (19) !0 %!−h !−h !−h & C u ∂ v 1/2 ∂ v 1/2 φ . ≤ | || z | ' z ' ' '
It remains to examine the stochastically forced term g = gk k≥1 in order to make precise the Lipschitz condition alluded to above. For this purpose{ } we introduce some notation. Suppose U is any (separable) Hilbert space. One defines (2(U)viathe inner product:
(h, g)"2(U) = (hk,gk)U . "k For any normed space Y ,wesaythatg : Y [0,T] Ω (2(U)isuniformly × × → Lipschitz with constant KY if:
g(x, t, ω) g(y,t,ω) 2 K x y for x, y Y (20) | − |" (U) ≤ Y | − |Y ∈ and g(x, t, ω) 2 K (1 + x )(21) | |" (U) ≤ Y | |Y where KY is independent of t and ω.Wedenotethecollectionofallsuchmap- 2 pings Lipu(Y,( (U)). For the analysis below we will frequently assume that g 2 2 ∈ Lipu(H,( (H)) Lipu(X,( (X)). It is worth noting at this juncture that the con- dition imposed∩ on g is not overly restrictive by considering some examples where the above conditions are satisfied: Example 3.2.
(Independently Forced Modes)Suppose(κk(t, ω)) is any sequence uniformly • bounded in L∞([0,T] Ω). We force the modes independently defining: × gk(v, t, ω)=κk(t, ω)(v, ek)ek.
In this case the Lipschitz constants can be taken to be KH = KX = KV = sup κ (t, ω) . ω,k,t | k | (Uniform Forcing)Givenauniformlysquaresummablesequenceak(t, ω)we • can take: gk(v, t, ω)=ak(t, ω)v 2 1/2 with KH = KX = KV =(supt,ω k ak(t, ω) ) as the Lipschitz constants. (Additive Noise)Wecanalsoincludethecasewhenthenoisetermdoesnot • depend on the solution: ,
gk(v, t, ω)=gk(t, ω)
2 1/2 Here the uniform constants can be taken to be KU := supt,ω k gk(t, ω) U for U = H, X, V as desired. | | -, . 808 NATHANGLATT-HOLTZANDMOHAMMEDZIANE
With the above framework in place we now give a variational definition for so- lutions of the system (1). Note that weak refers to the spatial-temporal regularity of the solutions. Strong refers to the fact that the probabilistic basis is given in advance (see Remark 2 below).
Definition 3.3 (Weak-Strong Solutions). Suppose that (Ω, P, , ( t)t≥0, (βk)) is a F F p fixed stochastic basis, T>0andp [2, ]. For the data assume that u0 L (Ω; H) ∈ ∞ ∈ 2 and is 0-measurable. We suppose that f and g are respectively H and ( (H) valued,F predictable processes with: f Lp(Ω,L2(0,T; V $)),gLip (H,(2(H)). (22) ∈ ∈ u We say that an t adapted process u is a weak-strong solution to the stochastically forced primitiveF equation if: u C([0,T]; H)a.s. u Lp(Ω; L∞([0,T]; H) L2([0,T]; V )) (23) ∈ ∈ ∩ and satisfies: ∞
d u(t),v + νAu + B(u),v dt = f,v dt + gk(u, t),v dβk ( ) ( ) ( ) ( ) (24) k"=1 u(0),v = u ,v ( ) ( 0 ) for any v V . ∈ Several remarks are in order regarding this definition: Remark 2. Note that, as in the theory of the Navier-Stokes equations, the pressure dis- • appears in the variational formulation. Suppose that ∂xp in (1)isintegrable and does not depend on the vertical variable z.Then: L 0 L 0 ∂xpv dz dx = ∂xp vdz dx =0 (25) !0 !−h !0 %!−h & for every v V . For the probabilistically∈ ’strong’ solutions we consider, the stochastic basis • is given in advance. Such solutions can be understood pathwise. This is in contrast to the theory of Martingale solutions considered for many non linear systems where the underlying probability space is constructed as part of the solution. See [10]chapter8or[24]. One has to check that each of the terms given in (24)arewelldefined.In • particular, the stochastic terms deserve special attention. Recall that the collection 2(0,T)ofcontinuoussquareintegrablemartingalesisaBanach space underM the norm: 1/2 2 X 2 = E sup X(t) . ' 'M | | % 0≤t≤T & Applying standard Martingale inequalities and making use oftheuniform Lipschitz assumption one establishes that: g (u, t),v dβ for all v V ( k ) k ∈ "k converges in this space. See [16]or[10]forfurtherdetailsonthegeneral construction of stochastic integrals. THE STOCHASTIC PRIMITIVE EQUATIONS IN TWO SPACE DIMENSIONS 809
4. The Galerkin systems and a priori estimates. We now introduce the Galerkin systems associated to the original equation and establish some uniform aprioriestimates.
(n) Definition 4.1 (The Galerkin System). An adapted process u in C(0,T; Hn)is asolutiontotheGalerkin System of Order n if for any v H : ∈ n ∞ (n) (n) (n) (n) d u ,v + νAu + B(u ),v dt = f,v dt + gk(u ,t),v dβk ( ) ( ) ( ) ( ) (26) k"=1 u(n)(0),v = u ,v . ( ) ( 0 ) These systems also be written as equations in Hn: ∞ (n) (n) (n) (n) du +(νAu + PnB(u ))dt = Pnfdt+ Pngk(u ,t)dβk (27) k"=1 (n) u (0) = Pnu0.
We note that the second formulation (27)allowsonetotreatu(n) as a process in Rn.AssuchonecanapplythefinitedimensionalItˆocalculustothe Galerkin systems above. We next establish some uniform estimates on u(n) (independent of n). To simplify notation we drop the (n)superscriptfortheremainderofthesection. Lemma 4.2 (A Priori Estimates). (i) Assume that u is the solution of the Galerkin System of Order n.Suppose that p 2 and: ≥ g Lip (H,(2(H)),fLp(Ω; L2(0,T; V $)),u Lp(Ω,H)(28) ∈ u ∈ 0 ∈ then: T E sup u p + u 2 u p−2dt C (29) | | ' ' | | ≤ W )t∈[0,T ] !0 * and: p/2 T E u 2dt C (30) ' ' ≤ W )!0 * for an appropriate constant
p C = C (p, ν,λ , E u ,T, f p 2 ! ,K ), W W 1 | 0| | |L (Ω;L (0,T ;V )) H that does not depend on n. (ii) Given the additional assumptions on the data: g Lip (X,(2(X)),∂f Lp(Ω; L2(0,T; V $)),∂u Lp(Ω; H)(31) ∈ u z ∈ z 0 ∈ then: T E sup ∂ u p + ∂ u 2 ∂ u p−2dt C (32) | z | ' z ' | z | ≤ I )t∈[0,T ] !0 * p where CI = CI (p, ν,λ1, E ∂zu0 ,T, ∂zf Lp(Ω;L2(0,T ;V !)),KX),independentof n. | | | | 810 NATHANGLATT-HOLTZANDMOHAMMEDZIANE
(iii) Finally assume that in addition to (28): g Lip (V,(2(V )),f Lp(Ω; L2(0,T; H)),u Lp(Ω,V). (33) ∈ u ∈ 0 ∈ If τ is a stopping time taking values in [0,T],andM apositiveconstantso
that: τ ( u 2 + ∂ u 2 ∂ u 2)dt M (34) ' ' | z | ' z ' ≤ !0 then: τ E sup u p + Au 2 u p−2dt C eCS M (35) ' ' | | ' ' ≤ S )t∈[0,τ] !0 * p where C = C (p, ν, E u ,T, f p 2 ,K ). S S ' 0' | |L (Ω;L (0,T ;H)) V Proof. By applying Itˆo’s formula one finds a differential for u 2 and then for eφ u p: | | | | deφ u p + pνeφ u 2 u p−2dt | | ' ' | | ∞ p =peφ f,u u p−2dt + eφ P g (u, t) 2 u p−2dt ( )| | 2 | n k | | | k=1 ∞ " p(p 2) (36) + − eφ g (u, t),u 2 u p−4dt 2 ( k ) | | k=1 ∞ " + peφ g (u, t),u u p−2dβ + φ$ u peφdt. ( k )| | k | | k"=1 Here, φ is a non-positive element in C1(0,T)tobedeterminedbelow.Thisfunction T φ p will be used to cancel offterms involving 0 e u dt.Forthedeterministicexternal forcing term we estimate: | | / T peφ f,u u p−2dt ( )| | !0 T T φ 2 p−2 νp φ 2 p−2 C(ν, p, λ ) e f ! u dt + e u u dt ≤ 1 | |V | | 2 ' ' | | !0 !0 T (p−2)φ/p p−2 2 C(ν, p, λ ) sup (e u ) f ! dt ≤ 1 | | | |V )t∈[0,T ] * !0 (37) νp T + eφ u 2 u p−2 dt 2 ' ' | | !0 1 pν T sup (eφ u p)+ eφ u 2 u p−2 dt ≤ 4 | | 2 ' ' | | t∈[0,T ] !0 p + C(ν, p, λ ) f 2 ! . 1 | |L (0,T ;V ) Taking advantage of the Lipschitz condition assumed for g: ∞ T eφ g (u, t) 2 u p−2dt | k | | | k=1 !0 " T K eφ(1 + u 2) u p−2dt (38) ≤ H | | | | !0 1 T sup (eφ u p)+C(p, K ,T) 1+ eφ u p dt . ≤ 4 | | H | | t∈[0,T ] ) !0 * THE STOCHASTIC PRIMITIVE EQUATIONS IN TWO SPACE DIMENSIONS 811
Applying the above estimates, absorbing terms and rearranging one deduces:
T sup (eφ u p)+pν eφ u 2 u p−2dt | | ' ' | | t∈[0,T ] !0 p p C(p, ν,λ ) u + f 2 ! +1 ≤ 1 | 0| | |L (0,T ;V ) ' T T( (39) + C (p, K ,T) eφ u pdt +2 φ$eφ u pdt 1 H | | | | !0 !0 t ∞ φ p−2 +2p sup e gk(u, t),u u dβk . t∈[0,T ] + 0 ( )| | + +! k=1 + + " + For the final term involving the+ Itˆointegral we apply the Burkholder-Davis-Gundy+ + + (BDG) inequality (see [16]). This yields the following:
∞ t φ p−2 2pE sup e gk(u, s),u u dβk t∈[0,T ] + 0 ( )| | + +"k=1 ! + + + 1/2 +T ∞ + + 2φ 2 2(p−2) + C(p)E e gk(u, t),u u dt ≤ ) 0 ( ) | | * ! k"=1 (40) 1/2 T C(K ,p)E e2φ(1 + u 2) u 2(p−1)dt ≤ H | | | | )!0 * 1 T E sup (eφ u p) + C (p, T, K )E 1+ eφ u pdt ≤ 4 | | 2 H | | )t∈[0,T ] * ) !0 * Note that the constants in estimates above are independent of φ.Setφ(t)= (C + − 1 C2)t,whereC1,C2 are the constants arising in (39)and(40)respectively.This choice, used in conjunction with the preceding estimates implies (29). From (36)withp =2andφ =0,onededucesthatforanyr>2:
r/2 T 2 r r r E u dt CE 1+ u0 + f L2(0,T ;V !) +sup u ) 0 ' ' * ≤ ) | | | | t∈[0,T ] | | * ! (41) r/2 t ∞ + CE sup gk(u, t),u dβk . t∈[0,T ] + 0 ( ) + +! k=1 + + " + For the second term we again employ BDG+ and infer: + + + r/2 ∞ t CE sup gk(u, s),u dβk t∈[0,T ]+ 0 ( ) + +"k=1 ! + + r/4 + T +∞ + + 2 + CE gk(u, t),u dt (42) ≤ ) 0 ( ) * ! "k=1 r/2 1 T CE sup u r +1 + E u 2 . ≤ | | 2 ' ' )t∈[0,T ] * )!0 * Absorbing terms and applying estimates already obtained in (29)yields(30). 812 NATHANGLATT-HOLTZANDMOHAMMEDZIANE
To obtain the desired estimates in (ii) we make use of the commutativity of Pn and ∂zz on D(A)(seeRemark1). In particular, note that vertical cancellation property established in Lemma 3.1 along with this commutativity means that: P B(u),∂ u = B(u),∂ u =0. ( n zz ) ( zz ) 2 Thus, when we apply Itˆo’s formula for ∂zv ,thenonlineartermdisappearsas above. Bootstrapping to p 2withasecondapplicationofItˆowearriveatthe| | differential: ≥ d ∂ u p + pν ∂ u 2 ∂ u p−2dt | z | ' z ' | z | =p f,∂ u ∂ u p−2dt ( zz )| z | ∞ p + ∂ P g (u, t) 2 ∂ u p−2dt 2 | z n k | | z | k=1 " ∞ (43) p(p 2) + − g (u, t), ∂ u 2 ∂ u p−4dt 2 ( k − zz ) | z | k=1 ∞ " + p g (u, t), ∂ u ∂ u p−2dβ . ( k − zz )| z | k "k=1 We bound the first term on the right hand side of (43)asin(37). For the second term we utilize the Lipschitz condition imposed in (31)andtheuniformbound(30) established in the previous case:
T ∞ p 2 p−2 E ∂zPngk(u, t) ∂zu dt 2 0 | | | | ! k"=1 T ∞ 2 p−2 C(p)E ∂zgk(u, t) ∂zu dt ≤ 0 | | | | ! k=1 (44) "T C(K ,p)E ( u 2 +1)∂ u p−2dt ≤ X ' ' | z | !0 p/2 T 1 C(K ,p)E ( u 2 +1)dt + E sup ∂ u p . ≤ X ' ' 8 | z | )!0 * )t∈[0,T ] * The third term in (43)isestimatedinthesamemanner.Thefinaltermishandled using BDG:
∞ t p−2 pE sup gk(u, s), ∂zzu ∂zu dβk t∈[0,T ] + 0 ( − )| | + +k"=1 ! + + + 1/2 + T ∞ + + 2 2(p−2) + C(p)E ∂zgk(u, t),∂zu ∂zu dt ≤ ) 0 ( ) | | * ! "k=1 1/2 (45) T C(p, K )E (1 + u 2 ) ∂ u 2(p−1)dt ≤ X | |X | z | )!0 * p/2 1 T E sup ∂ u p + C(p, K )E (1 + u 2)dt . ≤ 8 | z | X ' ' )t∈[0,T ] * )!0 * THE STOCHASTIC PRIMITIVE EQUATIONS IN TWO SPACE DIMENSIONS 813
For the (iii), we do not have cancellation in the nonlinear term. Here the differ- ential reads: deφ u p + pνeφ Au 2 u p−2dt ' ' | | ' ' =peφ f B(u),Au u p−2dt ( − )' ' ∞ p + eφ P g (u, t) 2 u p−2dt 2 |∇ n k | ' ' k=1 " ∞ (46) p(p 2) + − eφ ( g (u, t), u)2 u p−4dt 2 ∇ k ∇ ' ' k=1 ∞ " + peφ ( g (u, t), u) u p−2dβ + φ$ u peφdt. ∇ k ∇ ' ' k ' ' k"=1 Once again φ will be a non-positive function chosen further on to cancel offterms. Integrating up to t τ,thentakingasupremumint and estimating terms as previously reveals: ∧
τ E sup ( u peφ)+pν eφ Au 2 u p−2dt ' ' | | ' ' )t∈[0,τ] !0 * p p CE( u + f 2 ) ≤ ' 0' | |L (0,T ;H) τ (47) φ p−2 +2E e (B(u),∂xxu) u dt 0 | |' ' ! τ τ + C (p, K )E eφ u pdt +2E φ$eφ u pdt. 3 V ' ' ' ' !0 !0 We apply Lemma 3.1,(v)with' = pν/4inferring: τ eφ (B(u),∂ u) u p−2dt | xx |' ' !0 pν τ eφ Au 2 u p−2dt (48) ≤ 4 0 | | ' ' ! τ + C (ν, p) eφ u p( u 2 + ∂ u 2 ∂ u 2)dt. 4 ' ' ' ' | z | ' z ' !0 Taking the previous estimates into account we set: t φ(t)= C ( u 2 + ∂ u 2 ∂ u 2) tC − 4 ' ' | z | ' z ' − 3 !0 where C3 and C4 are the constants appearing in (47)and(48)respectively.Given the assumption (34), eφ(τ) C(ν, p, T )e−CM.Withthiswecanapply(48)to(47) and conclude the final bound≥ (35).
Remark 3. If one could find a subsequence nk,astoppingtimeτ with P(τ>0) = 1 and a positive constant M such that: τ sup ( u(nk) 2 + ∂ u(nk) 2 ∂ u(nk) 2)ds M a.s (49) ' ' | z | ' z ' ≤ nk !0 then the existence of solutions taking values in Lp(Ω; L2([0,T]; D(A)) L∞([0,T]; V )) would follow. ∩ 814 NATHANGLATT-HOLTZANDMOHAMMEDZIANE
We conclude this section with some comments concerning the existence and uniqueness of solutions to the Galerkin systems. Regarding existence one uses that B is locally Lipschitz in conjunction with the a priori bounds established above. These properties taken together allows one to establish global existence on any compact time interval via Picard iteration methods. See [12]fordetailedproofs. Uniqueness is established as below for the full infinite dimensional system.
5. Existence and uniqueness of solutions. With the uniform estimates on the solutions of the Galerkin systems in hand, we proceed to identify a (weak) limit u.Thiselementisshowntosatisfyastochasticdifferential(54)withunknown terms corresponding to the nonlinear portions of the equation. Next we prove a comparison lemma that establishes a sufficient condition (67)fortheidentification of the unknown portions of the differential. This lemma, in conjunction with some further estimates, provides the final step in the main theoremconcerningexistence below. We will assume the following conditions on the data throughout this section: p 2 $ 2 2 f,∂zf L (Ω; L (0,T; V )),g Lip (H,( (H)) Lip (X,( (X)) ∈ ∈ u ∩ u (50) u ,∂ u Lp(Ω; H). 0 z 0 ∈ (n) Here p 4sothatthesequencePnB(u )willhaveaweaklyconvergentsubse- quence.≥ These assumptions may be weakened slightly for several of the lemmas leading up to the main result. In particular the limit system (54)canbeobtained by merely assuming: u Lp(Ω; H),f Lp(Ω; L2(0,T; V $)),gLip (H,(2(H)). (51) 0 ∈ ∈ ∈ u Lemma 5.1 (Limit System). There exists adapted processes u, B∗ and g∗ with the regularity: u Lp(Ω,L2(0,T; V ) L∞(0,T; H)) ∈ p 2 ∩ ∞ ∂zu L (Ω,L (0,T; V ) L (0,T; H)) (52) ∈ u C(0,T; H)∩a.s. ∈ and: B∗ L2(Ω; L2(0,T; V $)),g∗ L2(Ω; L2(0,T; (2(H))) (53) ∈ ∈ such that u, B∗ and g∗ satisfy: ∞ ∗ ∗ d u, v + νAu + B ,v dt = f,v dt + gk(t),v dβk ( ) ( ) ( ) ( ) (54) "k=1 u(0),v = u ,v ( ) ( 0 ) for any test function v V . ∈ Remark 4. We use the following elementary facts regarding weakly convergent sequences in the proof below. (i) Suppose B ,B are Banach spaces and that L : B B is a bounded linear 1 2 1 → 2 mapping. If xn ,xin B1 then Lxn ,Lxin B2. (ii) For p [1, )take: ∈ ∞ t L (w)(t)= wds w Lp(Ω [0,T]). 1 ∈ × !0 If x ,xin Lp(Ω (0,T)) and then L (x ) ,L (x)inthesamespace. n × 1 n 1 THE STOCHASTIC PRIMITIVE EQUATIONS IN TWO SPACE DIMENSIONS 815
(iii) Take: t k L2(v)(t)= v dβk 0 "k ! k 2 2 2 for v = v L (Ω,L (0,T; ( )). Given that vn ,vin this space then { }∈ 2 2 L2(vn) ,L2(v)inL (Ω; L (0,T)) (Proof- Lemma 5.1). Applying the estimate (29)withAlaoglu’stheoremwede- duce the existence of a subsequence of Galerkin elements u(n) and an element u Lp(Ω,L2(0,T; V ) L∞(0,T; H)) so that: ∈ ∩ u(n) ,u in Lp(Ω; L2(0,T; V )) (55) and: u(n) ,∗ u in Lp(Ω; L∞(0,T; H)). (56)
To deduce the desired regularity for ∂zu we apply the uniform estimates given in p 2 ∞ (32)andthinoursubsequenceagainsothat∂zu L (Ω,L (0,T; V ) L (0,T; H)) with: ∈ ∩ (n) p 2 ∂zu ,∂zu in L (Ω,L (0,T; V )) (57) as well as: (n) ∗ p ∞ ∂zu , ∂zu in L (Ω,L (0,T; H)). (58) By an application of (13): T (n) 2 E P B(u ) ! dt | n |V !0 2 T (59) CE sup ( u(n) 4 + ∂ u(n) 4)+ u(n) 2dt . ≤ | | | z | ' ' t∈[0,T ] )!0 * The later quantity is uniformly bounded as a consequence of (30)and(32). Thinning u(n) as necessary we find an element B∗ L2(Ω; L2(0,T; V $)) so that:2 ∈ (n) ∗ 2 2 $ PnB(u ) ,B in L (Ω; L (0,T; V )). (60) Finally the Lipschitz assumption along with the Poincar´einequality imply:
T ∞ T (n) 2 (n) 2 E Pngk(t, u ) dt CE ( u +1)dt . (61) 0 | | ≤ ) 0 ' ' * ! k"=1 ! This gives the uniform bounds needed to infer g∗ L2(Ω; L2(0,T; (2(H))) and implies that: ∈ (n) ∗ 2 2 2 Png(u ) ,g in L (Ω; L (0,T; ( (H))). (62) We next establish that u satisfies (54). To this end, fix any measurable set E Ω [0,T]andv V .Byapplying(55)andusingthateachu(n) is a solution ⊂ × ∈
2Given only the weaker assumptions on the initial data (51)onecanstillshowthatB∗ ∈ L4/3(Ω; L2(0,T; V ")) with the estimate:
T T 3/2 E P B u(n) 4/3dt CE u(n) 2 u(n) 2dt . | n ( ))|V ! ≤ sup | | + # # !0 4t∈[0,T ] %!0 & 5 816 NATHANGLATT-HOLTZANDMOHAMMEDZIANE to an associated Galerkin system we infer: T T (n) E χE(u, v)dt =limE χE(u ,v)dt n→∞ !0 !0 T =lim E χE(Pnu0,v)dt n→∞ % !0 T t E χ νAu(n) + P B(u(n)) P f,v ds dt − E ( n − n ) !0 6!0 7 T ∞ t (63) + E χ P g (u(n),s),v dβ (s) dt E ( n k ) k !0 4k=1 !0 5 & T " t =E χ (u ,v) νAu + B∗ f,v ds dt E 0 − ( − ) !0 6 !0 7 T ∞ t ∗ + E χE gk(s),v dβk(s) dt 0 4 0 ( ) 5 ! "k=1 ! For the final equality above we use Remark 4.Observethat: P B(u(n)),v , B∗,v in L2(Ω [0,T]) (64) ( n ) ( ) × Au(u),v , Au, v in L2(Ω [0,T]) (65) ( ) ( ) × along with: (n) ∗ 2 2 2 [(Pngk(u ),v)]k≥1 , [(gk,v)]k≥1 in L (Ω; L ([0,T],( )). (66) Since E is arbitrary in (63), the equality (54)followsuptoasetofmeasurezero. Referring then to results in [29], chapter 2 we find that u has modification so that u C([0,T]; H)a.s. ∈ With a candidate solution in hand, it remains to show that B(u)=B∗ and g∗ = g(u). A sufficient condition for these equalities, at least up to a stopping time, is captured in the following: Lemma 5.2. If 0 τ T is any stopping time such that: ≤ ≤ τ E u u(n) 2dt 0(67) ' − ' → !0 then, λ P-a.e.3: × ∗ B(u(t))11t≤τ = B (t)11t≤τ (68) and for every k: ∗ gk(u, t)11t≤τ = gk(t)11t≤τ . (69) Proof. Let E,ameasurablesubsetofΩ [0,T]andv V be given. It is sufficient to show that: × ∈ T ∗ E χ (B(u(t)) B (t))11 ≤ ,v dt =0 (70) E( − t τ ) !0 and that for any k: T ∗ E χ (g (u(t),t) g (t))11 ≤ ,v dt =0 (71) E( k − k t τ ) !0 3On [0,T]weusetheLebesguemeasureλ. THE STOCHASTIC PRIMITIVE EQUATIONS IN TWO SPACE DIMENSIONS 817 where χE is the characteristic function of E.Duetotheweakconvergence(60) established above: T (n) ∗ E χ (P B(u (t)) B (t))11 ≤ ,v dt 0asn . (72) E( n − t τ ) → →∞ !0 Using that: (n) B(u) PnB(u ),v |( − )| (73) C u 2 Q v + C( u u u(n) + u(n) u u(n) ) v ≤ ' ' ' n ' ' '' − ' ' '' − ' ' ' we estimate: T (n) E χ (P B(u ,t) B(u, t))11 ≤ ,v ds E( n − t τ ) + !0 + + + + T + +C (I P )v E u 2ds + ≤ ' − n ' ' ' (74) 4!0 5 1/2 T τ 1/2 + C v E ( u + u(n) )2ds E u u(n) 2ds ' ' ' ' ' ' ' − ' 4 !0 5 6 !0 7 which, also vanishes as n .Theseestimatesimplythefirstitem. For the second item (71→∞)weagainexploittheweakconvergencein(62)toinfer that for every k: T (n) ∗ E χ P (g (u ,t) g (t))11 ≤ ,v dt 0asn . (75) E( n k − k t τ ) → →∞ !0 Also: T T E χE Qng(u, t)11t≤τ ,v dt KH Qnv E (1 + u )dt 0asn . (76) + 0 ( ) + ≤ | | 0 | | → →∞ + ! + ! + + Finally+ by the assumption (67),+ the Lipschitz continuity of g and the Poincar´e + + Inequality: T (n) E χ P (g (u ,t) g (u, t))11 ≤ ,v dt E( n k − k t τ ) !0 (77) τ 1/2 C(K ,T) v E u(n) u 2 dt 0 ≤ H | | | − | → % !0 & as n .Combining(75), (76)and(77)providesthesecondequality(69). →∞ With this lemma in mind we compare u to the sequence u(n) of Galerkin estimates to show that (67)issatisfiedforasequenceofstoppingtimesτn.Sinceweareable to choose τn so that τn T a.s., this is sufficient to deduce the existence result. Here we are adapting techniques↑ used in [3]. Theorem 5.3 (Existence of Weak-Strong Solutions). Suppose that p 4 and p 2 $ 2 2 ≥ f,∂zf L (Ω; L (0,T; V )), g Lipu(H,( (H)) Lipu(X,( (X)) and u0,∂zu0 Lp(Ω; H∈).Thenthereexistsan∈H continuous weak-strong∩ solution u with the ad-∈ ditional regularity: ∂ u Lp(Ω,L2(0,T; V ) L∞(0,T; H)). (78) z ∈ ∩ 818 NATHANGLATT-HOLTZANDMOHAMMEDZIANE
Proof. The needed regularity conditions for u follow from Lemma 5.1.Moreoverwe found that u satisfies the differential (54). Thus it remains only to show that for almost every t, ω:
∗ ∗ B(u)=B gk(u)=gk. (79)
To this end, for R>0, define:
r 2 2 2 2 τR =inf sup u +sup ∂zu + u + ∂zu ds > R T. (80) r∈[0,T ] | | | | ' ' ' ' ∧ 8s∈[0,r] s∈[0,r] !0 9
Notice that τR is increasing as a function of R>0andthatmoreover:
T P(τ T ) P sup u 2 +sup ∂ u 2 + u 2 + ∂ u 2 ds R . (81) R ≤ ≤ | | | z | ' ' ' z ' ≥ )s∈[0,T ] s∈[0,T ] !0 *
Thus, as a consequence of (52) τ T ,almostsurely. R → We now fix R and show that τR satisfies (67). Since by the dominated convergence theorem:
τR 2 lim u Pnu dt =0 (82) n→∞ ' − ' !0
(n) it is sufficient to compare Pnu and u .Thedifferenceofthesetermssatisfiesthe differential:
d(P u u(n))+[νA(P u u(n))+P B∗ P B(u(n))]dt n − n − n − n ∞ = P (g∗ g (u(n)))dβ . n k − k k "k=1
By applying Itˆo’s lemma we find that:
d( P u u(n) 2eψ)+2ν P u u(n) 2eψdt | n − | ' n − ' =2 B(u(n)) B∗,P u u(n) eψdt ( − n − ) ∞ +2 g∗(t) g (u ,t),P u u(n) eψdβ ( k − k n n − ) k k=1 (83) ∞" + P (g∗(t) g (u ,t)) 2eψdt | n k − k n | k"=1 + ψ$ P u u(n) 2eψdt. | n − |
As in the a priori estimates for Lemma 4.2, ψ is a C1 function chosen further on to cancel offterms. The first term on the right hand side of (83)isestimatedusing THE STOCHASTIC PRIMITIVE EQUATIONS IN TWO SPACE DIMENSIONS 819 the cancellation property, (13)and(15): B(u(n)) B∗,P u u(n) ( − n − ) = B(u(n) P u, P u)+(B(P u) B(u)) + (B(u) B∗),P u u(n) ( − n n n − − n − ) C( u(n) P u u(n) P u P u ≤ | − n |' − n '' n ' + u(n) P u 1/2 u(n) P u 3/2 ∂ P u ) | − n | ' − n ' | z n | (n) +(B(P u u, P u) ! + B(u, P u u) ! ) P u u (84) | n − n |V | n − |V ' n − ' + B(u) B∗,P u u(n) ( − n − ) ν P u u(n) 2 + C (ν) un P u 2( u 2 + ∂ u 4) ≤ ' n − ' 1 | − n | ' ' | z | 2 2 + C (ν)( B(P u u, u) ! + B(u, P u u) ! ) 2 | n − |V | n − |V + B(u) B∗,P u u(n) . ( − n − ) For the last term involving g and g∗,usingthebasicpropertiesofthe(2 inner product and the Lipschitz conditions for g we infer: ∗ (n) 2 P (g g(u )) 2 | n − |" (H) ∗ ∗ (n) =2(g g(u),Pn(g g(u )))"2 (H) − − (85) (n) 2 ∗ 2 + P (g(u) g(u )) 2 P (g g(u)) 2 | n − |" (H) −| n − |" (H) ∗ ∗ (n) 2 (n) 2 2 2(g g(u),P (g g(u ))) 2 +4K ( P u u + P u u ). ≤ − n − " (H) H | n − | | n − | With estimates (84)and(85)inmindwenowtake: t ψ(t)= C ( u 2 + ∂ u 4)ds 4K2 t (86) − 1 ' ' | z | − H !0 where C1 is the constant from the last inequality in (84)andKH is the Lipschitz constant associated with g.Byexamining(80)wenotice: 2 2 − 1 − eψ(τR) e C (R+R ) 4KH T a.s. (87) ≥ The estimates given in (84)and(85)cannowbeappliedto(83). After integrating up to the stopping time τR,takingexpectationsandrearrangingonefinds: τR E P u u(n) 2dt ' n − ' % 0 & ! τR ∗ (n) ∗ ∗ n ψ CE B(u) B ,Pnu u +(g g(u),Pn[g g(u )])"2 e dt (88) ≤ 0 ( − − ) − − ! 'τR ( 2 2 2 + CE ( Q u + B(P u u, u) ! + B(u, P u u) ! )dt | n | | n − |V | n − |V !0 where the numerical constant C depends on R and T .Notethatsince:
τR τR E eψ(τR) P u u(n) 2dt E eψ(t) P u u(n) 2dt. (89) ' n − ' ≤ ' n − ' % !0 & !0 and taking into account (87)weseethatthetermeφ can be absorbed into the constants on the right hand side of (88). (n) ∗ p 2 Due to (55)oneinfersthatPnu u , 0inL (Ω; L (0,T; V )). Similarly (62) ∗ (n) − 2 2 2 implies that Png Png(u ) , 0inthespaceL (Ω; L (0,T; ( (H))). As such, the first terms on the− right hand side of (88)vanishinthelimitasn .Theterm involving Q u approaches zero in this limit as a consequence of→∞ the dominated | n | 820 NATHANGLATT-HOLTZANDMOHAMMEDZIANE convergence theorem. For the final terms we apply the estimateonB in V $ given in (15)andmakefurtheruseofτR to conclude:
τR 2 E B(Pnu u, u) V ! dt 0 | − | ! τR 2 2 2 CE ( ∂z(Pnu u) + Pnu u ) u dt (90) ≤ 0 | − | | − | ' ' ! τR + CE (P u u) 2 ∂ u 2dt. | n − | ' z ' !0 Moreover:
τR 2 E B(u, Pnu u) V ! dt 0 | − | ! τR CE ( ∂ u 2 + u 2) ∂ (P u u) 2 + u 2 ∂ (P u u) 2dt ≤ | z | | | | x n − | | | ' z n − ' !0 - τR . CE (sup ∂ u 2 +sup u 2) (P u u) 2dt ≤ | z | | | ' n − ' (91) ) t∈[0,τR] t∈[0,τR] !0 * τR + CE sup u 2 ∂ (P u u) 2dt | | ' z n − ' )t∈[0,τR] !0 * τR C(R)E ( P u u 2 + ∂ (P u u) 2) dt . ≤ ' n − ' ' z n − ' %!0 & Thus, again by the dominated convergence theorem, the final two terms also con- verge to zero as n .WecannowconcludethatτR satisfies Lemma 5.2. Fixing arbitrary→∞ (t, ω)offofanappropriatelychosensetofmeasurezero,there is an R so that t τR(ω). As such (79)forholdsthegivenpair,completingthe proof. ≤
Having established existence we next turn to the question of uniqueness:
Theorem 5.4 (Uniqueness). Suppose that u1 and u2 are weak-strong solutions in the sense of Definition 3.3 and that u1(0) = u2(0) a.s. in H.Assumemoreover that: ∂ u L4([0,T]; H) a.s. (92) z 2 ∈ Then: P(u (t)=u (t) t [0,T]) = 1. (93) 1 2 ∀ ∈ In particular, the solutions constructed in Theorem 5.3 are unique since they belong (almost surely) to L∞([0,T]; H) which is included in L4([0,T]; H)
Proof. Subtracting we find the u u satisfies the differential: 1 − 2 d u u ,v + νA(u u ),v dt ( 1 − 2 ) ( 1 − 2 ) = B(u ,u u )+B(u u ,u ),v dt ( 1 2 − 1 2 − 1 2 ) + g (u ,t) g (u ,t),v dβ (94) ( k 1 − k 2 ) k "k u (0) u (0),v = u u ,v . ( 1 − 2 ) ( 0,1 − 0,2 ) THE STOCHASTIC PRIMITIVE EQUATIONS IN TWO SPACE DIMENSIONS 821
We apply Itˆo’s lemma and deduce: d( u u 2eφ)+2ν u u 2eφdt | 1 − 2| ' 1 − 2' =2 B(u u ,u ),u u eφdt ( 1 − 2 2 2 − 1) φ + gk(u1,t) gk(u2,t),u1 u2 e dβk ( − − ) (95) "k + g (u ,t) g (u ,t) 2eφdt | k 1 − k 2 | "k + φ$ u u 2eφ dt. | 1 − 2| Once again φ will be chosen judiciously to cancel offterms below. Applying(13) with Young’s inequality one finds:
B(u1 u2,u2),u2 u1 |( − − )| (96) ν u u 2 + C(ν) u u 2( u 2 + ∂ u 4). ≤ ' 1 − 2' | 1 − 2| ' 2' | z 2| Taking KH to be the Lipschitz constant associated with g and C(ν)fromthepre- ceding inequality, set: t φ(t)= C(ν) ( u 2 + ∂ u 4) dt K2 t. (97) − ' 2' | z 2| − H !0 Given the regularity conditions assumed for u2 we infer, as in Theorem 5.3 that eφ(t) > 0almostsurely.Integrating(95)uptot taking expected values and making use of the estimates on B and g one concludes that for any t [0,T]: ∈ E( u (t) u (t) 2eφ(t)) E( u (0) u (0) 2)=0. (98) | 1 − 2 | ≤ | 1 − 2 | This implies: P(u (t)=u (t), t [0,T] Q)=1. 1 2 ∀ ∈ ∩ However since both u1 and u2 take values in C([0,T]; H)(93)followimmediately.
Acknowledgments. We are grateful to the anonymous referees for their many valuable comments and suggestions. This work has been supported in part by the NSF grant DMS-0505974.
REFERENCES
[1]A.BensoussanandJ.Frehse,Local solutions for stochastic Navier Stokes equations,M2AN Math. Model. Numer. Anal., 34 (2000), 241–273. [2]A.BensoussanandR.Temam,Equations´ stochastiques du type Navier-Stokes,J.Functional Analysis, 13 (1973), 195–222. [3]H.Breckner,Galerkin approximation and the strong solution of the Navier-Stokes equation, J. Appl. Math. Stochastic Anal., 13 (2000), 239–259. [4]D.Bresch,A.KazhikhovandJ.Lemoine,On the two-dimensional hydrostatic Navier-Stokes equations,SIAMJ.Math.Anal.,36 (2004/05), 796–814. [5]Z.Brze´zniakandS.Peszat,Strong local and global solutions for stochastic Navier-Stokes equations,in“InfiniteDimensionalStochasticAnalysis”(eds.P.Clment, F. den Hollander J. van Neerven, and B. de Pagter), R. Neth. Acad. Arts Sci., Amsterdam, (2000), 85–98. [6]C.CaoandE.Titi,Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics,Ann.ofMath.(2),166 (2007), 245–267. [7]M.Capi´nskiandD.Gatarek,Stochastic equations in Hilbert space with application to Navier- Stokes equations in any dimension,J.Funct.Anal.,126 (1994), 26–35. [8]A.B.Cruzeiro,Solutions et mesures invariantes pour des ´equations d’´evolution stochastiques du type Navier-Stokes,Exposition.Math.,7 (1989), 73–82. 822 NATHANGLATT-HOLTZANDMOHAMMEDZIANE
[9] B. Cushman-Roisin and J.-M. Beckers, “Introduction to Geophysical Fluid Dynamics: Phys- ical and Numerical Aspects,” To be published by Academic Press. [10]G.DaPratoandJ.Zabczyk,“StochasticEquationsinInfiniteDimensions,”Cambridge University Press, Cambridge, 1992. [11]G.DaPratoandJ.Zabczyk,“ErgodicityforInfinite-Dimensional Systems,” Cambridge Uni- versity Press, Cambridge, 1996. [12] F. Flandoli, “Lectures on 3D Fluid Dynamics,” C.I.M.E. Lecture Notes, 2005. [13]F.FlandoliandD.Gatarek,Martingale and stationary solutions for stochastic Navier-Stokes equations,Probab.TheoryRelatedFields,102 (1995), 367–391. [14]F.Guill´en-Gonz´alez,N.MasmoudiandM.A.Rodr´ıguez-Bellido, Anisotropic estimates and strong solutions of the primitive equations,DifferentialIntegralEquations,14 (2001), 1381– 1408. [15]C.Hu,R.TemamandM.Ziane,Regularity results for linear elliptic problems related to the primitive equations,in“FrontiersinMathematicalAnalysisandNumericalMethods” (ed T. Li), World Sci. Publ., River Edge, NJ, (2004), 149–170. [16]I.KaratzasandS.E.Shreve,“BrownianMotionandStochastic Calculus,” 2nd edition, Springer-Verlag, New York, 1991. [17]G.M.Kobelkov,Existence of a solution ‘in the large’ for the 3D large-scale ocean dynamics equations,C.R.Math.Acad.Sci.Paris,343 (2006), 283–286. [18]I.KukavicaandM.Ziane,On the regularity of the primitive equations of the ocean,Nonlin- earity, 20 (2007), 2739–2753. [19]J.-L.Lions,R.TemamandS.Wang,Models for the coupled atmosphere and ocean,Comput. Mech. Adv., 1 (1993), 1–120. [20]J.-L.Lions,R.TemamandS.Wang,New formulations of the primitive equations of atmo- sphere and applications,Nonlinearity,5 (1992), 237–288. [21]J.-L.Lions,R.TemamandS.Wang,On the equations of the large-scale ocean,Nonlinearity, 5 (1992), 1007–1053. [22]J.-L.MenaldiandS.S.Sritharan,Stochastic 2-D Navier-Stokes equation,Appl.Math.Op- tim., 46 (2002), 31–53. [23]R.MikuleviciusandB.L.Rozovskii,Global L2-solutions of stochastic Navier-Stokes equa- tions,Ann.Probab.,33 (2005), 137–176. [24]R.MikuleviciusandB.L.Rozovskii,Martingale problems for stochastic PDE’s,in“Stochastic partial differential equations: six perspectives” (eds. R. A. Carmona and B. L. Rozovskii), Amer. Math. Soc., Providence, RI, (1999), 243–325. [25]R.MikuleviciusandB.L.Rozovskii,On equations of stochastic fluid mechanics,in“Stochas- tics in Finite and Infinite Dimensions” (eds. T. Hida, R. L. Karandikar, H. Kunita, B. S. Rajput, S. Watanabe and J. Xiong), Birkh¨auser Boston, Boston, MA, (2001), 285–302. [26]R.MikuleviciusandB.L.Rozovskii,Stochastic Navier-Stokes equations for turbulent flows, SIAM J. Math. Anal., 35 (2004), 1250–1310. [27] J. Pedlosky, “Geophysical Fluid Dynamics,” Springer-Verlag, New York, 1987. [28]M.Petcu,R.TemamandD.Wirosoetisno,Existence and regularity results for the primitive equations in two space dimensions,Commun.PureAppl.Anal.,3 (2004), 115–131. [29]B.L.Rozovskii,“StochasticEvolutionSystems,”KluwerAcademic Publishers Group, Dor- drecht, 1990. [30]R.TemamandM.Ziane,Some mathematical problems in geophysical fluid dynamics,in “Handbook of Mathematical Fluid Dynamics. Vol. III” (eds. S.FriedlanderandD.Serre), North-Holland, Amsterdam, (2004), 535–657. Received August 2007; revised March 2008 E-mail address: [email protected] E-mail address: [email protected]