Martingale solutions of the stochastic 2D primitive equations with anisotropic viscosity

. The stochastic 2D primitive equations with anisotropic viscosity are studied in this paper. The existence of the martingale solutions and pathwise uniqueness of the solutions are obtained. The proof is based on anisotropic estimates, the compactness method, tightness criteria and the Jakubowski version of the Skorokhod theorem for nonmetric spaces.


Introduction
The primitive equations (PE) for oceanic and atmospheric dynamics are fundamental models in meteorology, which derived from the Navier-Stokes equations assuming a hydrostatic balance for the pressure term in the vertical direction.For more information on physical background and geophysical applications of the primitive equations, we refer the reader to [9,25,46], for example.
Known results: For the deterministic primitive equations with full viscosity, i.e., the diffusion term −∆ is defined by −∂ 2 x − ∂ 2 y − ∂ 2 z , the mathematical analysis of the initial value problem has been started by Lions, Temam and Wang [40][41][42], where the notions of weak and strong solutions were defined and the global existence of weak solutions was proved, however, the uniqueness of weak solutions is still unclear.By decomposing the velocity into barotropic and baroclinic components, a breakthrough result has been proven by Cao-Titi [8], where the global well-posedness of strong solutions in H 1 to the three dimensional primitive equations has been obtained, see also Kobelkov [33] and Kukavica-Ziane [36].One can see some other literatures (for instance [21,26,30]- [34,48,49]) for the well-posed results with different space dimensions, initial data and boundary conditions.In particular, about uniqueness of weak solutions, by introducing the notion of z-weak solution (see [2,47]), i.e., weak solutions with additional regularity in the vertical direction, researchers have found some results [31,34,37,44].
In the models of geophysical flows, due to uncertainties in the physics derivations and intrinsic heat fluctuations, white noise driven random term was introduced to the geophysical equations (see for instance [45]).Research on stochastic primitive equations with full viscosity is a classical topic and has been studied extensively in a number of literatures.For the well-posedness, regularity, random attractor and existence and regularity of invariant measures, we refer the readers to [10]- [18,22,51].For the deviation principles and small time asymptotics of the primitive equations, see [12,13].
Due to the presence of strong turbulent mixing in the horizontal direction in the large scale atmosphere, the viscosity in the horizontal direction is much stronger than that in the vertical direction.As a result, it is necessary to investigate the primitive equations with anisotropic viscosity, i.e., the diffusion term −∆ is replaced by , where ν 1 , ν 2 , ν 3 ≥ 0, and, in particular, the system that with only horizontal viscosities, i.e., the diffusion term −∆ is replaced by −∆ H , where ∆ H = ν 1 ∂ 2 x + ν 2 ∂ 2 y , ν 1 , ν 2 > 0 (see [6,7,38]).Mathematically, for the primitive equations with partial viscosity, we would like to emphasis that the system is not purely parabolic anymore.Cao et al. [6,7] studied the 3D deterministic primitive equations with only horizontal viscosity analytically.They tackled this problem in a periodical setting by considering a vanishing vertical viscosity limit and obtained global strong well-posedness results for initial data with regularity near H 1 , and local well-posedness for initial data in H 1 .Instead of considering a vanishing vertical viscosity limit, by a direct approach which in particular avoids unnecessary boundary conditions on top and bottom, Hussein et al. [28] studied the initial value and the time-periodic problem for the 3D deterministic primitive equations with horizontal viscosity and obtained existence and uniqueness of local z-weak solutions for initial data in H 1 z L 2 xy and local strong solutions for initial data in H 1 .Furthermore, if ∂ z v 0 ∈ L q for q > 2, the local z-weak solutions extended to a global strong solution.For the case of full hyper-viscosity or only horizontal hyper-viscosity, i.e., the diffusion term −∆ is replaced by −∆ + ε(−∆) l or by −∆ + ε(−∆ H ) l , respectively, where ε > 0, l > 1, strong convergence for ε → 0 of hyper-viscous solutions to a weak solution of the 3D deterministic Navier-Stokes and primitive equations, respectively, was obtained by Hussein in [27].
For the 3D deterministic primitive equations without viscosity, i.e., ν 1 = ν 2 = ν 3 = 0, blow-up result was obtained by Cao et al. in [5], see also Wong [52].Ill-posedness result was obtained in Sobolev spaces by Han-Kwan and Nguyen in [24], where the solution map was not (Hölder) continuous with respect to initial data.Local well-posedness has been proven only for analytical data by Kukavica et al. in [35].

Stochastic 2D primitive equations with only horizontal viscosity:
The primary goal of this paper is to study the well-posedness of 2D stochastic primitive equations with horizontal viscosity driven by multiplicative white noise.For simplicity, a period setting is considered here.We consider the domain M := (0, 1) × (−1, 0) and denote by x ∈ (0, 1) the horizontal coordinate and by z ∈ (−1, 0) the vertical one.Let (v, w) be the velocity with horizontal component v and vertical component w, and p be the pressure.Then the 2D stochastic primitive equations are given by where σ is the random external forces, f is an external force term and W is a cylindrical Wiener process, the definitions of which will be introduced in Section 2. As in [27], note that the vertical periodicity and parity conditions in (1.1) correspond in to an equivalent set of equations on (0, 1) × (−1, 0) with lateral periodicity and With these boundary conditions, we suppose with no loss of generality that (see also in [19]) In the rest of paper, in order to focus our attention on the difficulties arising from the nonlinear term, we ignore the external force term f .The whole process of proof shows that the existence of f does not affect our conclusion.
Studies on stochastic geophysical fluid equations with anisotropic viscosity have attracted more and more attention in recent years.Especially, for the stochastic Navier-Stokes equations, by adding a term of Brinkman-Forchheimer type, Bessaih and Millet [1] established the existence and uniqueness of global weak solutions (in the PDE sense) in the whole space R 3 .Liang, Zhang and Zhu [39] investigated the existence of the martingale solutions and pathwise uniqueness of the solutions in a given anisotropic Sobolev space on R 2 or on the two dimensional torus T 2 .Comparing with the Navier-Stokes equations, it is worth to point out that the system of primitive equations is generally harder to deal with, the nonlinear term w∂ z v is a more difficult version in contrast to the nonlinearity of the Navier-Stokes equations since w = w(v) given by (1.4) involves a first order derivative.Therefore, no matter which type of viscosity is considered, how to deal with the estimates of nonlinear term is a challenge.
Returning to the study of this paper, for the stochastic primitive equations with partial viscosity, fewer works have been done.To the best of our knowledge, only the literature [50] was involved on this topic.In [50], the existence and uniqueness of pathwise solutions in H 1 to the stochastic 3D primitive equations with only horizontal viscosity and diffusivity driven by transport noise were established.Firstly, the global existence of martingale solutions was established for a modified equations with a cut-off acting on L ∞ z L 4 xy -norm of the solution.After a standard argument using the theorems by Prokhorov and Skorokhod, the existence of maximal solutions up to a strictly positive stopping time was established.Then, to establish uniqueness, more regular initial data was needed in the vertical direction.Finally, the global existence was established using the logarithmic Sobolev embedding and an iterated stopping time argument.
Generally, what works in 3D does not necessarily work in 2D, in this paper, we study the stochastic 2D primitive equations with horizontal viscosity.By new anisotropic estimations and compactness method differently from the above mentioned methods in [50], global existence of martingale solutions has been obtained.In the proof of the tightness argument of Galerkin schemes, an Aldous's condition (first introduced in [3]) is applied to deal with the H −1 xz norm.On the other hand, compared to [50], where the solutions were strong in the PDE sense, the solutions considered here are z-weak meanings (in the PDE sense), moreover, to obtain the uniqueness of z-weak solutions, we don't need more regular initial value.
This paper is organized as follows: In Section 2, some basic definitions and notations are given for a periodic setting such as function spaces, assumptions and the definition of martingale solutions.In Section 3, the main results with proofs are formulated.

Preliminaries
For s ∈ [0, +∞), one defines the Bessel potential spaces where C ∞ per (M ) denotes the spaces of smooth functions which are periodic of any order (cf.[27]) in all directions on ∂M and (•) denotes the dual of the corresponding space.Moreover, H s per (M ) can be characterized by means of Fourier series as Furthermore, we recall the anisotropic Sobolev norms, Ḣs,s For the sake of simplicity, we abbreviate Ḣs,s , respectively.We will be working on these spaces, The Leray operator P H is the orthogonal projection of L 2 (M ) on to H.The action of this operator is given by Let e k , k ≥ 1 be an orthonormal basis of H whose elements belong to H 2 and orthogonal in H 0,1 and H 1,0 (hence also H 1,1 ).For integers k, l ≥ 1 with k = l, we deduce that for i = x, z, Therefore, ∂ 2 i e k is a constant multiple of e k .For example, for k = (k 1 , k 2 ), the eigenfunctions and associated eigenvalues can be identified, .
In accordance with equality (1.4), we take And let Due to boundary conditions (1.2), we have the following lemma.
Lemma 2.1 (Anisotropic estimates).The trilinear forms b and B have the following properties.There exists a constant C > 0 such that (2.1) Proof.The properties of (2.2) and (2.3) have been obtained in [19], here, by Hölder inequality, we give the estimate of (2.1) anisotropically.
where we make use of the boundary conditions and get the following estimates, This completes the proof.

Main results
In this section, we state two theorems about the well-posedness of equation (1.1), which will be proved in the following subsections.Theorem 3.1.Assume that v 0 is a random variable in H0,1 and suppose that σ satisfies Hypothesis 2.3 with K 2 < 2 11 and K1 < 2 7 .Then (1.1) has a global martingale solution.
Theorem 3.2.(Pathwise uniqueness).Assume that v 0 is a random variable in H0,1 .Suppose that σ satisfies Hypothesis 2.3 with are two weak solutions on the same stochastic basis (Ω, F, P ).Then we have v 1 = v 2 P − a.s..

Galerkin approximation and a priori estimates
Let H n = span(e 1 , . . ., e n ) and P n (resp.Pn , P n ) denote the orthogonal projection from H (resp. H0,1 , H 1 ) to H n .As in [39], we have Furthermore, for u ∈ H n , we have ∂ 2 z u ∈ H n and for any v ∈ H0,1 , Hence, given v ∈ H0,1 , we have This proves that P n and Pn coincide on H0,1 .Similarly, we can prove P n , P n , and P n coincide on H 1 .We consider the following stochastic ordinary differential equations on Then for k = 1, . . ., n, t ∈ [0, T ], we have Note that since it is in finite dimensions, there exists a constant Hence by Theorem 3.1.1 in [43], there exists a unique global solution v n (t) to (3.1).Moreover, v ∈ C(R + , H n ), P − a.s.. Lemma 3.3.We have the following energy estimates under the hypothesis of Theorem 3.1, Proof.By Itô formula, we have The growth condition implies that The Burkholder-Davis-Gundy inequality and the Young inequality as well as the growth condition imply that, By (3.2)-(3.4)and dropping some negative terms, we deduce Gronwall's lemma implies that where C is a constant depending on K 0 , K 1 , K 2 , T but not n.
This completes the proof.However, we also need the L 4 (Ω) uniform estimates of v n .
Lemma 3.4.We have the following uniform estimates under the hypothesis of Theorem 3.1, Proof.Applying once more the Itô formula to the square of • 2 L 2 , we obtain where The growth condition implies that The Burkholder-Davis-Gundy inequality, the growth condition and the Young inequality imply that since K 2 < 2 11 , we can choose 0 < γ < 1, such that 6K 2 + 16 γ K 2 − 4 < 0. Thus, combining (3.7)-(3.8)and dropping some negative terms on the right of the inequality, we have: We complete the proof.
Lemma 3.5.We have the following uniform estimates under the hypothesis of Theorem 3.1, Proof.Using again the Itô formula to v n (t) 2 H 0,1 , we obtain where The growth condition implies that and Since K1 < 2 7 , we can choose 0 < β < 1 such that ( 6 β + 1) K1 − 2 < 0. By (3.10)-(3.12)and dropping some negative terms, we deduce Gronwall's lemma implies that which complete the proof.

Tightness of the family of laws for the Galerkin solutions
In this section, we will prove the tightness of the family of laws for the Galerkin solutions.Before we proceed further, we recall the following definition and theorem which we can refer to [3].
Let (S, ) be a separable and complete metric space.
Definition 3.7.A sequence (X n ) n≥1 satisfies the Aldous's condition [A] iff for any ε, η > 0, there exists δ > 0 such that for every sequence (τ n ) n≥1 of F t -stopping times with τ n ≤ T , we have Theorem 3.8 (see [3]).Assume that (X n ) satisfies Aldous's condition [A].Let P n be the law of X n on C([0, T ], S), n ∈ N. Then for every ε there exists a subset A ε ⊂ C([0, T ], S) such that We now state and prove the following crucial result.
Lemma 3.9.The family Proof.For any sequence (τ n ) n∈N of F t -stopping times with τ n ∈ [0, T ], we have Using Lemma 3.3, the definition of • H −1 and the Hölder inequality, we infer that E I (1)   n By Lemma 3.4, Lemma 3.6 and (2.1), we have By Burkholder-Davis-Gundy inequality and (2.4), we have Collecting these estimates altogether and plugging them in (3.19), we obtain that there exists a constant C > 0 such that for any δ ∈ (0, ∞), By Chebyshev's inequality, for any ε > 0 and ϑ > 0 Choosing δ in such a way that δ < C −1 ϑε, we infer that In the next lemma we will prove that the family of laws of (v n ) n∈N , denoted by Pn is tightness.
Proof.In order to get the tightness of Pn , we should prove that there exists a compact subset K ε of X such that By the Chebyshev inequality and Lemma 3.3, Lemma 3.5, we infer that for any n ∈ N and any r 1 , r 2 > 0, ≤ ε 3 , we denote Then By the definition, K ε is uniformly bounded hence relatively compact in L 2 w ([0, T ]; H 1,1 ) and L ∞ w * ([0, T ]; H 0,1 ).Since L 2 could be embedded compactly in H −1 , as n, m → ∞, we have (3.26) By (3.25), argument analogously to the proof of the classical Arzelá-Ascoli Theorem, we can obtain that v n ∈ A ε 3 is compact in C([0, T ], H −1 ).To sum up, we conclude that K ε is a compact subset of X .The proof is thus completed.We will use the following Jakubowski's version of the Skorokhod theorem in the form given by Brzezniak and Ondrejat [4], see also [29].
Theorem 3.11.Let Y be a topological space such that there exists a sequence f m of continuous functions f m : Y → R that separates points of Y. Let us denote by S the σ -algebra generated by the maps f m .Then (j1) every compact subset of Y is metrizable; (j2) if (µ m ) is tight sequence of probability measures on (Y, S), then there exists a subsequence (m k ), a probability space (Ω, F, P ) with Y -valued Borel measurable variables ξ k , ξ such that µ m k is the law of ξ k and ξ k converges to ξ almost surely on Ω.Moreover, the law of ξ is a Radon measure.
As in [39], we can obtain that on each space appearing in the definition of X there exists a countable set of continuous real-valued functions separating points.Then all the conditions of the above Skorokhod theorem are satisfied.By Theorem 3.2, there exists another probability space ( Ω, F, P) and a subsequence Pn k as well as random variables ṽn k in the space ( Ω, F, P), such that (i) ṽn k has the law Pn k , (ii) Pn k converges weakly to some P , (iii) ṽn k → ṽ in X P − a.s. and ṽ has the law P ∈ P(C([0, T ]; H −1 )).
Remark 3.12.since ṽn k has the same law as v n k , we immediately have Remark 3.13.Similarly, for L 4 (Ω) estimates, we have (3.28)

Pass to the limit and the proof of main theorems
Let us denote the subsequence (ṽ n k ) again by (ṽ n ) and pass the limit as n → ∞.
Lemma 3.16.Assume that u, v ∈ H1,1 , we have Proof.Set Φ = u − v.We deduce Integrating by parts, using a similar method as in Lemma 2.1, Hölder inequality and Young inequality imply Combining I 1 and I 2 , we end the proof.Proof of Theorem 3. Taking expectation on both sides, by the martingales have zero averages, for L 1 < 1, we have Thus we obtain the uniqueness of the solution.
With martingale solutions and pathwise uniqueness of the solutions in hand, we obtain the existence of the pathwise solutions (meaning that the solutions are defined on the prescribed probability space) to this system by the infinite-dimensional-space extension of the Yamada-Watanabe theorem (see [23]).