Time-splitting methods to solve the Hall-MHD systems with Lévy noises

In this paper, we establish the existence of a martingale solution to the stochastic incompressible Hall-MHD systems with Levy noises in a bounded domain. The proof is based on a new method, i.e., the time splitting method and the stochastic compactness method.

where the n-dimensional vector fields u and B are the fluid velocity and the magnetic field, respectively. The scalar field p is the pressure. (u · ∇) = n i=1 u i ∂ i and the symbol × stands for the usual n-dimensional cross-product. The given vector fields f 1 and ∇×f 2 are external forces on the magnetically charge fluid flows. The system (1.1) has been studied in the physics literature for decades (see e.g. [1], [15] and their references) and has application in a number of physics fields as geo-dynamo (see e.g. [17]), neutron stars (see e.g. [23]) and magnetic reconnection in plasmas (see e.g. [10]). The Hall-MHD system was first studied by Lighthill [15] in 1960. In comparison with the usual incompressible magnetohydrodynamic (MHD) system, we have the new term ∇ × ((∇ × B) × B) (Hall term) which is due to Hall effect and prevents straightforward adaptations from arguments used in the mathematical analysis of Navier-Stokes and related models. This term is very important when the magnetic shear is large, where the magnetic reconnection happens. The magnetic reconnection means the change of the topology of magnetic field lines. We refer [10,24] for the physical background of the magnetic reconnection and the Hall-MHD.
From several points of view it is reasonable to add the stochastic parts to the equation of motion.
• It can be understood as a turbulence in the fluid motion.
• It can be interpreted as a perturbation from the physical model.
• Apart from the force f 1 and ∇ × f 2 we are observing there might be further quantities with a (usually small) influence on the motion.
In the case of laminar flows where the shear is weak, one ignores the Hall term, and the system reduces to the usual MHD. The stochastic MHD system was considered in many papers. Barbu and Da Parto [3] obtained the existence of strong solutions to the two-dimensional MHD equations driven by random exterior forcing terms both in velocity field and in magnetic field, and they proved the existence and uniqueness of an invariant measure via the coupling methods. The stochastic magneto-hydrodynamic equations driven by multiplicative noise in two-dimensional domains were studied by Chueshov and Millet [8]. Concerning the MHD system perturbed by white noises in bounded domains, the analysis of the existence of solutions was provided by Sritharan and Sundar [25]. Sango [21] proved the existence of martingale solutions to the stochastic MHD equations by Galerkin approximation method with Prokhorov and Skorokhod's compactness results in [9,14]. As for the existence and uniqueness of strong solutions to the two-dimensional stochastic MHD equations driven by a multiplicative or an additive noise, we refer to Sundar [26]. Motyl [19] proved the existence of martingale solutions of the hydrodynamic-type equations driven by Lévy noise in 3D possibly unbounded domains. Tan, Wang and Wang [27] showed the global existence of strong solutions to the stochastic MHD equations driven by a multiplicative noise in probability if the initial data are sufficiently small.
To the best of our knowledge, the analysis of the global existence of martingale solutions to the stochastic Hall-MHD equations driven by Lévy noises is not solved yet. The aim of this paper is to prove the global existence of martingale solutions to the stochastic Hall-MHD equations. Comparing with the usual stochastic MHD, stochastic Hall-MHD will be more complex. Thanks to the effect of the Hall term, to complete our energy estimates and establish tightness property of the approximate solutions, we need to estimate the norm of the Hall term which is actually a nonlinear term. Furthermore, when we identify the convergence of the approximate solutions, we also need to identify the convergence of the Hall term. In the study of the existence of a martingale solution, one often use the classical Faedo-Galerkin approximation method, such as [3,8,25,26,19]. Here we perform a time splitting method which provide a new construction of the solutions. By using the time splitting method we can see how the stochastic part influence the Hall-MHD in the evolutionary process (see Remark 4.2).

Hypotheses and main result.
2.1. Hypotheses. Let (Ω, F , P) be a probability space. To solve the Hall-MHD systems (1.2) in Ω × R + × D, we assume that the initial data are divergence-free and possess certain regularity and the following boundary conditions: where n is the unit outer normal on ∂D. Moreover, we give the precise assumptions on each of the stochastic term appearing in systems (1.2): Assumption (A). Let us denote L HS (Z W ; H) the space of Hilbert-Schmidt operators from Z W into H = H u × H B (see (3.2) and (3.5) for details). Assume that σ i : (0, T ) × V → L HS (Z W , H), i = 1, 2 are nonlinear mapping, which satisfies the following condition: there exists a positive constant C > 0 such that for all u ∈ H u and B ∈ H B . H u , V u , V B and H B are defined in Section 3. More precisely, see (3.2), (3.3), (3.5) and (3.6). We define a continuous mapping from for each p ≥ 2 and all t ∈ [0, T ], there exists a positive constant C such that Here, we define a continuous mapping from First, we define the concept of solutions for the problems (1.5)-(2.2) as follows.
Definition 2.1. A martingale solution of (1.5)-(2.2) is a system ((Ω, F , F t , P), W, π, u, B), which satisfies (1) U = (Ω, F , F t , P) is a filtered probability space with a filtration F t , i.e., a set of sub σ-fields of F with F s ⊂ F t ⊂ F for 0 ≤ s < t < ∞, (2) W (t) is a cylindrical Wiener process with positive symmetric trace class covariance operator Q. The process W is independent of π.
In the above, all stochastic integrals are defined in the sense of Itô, see [2,9,14,20]. Theorem 2.2 will be proved through the following steps. First, we use the time splitting method to construct the approximate solutions to the problems (1.5)-(2.2). More precisely, on the probability space (Ω, F , P) with cylindrical Wiener process W and Poisson random measures π, we solve alternatively the deterministic part of (1.5) on time intervals [t 2k , t 2k+1 ) and the stochastic part of (1.5) on time intervals [t 2k+1 , t 2k+2 ). For details, see (4.1)-(4.4). Meanwhile, we can get the energy estimates by using the Itô formula to the function |U τ | p H and the Burkholder-Davis-Gundy inequality. Thanks to the orthogonality in L 2 of the Hall term with B, the energy estimates of the deterministic part, similar to the usual MHD case, hold true. However, for the stochastic part of (1.5) on time intervals [t 2k+1 , t 2k+2 ), we need to estimate it in the view of stochastic partial differential equations.
The second step is to take a limit as τ → 0 and prove the existence of martingale solutions. From energy estimates, the approximate solutions (W τ ,π τ , u τ , B τ ) may converge on [0, T ]. However the convergence is too weak to guarantee that the limit is a solution on [0, T ]. In the two-dimensional case, it can be shown by using certain monotonicity principle that the nonlinear terms converge to the right limit and hence a global strong solution can be obtained [16]. But when the space dimension is three, the monotonicity does not hold and to the best of our knowledge there is no result on the global strong solutions. This is why we pursue the martingale solutions instead. As is explained, the main issue is the convergence of the nonlinear terms.
To this end, we relax the restriction on the probability space and aim to prove a tightness result of the random variables (W τ , π τ , u τ , B τ ). Since we already have the energy estimates, this can be obtain by showing that U τ = (u τ , B τ ) satisfies the Aldous condition. Then from the Jakubowski-Skorokhod Theorem there exist a probability space (Ω,F ,P) and random variables (W τj ,π τj ,ū τj ,B τj , ) → (W ,π , u , B )P-a.s., with the property that the probability distribution of (W τj , π τj ,ū τj ,B τj ) is the same as that of (W τ ,π τ , u τ , B τ ). When passing to a limit as τ → 0, the usual method is to show that the limit process of the stochastic integral is a martingale, and to identify its quadratic variation. Then they usually apply the representation theorem for martingales or the revised representation theorem (see [12]) to prove that it solves the equations. But here instead, we choose to prove Lemma 5.1 and Lemma 5.2 to obtain that (W ,π , u , B ) satisfies the equations (1.5)-(2.2) by passing to the limit directly. Therefore it is a martingale solution of (1.5)-(2.2) in the sense of Definition 2.1.
The rest of the paper is organized as follows. We recall some analytic tools in Sobolev spaces and some basic theory of stochastic analysis in Section 3. In Section 4, we construct the solutions to an approximate scheme by the time splitting method and we prove the tightness property of the approximate solutions (W τ , π τ , u τ , B τ ). In Section 5, we pass to the limit as τ → 0 to get a martingale solution of (1.5)-(2.2) in the sense of Definition 2.1.
Notation. Throughout this paper we drop the parameter ω ∈ Ω. Moreover, we use C to denote a generic constant which may vary in different estimates. For simplicity, we will write A B if A ≤ CB.

Preliminaries.
Let H 1 (D) stand for the Sobolev space of all ϕ ∈ L 2 (D) for which there exist weak derivatives ∂ϕ In the space H u , H B and H = H u × H B , we consider the scalar product and the norm inherited from L 2 (D) and (for brevity, we omit subscript "u" and "B") denote them by ·, · H and | · | H respectively, i.e.
In the space V u and V B we consider the scalar product inherited from where ·|· denotes the duality pairing between V and V. Moreover, we define a bilinear map B by B(ϕ, ψ) :=b(ϕ, ψ, ·), ∀ϕ, By the Sobolev embedding theorem and Hölder's inequality, one has and in particularb (ϕ 1 , ϕ 2 , ϕ 2 ) = 0, ∀ϕ 1 , ϕ 2 ∈ V. Meanwhile, we infer that B(ϕ, ψ) ∈ V for all ϕ, ψ ∈ V and we use the notation B(ϕ) := B(ϕ, ϕ). For B, we have the following property (see Lemma 6.4 in [19]): (1) There exists a constant C > 0, such that (2) In particular, the form B : V × V → V is bilinear and continuous. Moreover, (3) The mapping B is locally Lipschitz continuous on the space V, i.e. for every For f (s), by the Sobolev embedding theorem, we have We now recall some preliminaries of stochastic analysis and useful tools for the sake of convenience and completeness. For details, we refer the readers to [2,9,14,20] and the references therein. In particular, we shall state the definitions of Wiener process, time homogenous Poisson random measure, Lévy process, Itô's formula and the BDG inequality and so on.
If W is an Z W -valued Wiener processes with covariance operator Q with T rQ < ∞, then W is a Gaussian process on Z W and E(W (t)) = 0, Cov(W (t)) = tQ, (2) π is independently scattered, that is, if the sets A j ∈ Z ⊗ B(R + ), j = 1, 2, . . . , n are pair-wise disjoint, then the random variables π(A j ), j = 1, 2, . . . , n are pair-wise independent; (3) for all A ∈ Z and Note that we can construct a corresponding Poisson random measure from a Lévy process. For example, given an E-valued Lévy process over (Ω, F , F t , P), one can construct an integer valued random measure in the following: for each  (1) L(t) is F t -measurable for any t ≥ 0; (2) the random variable L(t) − L(s) is independent of F s for any 0 ≤ s < t; Let us now recall the Itô formula for general Lévy-type stochastic integrals, see [2,20]. We define P 2 (T, E) to be the set of all equivalence classes of mappings f : [0, T ] × E × Ω → R which coincide almost everywhere with respect to × P and which satisfy the following conditions: (1) f is predictable; Now we are ready to give Itô's formula for general Lévy-type stochastic integrals, let X be such a process in the following: where for each t ≥ 0, z ∈ Z, |G| 1 2 , F ∈ P 2 (T ) and H ∈ P 2 (T, E). Furthermore, we take K to be predictable and E = {z ∈ R d : 0 < |z| < 1}. Denote so that for each t ≥ 0, we have Assume that for all t > 0, sup 0≤s≤t sup 0<|z|<1 H(s, z) < ∞ a.s., then one has (see Theorem 5.1 in [11]). Lemma 3.4. Let Φ be a function of class C 2 on R n and X(t) a n-dimensional semi-martingale given as above. Assume that H i (t, z)H j (t, z) = 0, i, j = 1, ..., n. Then the stochastic process Φ(X(t)) is also a semi-martingale and the following formula holds We now recall an important inequality in the stochastic analysis: Lemma 3.5 (Burkholder-Davis-Gundy inequality). Let T > 0, for every fixed p ≥ 1, there is a constant C p ∈ (0, ∞) such that for every real-valued square integrable càdlàg martingale M t with M 0 = 0, and for every T ≥ 0, where M t is the quadratic variation of M t and the constant C p does not depend on the choice of M t .
Definition 3.6. Let X be a separable Banach space and let B(X) be its Borel sets. A family of probability measures P on (X, B(X)) is tight if for any ε > 0, there exists a compact set K ε ⊂ X such that Π(K ε ) ≥ 1 − ε for all Π ∈ P. A sequence of measures {Π n } on (X, B(X)) is weakly convergent to a measure Π if for all continuous and bounded functions h on X Lemma 3.7 (Jakubowski-Skorokhod Theorem [12]). Let X be a topological space such that there exists a sequence {h m } of continuous functions h m : X → R that separate points of X . Denote by S the σ-algebra generated by the maps {h m }. Then (1) every compact subset of X is metrizable.
(2) every Borel subset of a σ-compact set in X belongs to S .
(3) every probability measure supported by a σ-compact set in X has a unique Radon extension to the Borel σ-algebra on X .
(4) if Π m is a tight sequence of probability measures on (X , S ), there exist a subsequence Π m k converging weakly to a probability measure Π, and a probability space (Ω, F , P) with X valued Borel measurable random variables X k and X such that, Π m k is the distribution of X k , and X k → X a.s. on Ω. Moreover, the law of X is a Radon measure.

4.
Time splitting method and energy estimates. In order to solve (1.5)-(2.2) with given cylindrical Wiener process W and Poisson random measures π on the probability space (Ω, F , P), we first consider X which is a Hilbert space and satisfy In particular, X is compactly embedded into the space H.    where T τ is assumed to be an even integer. We partition the time interval [0, T ] into T τ time intervals of length τ . Meanwhile, we denote t j = jτ . We shall look for the pair of sequences U τ which satisfy the following equation: , and these rescaling procedures should yield a solution U τ is consistent with the solution U to (1.5) when τ → 0. Actually in the stochastic equation (4.3), we can see the term σ τ (t, U τ )dW (t) is "faster" than the other term caused by the Wiener process W , and this phenomenon can not be seen if we use the classical Faedo-Galerkin approximation method. In (4.3) we have also regularized the coefficients σ τ and g τ , where σ τ and g τ are globally Lipschitz continuous and satisfies Assumption (A) and Assumption (B) uniformly in τ and converge punctually to σ and g when τ → 0, respectively. We note that the derivatives of the coefficients σ τ and g τ are also Lipschitz continuous. For , Ψ and Φ is the non-negative smooth density of a probability measure or a mollifier.
Let us introduce the stopping times Applying Itô's formula in Lemma 3.4 For all a, b ∈ H and p ≥ 2, from Taylor's formula, it holds that (4.5) By using these above inequalities (4.5), when p = 2, we have Taking supremum t ∈ [0, T ∧ ς N ] in the above inequality (4.6) and expectation on the For the second term on the right hand side of (4.8), integrating by parts, one has For the last term on the right hand side of (4.8), by using (2.4) in Assumption (B), we have For the fourth term on the right hand side of (4.8), thanks to Assumption (A), one has (4.11) For the stochastic terms in (4.8), by the Burkholder-Davis-Gundy inequality and Young's inequality, we can get Meanwhile, it holds that From (4.5), one has In summary, by using (4.8)-(4.15), Gronwall's Lemma and Fatou's lemma (passing to the limit as N → ∞), we have the following Proposition: After getting the energy estimates of the deterministic part, we need to estimate the stochastic part.
By using Assumption (A), Assumption (B) and the Burkholder-Davis-Gundy inequality, then we have We combine Proposition 1 and Proposition 2 to get the following estimate: Let L(U τ ) be probability measures of the solutions U τ of the splitting equations on the measurable space (G, Before proving the tightness of L(U τ ), we recall the following lemma.
Lemma 4.1. Let (U τ ) τ ∈(0,1) be a sequence of càdlàg (F t ) t≥0 -adapted X -valued processes such that (i) there exist a positive constant C > 0 such that (ii) (U τ ) satisfies the Aldous condition in X . Let P be the law of U τ on G. Then for every ε > 0 there exists a compact subset K ε of G such that Let us recall the Aldous condition in the form given by Métivier (see [13]).

Definition 4.2.
A sequence (U τ ) τ ∈(0,1) satisfies the Aldous condition in the space X if and only if for ∀ε > 0, ∀η > 0 ∃δ > 0 such that for every sequence ( τ ) τ ∈(0,1) of (F t ) t≥0 -stopping times with τ ≤ T , one has Now, we are going to prove the following lemma. Proof. In order to prove the tightness of measures {L(U τ )} on (G, G ), since we have already proved Proposition 1, applying Lemma 4.1, it is sufficient to prove that the sequence (U τ ) satisfies the Aldous condition in Definition 4.2. Note that For the term R 2 , since A n : V → V and |A n U n | V ≤ U n and the embedding V → X is continuous, by using inequality (4.18), we have For the term R 3 , since X → V m and by the property of B, we get where B stands for the norm of B : H × H → V m , and we have used the inequality (4.18) in the last inequality of (4.21). For the term R 4 , we use the fact that X → V m and (3.7) to obtain For the term R 5 , by the Itô isometry, Assumption (A), continuity of the embedding V → X and (4.18), we obtain the following inequality Finally, we consider the term R 6 . By Assumption (B), and (4.18), one has Combining (4.19)-(4.24), we know that there exist a constant C > 0 such that for every sequence τ of F -stopping times with τ < T and for every τ ∈ (0, 1) and θ ≥ 0 there holds Now, we can apply Lemma 9 in [18] to infer that the sequence (U τ ) τ ∈(0,1) satisfies the Aldous condition in the space X . And from lemma 4.1 we complete the proof.
Lemma 5.1. For all ψ ∈ X, we have the following equalities:  By applying (5.2) with p = 1 and the dominated convergence theorem, we have Furthermore, we use Hölder's inequality and (5.2) to get for every τ ∈ (0, 1) and every p ∈ (1, ∞) where C(p) is a positive constant. Hence, we get the assertion (1) from (5.6), (5.7) and the Vitali theorem.
According to (5.12) and the dominated convergence theorem, we complete the proof of assertion (4).