Existence and Uniqueness of the Local Smooth Solution to 3D Stochastic MHD Equations without Diffusion

In this paper, we consider the existence of local smooth solution to stochastic magneto-hydrodynamic equations without diffusion forced by additive noise in R3. We first transform the system into a random system via a simple change of variable and borrow the result obtained for classical magneto-hydrodynamic equations, then we show that this random transformed system is measurable with respect to the stochastic element. Finally we extend the solution to the maximality solution. Due to the coupled construction of this system, we need more elaborate and complicated estimates with respect to stochastic Euler equation.


Introduction
The magneto-hydrodynamic (MHD) equations have a wide range of applications in geophysics, astrophysics, and plasma physics [1][2][3][4][5][6][7][8][9][10]. Herein we consider the existence and uniqueness of the local smooth solution of the Cauchy problem to the following 3-dimensional (3D) stochastic MHD equations without diffusion, where π, u = (u 1 , u 2 , u 3 ) and m = (m 1 , m 2 , m 3 ) denote, respectively, the pressure, velocity, and magnetic field. Assume that {β 1 j } ∞ j=1 and {β 2 j } ∞ j=1 are independent standard Brownian motions; they are defined in the filtered space (Ω, F , {F t } t≥0 , P), which satisfy the natural assumption. The white noise driven terms in the system are natural for solving practical and theoretical problems. The symbol ω ∈ Ω for stochastic quantities will be understood throughout, but will be written explicitly hereafter.
If g 2,j = 0 and m = 0, Equation (1) will be reduced to the stochastic Euler equation. There are numerous references on the mathematical theory for stochastic Euler equation [11,12]. Bessaih [13] established the existence of martingale solution on bounded domain in R 2 by a compactness method. Brzezniak and Peszat [14] established the existence of martingale solution in R 2 by a viscosity vanishing method. Bessaih and Flandoli [15] studied 2D Euler equation perturbed by noise. Menaldi and Sritharan [16] considered 2D stochastic Navier-Stokes equation. Kim [17] discussed the 2D random Euler equations in a simply-connected bounded domain. Kim [18] established the existence of local smooth solution to 3D stochastic Euler equation forced by additive noise. Kim [19] considered the strong solutions of stochastic 3D Navier-Stokes equations. Glatt-Holtz and Vicol [20] established the local existence of smooth solution to 2D stochastic Euler equation driven by multiplicative noise with slip boundary conditions, and obtained the global existence of smooth solution forced by additive noise. The stochastic MHD equations were considered by many authors. Barbu and Da Prato [21] proved the existence of strong solution to 2D stochastic MHD equations in bounded domains, and the existence and uniqueness of an invariant measure were also obtained by the coupling method. The study on stochastic MHD equations, see also in [22][23][24][25] and references therein. Kim [26] established the existence and uniqueness of a local smooth solution to the stochastic initial value problem with H α (R d )−initial data for α > d 2 + 2, d ≥ 2. If g 1,j = 0 and g 2,j = 0, Equation (1) is reduced to the deterministic MHD equations without diffusion which is a special kind of quasi-linear symmetric hyperbolic system, as we known that it only has the local existence of smooth solution, see [27].
No one has addressed the existence of the local smooth solution to the stochastic MHD equations without diffusion driven by additive noise when the initial data belongs to H α (R 3 ), α > 5 2 . Therefore, we will extend the well-known result for the deterministic MHD equations to the corresponding stochastic case. For the stochastic case, the main difficulty comes from the nonlinear coupling terms as in deterministic case. To overcome this difficulty, we add a cut-off function depending on the size of ∇(u, m) L ∞ (R 3 ) in front of the nonlinear convection term in the spirit of [18]. However, this cut-off function brings us an additional obstacle for uniqueness of the local smooth solution. Furthermore, we introduce a stopping time to overcome this new difficulty. Unlike the deterministic case, we need to show the measurability of solution obtained via a classical change of variable. To obtain suitable estimates, we need more elaborate and complicated estimates with respect to stochastic Euler equation due to the coupled construction of this system.
Our contributions are two-fold. First, we consider the stochastic MHD equations for initial data with spatial regularity actually only in H α (R 3 ), α > 5 2 , in contrast to Kim [26] with H α (R d )−initial data for α > d 2 + 2, d ≥ 2. Then, the system (1) does not contain any diffusion terms. We begin by reviewing some preliminaries associated with Equation (1) and then describe our main result. ∀s ∈ R, we define the usual Sobolev space wheref (ξ) is the Fourier transform of f , ·, · s and · s are the inner product and the norm of H s (R 3 ), respectively. The Sobolev space H s σ (R 3 ) is defined as follows, which is a closed subspace of H s (R 3 ). The projection Π : can be expressed by means of the Fourier transform: Define the trilinear formB : Now we can define B : where the (H 1 σ ) * denotes the dual of H 1 σ . The existence of such an operator is guaranteed by the Riesz representation theorem. The notation ·, · means the duality.
Denote the operators by the divergence free condition, system (1) can be rewritten as, in the sense of distributions in H 1 where U = (u, m) , U 0 = (u 0 , m 0 ) , denotes the transpose. Assume that {G j } j≥1 are H α -valued progressively measurable processes for some α > 5 2 such that for each T > 0, Now, we state the definition of a local smooth solution to the problem (1).

Definition 1.
A pair (U, τ) is called a local smooth solution of (1) if the following conditions are satisfied.
We now describe our main result. Theorem 1. Let U 0 ∈ L 2 (Ω; H α σ ) be F 0 -measurable random variable and {G j } j≥1 satisfy (7). Thus, there exists a unique local smooth solution of (1) in the sense of Definition 1. Then, we obtain the estimate of τ: where c > 0 is a constant independent of U and δ. The solution is unique in the following sense; Suppose that (U 1 , τ) and (U 2 , τ) are local smooth solutions to (1), respectively, if U 0,1 (x) = U 0,2 (x) = U 0 (x) almost surely, The paper is organized as follows. In Section 2, we construct the pathwise approximate smooth solution by introducing a cut-off function and controlling the nonlinear convection term in Equation (6) and regularizing the initial function and the noise with respect to the space variables. Equation (6) can be rewritten to a deterministic problem via a classical change of variable, then we apply the Kato method to get a smooth global solution for each fixed random element ω. To obtain the measurability of the solution, the continuous dependence on the initial data and the noise is established. In Section 3, by energy estimates and stopping time for fixed T > 0 and N ≥ 1 we first prove the existence of local smooth solution to the stochastic modified equations driven by an additive noise, then extend the existence interval by passing T → ∞ and N → ∞ where N ia a parameter in the cut-off function such that N = ∞ makes the cut-off function become an identity map. The limit function will be the solution. Finally, by the Chebyshev inequality and energy estimates, the probability of existence can be made arbitrarily close to one.

Construction of Approximate Solution
Let ρ ε = ρ ε (x), ε > 0 be the standard mollifier, and define where the convolution be taken with respect to the space variable. Then, Q ε is an H α σ -valued continuous square integral martingale for every α ≥ 1. For each integer N ≥ 1, we define Φ N as follows, ε and Y = (v,m), then we define a nonlinear operator and a function as follows, Therefore, Equation (1) can be rewritten as an abstract Cauchy problem Let Λ = (I − ∆) 1 2 and ∆ be the Laplacian operator in R 3 . To apply the Kato method to (13), we need to verify the properties of Λ, A(t, Y), F(t, Y) appearing in Theorem 6 in [27].
are certain non-negative functions defined for b ≥ 0 and T > 0, which are nondecreasing in r and T.
Property (1) follows from the fact that Λ commutes with Π. Property (3) was established in [27] when Φ N = 1 and Q ε = 1. Note that Φ N is independent of space variable and 0 ≤ Φ N ≤ 1. At the same time, Q ε ∈ C([0, ∞); H α+2 σ ) is a given function and it plays the same role as Y. Therefore, the method in [30] can be applicable to property (3). Property (4) holds due to the following inequality, By the fact that Q ε ∈ C([0, ∞); H α+2 σ ) and similar estimates, we can obtain property (5). With these properties in hand, we apply Theorem 6 in [27] to obtain a local existence to the Cauchy problem (13). In order to extend the local solution to a global solution in time, we will need the following lemmas. Lemma 1. [31] Let w be a Lipschitz continuous function in R 3 and v ∈ L 2 (R 3 ). Then, for some constant c independent of ε > 0, v, w, and ∀v, w, the left-hand side tends to zero as ε → 0. Then for a constant c > 0 independent of f , g.
Proof. Firstly, using Lemma 1 and Lemma 2, ∀u, v ∈ H α+1 σ , we have the estimates Due to the work in [27], there is a local solution By virtue of Equations (5) and (14)- (16), we have for some constant C independent of v,m, q 1 ε , q 2 ε , and δ. On the other hand, we can estimate directly for some constants C independent of v,m, q 1 ε , q 2 ε , and δ. Applying Λ α+1 on (17), (18) and multiplying by v * ρ δ andm * ρ δ , respectively; then, using (19)- (21) and passing δ → 0, we obtain for all t ∈ [0, T] and some positive constant C independent of T. By the Grönwall inequality, (22) yields that Y(t) α+1 is bounded on each bounded time interval. Therefore, we can extend the solution Y to a global one.
We prove the measurability of Y as a function of ω ∈ Ω.
we only need to show the continuous dependence of Y on U 0,ε and Q ε . Suppose that as n → ∞, . Define the nonlinear operators and functions as follows, Denote B n such that Then, for each Y ∈ H α+1 σ and t ≥ 0 we have Let Y n be the solution of the problem Due to Theorem 7 of [27], there is some (23). We can partition [0, T] into a number of smaller subintervals to get the continuous dependence of Y on U 0,ε and Q ε on [0, T].

Existence and Uniqueness of the Local Smooth Solution
We now construct the pathwise smooth solution by means of approximate solutions obtained in Section 2.
Step 1. Construct the local smooth solution for any fixed N ≥ 1 and T > 0.
Recalling (12) and (13), we choose a sequence {ε l } of decreasing positive numbers such that as ε l → 0 for almost all ω, where where Y l is a solution to (13) with ε = ε l . Then, we have U l ∈ C([0, T]; H α+1 σ ), for almost all ω and it is H α+1 σ -valued progressively measurable. It holds that in the sense of distributions over R 3 × [0, ∞), for almost all ω. We next define a stopping time T l,K for l, K = 1, 2, · · · by T, if the above set {· · · } is empty.
The Itô formula implies that for all t ≥ 0 and almost all ω. Using Equations (5) and (14), we can estimate the coupling terms for some constant C N > 0 independent of l. The Burkholder-Davis-Gundy inequality implies that where C > 0 is a constant independent of l, K and t. Combining (28)-(31), we have where the constant C N,T is independent of l and K. By the Fatou Lemma and passing K → ∞, we get thus we have P lim Therefore, we consider the following set Let Ω 2 be the set of all ω for which, as l → ∞, and (27) holds in the sense of distributions over R 3 × [0, ∞). Next, we define Ω * = Ω 1 ∩ Ω 2 . Then, P(Ω * ) = 1. Now, fixed any ω * ∈ Ω * . Then, there exists some L = L ω * ≥ 1 and a subsequence denoted by {U l j } such that for all l j . The choice of such a subsequence may depend on ω * . As U l j satisfies in the sense of distributions over R 3 × (0, ∞), thus where C > 0 is a constant independent of U l j and ω * . We choose a constant β > 0 such that (34) and (36), we extract a subsequence still denoted by {U l j } such that for every bounded open ball G in R 3 , as l j → ∞. Equation (38) uses the Corollary 8 in [32]. Next, we will prove that ∇U l j → ∇U strongly in C([0, T]; H β−1 (R 3 )).
for some constants C > 0 independent of h and R ≥ 1.
As the interaction of θ R with the projection operator Π is difficult to handle for our purpose, we remove Π by introducing the vorticity. Letũ l j = ∇ × u l j andm l j = ∇ × m l j . By Equation (27) and the equalities where the vector function we have Combining Equations (4), (5), (34); Lemma 3; and the Young inequality, we can estimate the coupling terms as follows, where C > 0 denotes constant independent of l j , R ≥ 1. By Equations (45)-(47), we obtain for all t ∈ [0, T], where C > 0 is a constant independent of l j , R ≥ 1, t ∈ [0, T) but depends on T. As We also have t 0 θ R f 2 0 ds → 0 as R → ∞ due to the definition of θ R and (34). Due to Grönwall inequality, it follows from (48) that θ RŨl j 0 → 0, as R → ∞, uniformly in l j and t ∈ [0, T]. (49) On the other hand, by Lemma 3 and (34), there is a constant C > 0 independent of l j , R ≥ 1 and t ∈ [0, T] such that By Equations (45) and (46) as well as the interpolation inequality Applying (51), Lemma 3 and the identity ∇ × (θ R U l j ) = θ R (∇ × U l j ) + (∇θ R ) × U l j , we obtain ∇ × (θ R U l j (t)) β−1 → 0 as R → ∞, uniformly in l j and t ∈ [0, T].
Due to the facts The Riesz transform is continuous from H s (R 3 ) into itself for any s and Equations (52)-(55), we have ∇(θ R U l j (t)) β−1 → 0, as R → ∞, uniformly in l j and t ∈ [0, T].
Next, we need to show thatτ N is a stopping time and U(· ∧τ N ) is H α σ -valued progressively measurable.

Proposition 3.τ N is a stopping time.
Proof. We need to show that the set {τ N > t} is F t -measurable for 0 ≤ t < T. We first claim that Let ω ∈ {τ N > t} ∩ Ω * . Then, according to the above procedure, there exists a subsequence {U l j } such that (34), (37)-(39) hold for some function U. Then,τ N can be defined in terms of the limit function U. Asτ N > t, it holds ∇U C([0,t];L ∞ (R 3 )) ≤ N − δ, for some δ. Therefore, we have ∇U l j C([0,t];L ∞ (R 3 )) ≤ N − δ 2 for all sufficiently large l j . Then, ω belongs to the right-hand set. Next, ω belongs to the right-hand set. As ω ∈ Ω * , there exists a subsequence {U l j } such that Equations (34) and (37)-(39) hold for some function U andτ N can be defined in terms of this limit function U. Simultaneously, there exists another subsequenceÛ l j , such that for some L ≥ 1 and ν ≥ 1 for all l j . Applying the above procedure on the interval [0, t], we can further extract a subsequence still denoted byÛ l j which satisfies (35), (37)-(39) hold for some functionÛ, which satisfies that SinceÛ ∈ C([0, t]; H β σ ), U ∈ C([0, T]; H β σ ) and repeating the above procedure on the uniqueness, we obtainÛ Therefore,τ N > t follows from (65). Thus, ω ∈ {τ N > t} ∩ Ω * . As the left-hand set is F t -measurable for 0 ≤ t < T. For t ≥ T, the left-hand set is empty, so it is F t -measurable. Thus,τ N is a stopping time. . Using θ R , we define φ R (x) = 1 − θ R (x), for all x. Then, for all ω ∈ Ω * , we have as R → ∞, for every 0 < T < ∞. As the continuity of time of U, it is enough to show the measurability of φ R U(· ∧τ N ) for each R in B(C([0, T]; H β (R 3 ))). Fixed any 0 < t < ∞ and let B r (p) be the closure of an open ball B r (p) with radius r > 0 and center p in C([0, T]; H β (R 3 )). We first claim that Let ω ∈ {φ R U(· ∧τ N ) ∈ B r (p)} ∩ Ω * . Then, according to the above procedure, there exists a subsequence {U l j } such that Equations (34) and (37)-(39) hold. By Equation (37), for any ν ≥ 1, φ R U l j (· ∧τ N ) ∈ B r+ 1 ν (p), for all sufficiently large l j . Thus, ω belongs to the right-hand set. Next, ω belongs to the right-hand set. Since ω ∈ Ω * , U(· ∧τ N (ω)) is well-defined. At the same time, for some L ≥ 1 such that for each ν ≥ 1, there exists a certain subsequence {U l j } such that (34), (37)-(39) hold for some functionŨ with T replaced by t ∧τ N (ω). Let t ∧τ N (ω), if the above set {· · · } is empty.
holds for all ν ≥ 1. Then, ω belongs to the left-hand set, and Equation (66) is valid. Asτ N is a stopping time, the right-hand set belongs to F t . Consequently, the map To improve the time regularity of U(· ∧τ N ), we need the next lemma which will be used in Proposition 5.

Lemma 4. It holds that
where C N,T is the same constant as in (34).
Proof. Choosing any constant K > 0. We first claim that for each ω ∈ Ω * , For any fixed ω ∈ Ω * . If the right-hand side is equal to K, then (68) holds. Suppose Then, there exists a subsequence {U l j } such that lim l→∞ U l j (·) L ∞ (0,T;H α σ ) = γ. By repetition of the above procedure, we can further extract a subsequence still denoted by {U l j } such that for some functionŨ. Combining Equations (69) and (71), we have Therefore, we obtain (68). Next, by applying the Fatou Lemma and (34), which yields for all K > 0. By passing K → ∞, we obtain (67).
Next, we improve the time regularity of U(· ∧τ N ).
Step 2. Extend the existence interval by passing the limit T → ∞ for any fixed N ≥ 1.
Let U T be the solution obtained with T = 1, 2, · · · .τ N,T is a stopping time defined bŷ T, if the above set {· · · } is empty.
Next, we pass N → ∞ to obtain the maximal time of existence of the local smooth solution. For each N = 1, 2, · · · , let U N be the solution obtained in step 2 andτ N be the stopping time associated with U N by (78). Let Ω 0 = ∞ N=1 Ω N , where Ω N be the same as in step 2 for each N.
Then, we have τ N =τ N , N = 1, 2, · · · and U(t ∧ τ N ) satisfies for all 0 ≤ t < ∞ and each ω ∈ Ω 0 . Thus, (U, τ) is a local smooth solution in the sense of Definition 1. We can show the uniqueness easily as in step 1.
Step 4. Estimate of the stopping time.
We now establish the estimate (11) of the stopping time. For the solution (U, τ) obtained in Step 3, fix any 0 < δ < 1, we have where K > 0 ia a constant defined by Since U(· ∧ τ N ) ∈ C([0, ∞); H α σ ) for almost all ω, we can define a stopping time τ N , if the above set {· · · } is empty.
Author Contributions: Z.Q. carried out the well-posedness of stochastic partial differential equations, and Y.T. carried out the perturbation of noise. Z.Q. and Y.T. carried out the proofs and conceived the study as well as read and agreed to the published version of the manuscript Funding: The project is supported by NSFC grants 11971188, 11471129. Z.Q. is supported by the CSC under grant No. 201806160015.