A well-posedness theory in Sobolev spaces for the stochastic magnetohydrodynamic equations in the whole space
aa r X i v : . [ m a t h . A P ] J u l A WELL-POSEDNESS THEORY IN SOBOLEV SPACES FOR THE STOCHASTICMAGNETOHYDRODYNAMIC EQUATIONS IN THE WHOLE SPACE
ILDOO KIM AND MINSUK YANGA
BSTRACT . We prove the existence of a mild solution to the three dimensional incompressible stochasticmagnetohydrodynamic equations in the whole space with the initial data which belong to the Sobolev spaces.Keywords: stochastic partial differential equations; magnetohydrodynamic equations; mild solution2010 MSC: 60H15; 35Q35
1. I
NTRODUCTION
Magnetohydrodynamics is the branch of physics that studies the dynamics for electrically conductingfluids influenced by a magnetic field. They are frequently generated in nature, for example, the sun,beneath the Earth’s mantle, plasmas in space, liquid metals, and so on. For the background knowledgewe refer the reader to Davidson’s monograph [ ] . The aim of this paper is to establish the existence ofa stochastic mild solution to the three dimensional incompressible magnetohydrodynamic (MHD) equa-tions driven by stochastic external forces, which can be described as the following system of stochasticpartial differential equations ( ∂ t − ∆ ) u + ∇ · ( u ⊗ u − b ⊗ b ) + ∇ π = g k dw k d t ( ∂ t − ∆ ) b + ∇ · ( u ⊗ b − b ⊗ u ) = g k dw k d t ∇ · u = ∇ · b = ( T ) × R with divergence-free initial vector fields u and b , where u , b , π , and dw k dt denote thevelocity field, the magnetic field, the pressure of fluid, and independent one-dimensional white noises ( k =
1, 2, . . . ) , respectively. We note that Einstein’s summation convention is used throughout the paper.In the absence of magnetic field, the MHD equations reduce to the Navier–Stokes equations. Thereare huge literature about the theory of the Navier–Stokes equations. For the deterministic Navier–Stokesequations, Fujita and Kato [ ] initiated the study for the existence of the mild solution with initial datain the critical Sobolev space ˙ H / . Many mathematicians have been interested in its stochastic versionsdue to the complicate dynamics of fluid motions. Naturally, there are many articles handling stochasticNaveri–Stokes equations extend the deterministic results. However, there are only a few results for the stochastic MHD equations. We refer the reader for background and history of these results to the intro-duction of [ ] , where the authors also considered well-posedness of three-dimensional incompressibleMHD equations with stochastic external forces.Before stating our main results, we motivate the definition of a mild solution and simplify the settingsby introducing a few notations. We denote by P the Leray projection operator onto divergence-free vectorfields. In R it can be expressed as P = I + R ⊗ R ,where R = ( R , R , R ) denotes the Riesz transforms and (( R ⊗ R ) u ) i = P j R i R j u j . By applying theLeray projection P to (1) formally, one can remove the pressure term since P ∇ π =
0. By introducingnew variables v = u + b , w = u − b , (2)one can rewrite the equations (1) as ( ∂ t − ∆ ) v = − P ∇ · ( w ⊗ v ) + G k dw k d t ( ∂ t − ∆ ) w = − P ∇ · ( v ⊗ w ) + G k dw k d t ∇ · v = ∇ · w = v = u + b and w = u − b and the stochasticexternal forces G k = P g k + P g k and G k = P g k − P g k .In order to neatly write the equations, we denote the heat semi-group by S ( t ) = e t ∆ and define thefollowing bilinear map B ( u , v )( t ) = Z t S ( t − s ) P ∇ · ( u ⊗ v )( s ) ds . (4)Solving the heat equation by Duhamel’s formula motivates the definition of a mild solution of the sto-chastic MHD equations. We say that ( v , w ) is a (mild) solution to (3) on ( T ) if it solves for 0 ≤ t < T the integral equations v = S ( t ) v − B ( w , v ) + Z t S ( t − s ) G k ( s ) dw ks w = S ( t ) w − B ( v , w ) + Z t S ( t − s ) G k ( s ) dw ks , (5)where w kt is the Brownian motion related to the white noisy dw k dt .In [ ] , well-posedness for the stochastic MHD equations was studied for the initial data v , w ∈ L € Ω , F ; ˙ H / + α σ ( R ) Š with 0 < α < /
2. We extend the well-posedness result for wider class of initial data v , w ∈ L € Ω , F ; ˙ H − / p Š , TOCHASTIC MHD EQUATIONS 3 but we admit that this paper does not cover the multiplicative noise case handled in [ ] with extraregularity condition on initial value. The precise statements of our main results is the following twotheorems. The exact notations and definitions for function spaces are presented in the next section. Theorem 1.
Let v , w ∈ L € Ω , F ; ˙ H / σ ( R ) Š and G , G ∈ ˙ H / p , σ ( ∞ , l ) .(i) There exists a positive number T such that the equation (3) with T = T has a solution v , w ∈ L € Ω , F ; C € [ T ] ; ˙ H / ŠŠ ∩ L € Ω × ( T ) , ¯ P ; ˙ H / Š . (ii) There exists a positive number ǫ such that if k ( v , w ) k L € Ω , F ; ˙ H / Š + k ( G , G ) k ˙ H / p ( ∞ , l ) < ε , then there exists a global in time solution ( v , w ) to the equation (3) with v , w ∈ L € Ω , F ; C € [ ∞ ) ; ˙ H / ŠŠ ∩ L € Ω × ( ∞ ) , ¯ P ; ˙ H / Š . Theorem 2.
Let v , w ∈ L € Ω , F ; ˙ H − / σ Š and G , G ∈ ˙ H − σ ( ∞ , l ) .(i) There exists a positive number T such that the equation (3) with T = T has a solution ( v , w ) ∈ L ( T ) × L ( T ) .(ii) There exists a positive number ε such that if k ( v , w ) k L € Ω , F ; ˙ H − / Š + k ( G , G ) k ˙ H − ( ∞ , l ) < ε , then there exists a global in time solution ( v , w ) to the equation (3) with v , w ∈ L ( ∞ ) . Remark 1. If X = L ( ∞ ) or X = L € Ω , F ; L (( ∞ ) ; ˙ H / ( R )) Š , then there is a positive constant C such that k B ( v , w ) k X × X ≤ C k v k X k w k X ∀ v , w ∈ X , which is proved in Proposition 2 and Proposition 3. We can obtain an upper bound ǫ < C i C ( i =
5, 6 ) in the second parts of main theorems, where constants C and C appear in Corollary 1 and Corollary 2,respectively. Moreover, in this case, by the uniqueness of the fixed point theorem, the solution is unique inthe closed subspace { ( u , v ) ∈ X × X : k ( u , v ) k X × X ≤ ǫ } (see Lemma 1). ILDOO KIM AND MINSUK YANG
Remark 2.
Although we took physical constants to be 1, it is possible to consider the general physicalconstants in the MHD equations, that is, ( ∂ t − ν ∆ ) v = − P ∇ · ( w ⊗ v ) + G k dw k d t ( ∂ t − ν ∆ ) w = − P ∇ · ( v ⊗ w ) + G k dw k d t ∇ · v = ∇ · w = where the kinematic viscosity ν and the magnetic resistivity ν are positive constants. In this paper, we focuson studying (3) for simplicity. The organization of this paper is as follows. In Section 2, we introduce notations and definitions usedthroughout this paper. In Section 3, we survey linear theories for stochastic partial differential equations.In Section 4, we prove bilinear estimates which play a crucial role. In Section 5, we complete the proofsof main theorems. 2. N
OTATIONS AND DEFINITIONS
The purpose of this section is to introduce notations and definitions which will be used throughoutthis paper. • Let N and Z denote the natural number system and the integer number system, respectively. Asusual R d , d ∈ N , stands for the Euclidean space of points x = ( x , ..., x d ) . • The gradient of a function f is denoted by ∇ f = ( D f , D f , · · · , D d f ) .where D i f = ∂ f ∂ x i for i =
1, ..., d and the divergence of a vector field v = ( v , . . . , v d ) is denotedby ∇ · v : = d X i = D i v i . • Let C ∞ ( R d ) denote the space of infinitely differentiable functions on R d . Let C ∞ c ( R d ) denotethe subspace of C ∞ ( R d ) with the compact support. Let C ∞ c , σ ( R d ) denote the subspace of C ∞ c ( R d ) with divergence free. Let S ( R d ) be the Schwartz space consisting of infinitely differentiable andrapidly decreasing functions on R d . We simply write C ∞ , C ∞ c , C ∞ c , σ , S by omitting ( R d ) . • For
O ⊂ R d and a normed space F , we denote by C ( O ; F ) the space of all F -valued continuousfunctions u : O → F with the norm | u | C : = sup x ∈ O | u ( x ) | F < ∞ . TOCHASTIC MHD EQUATIONS 5 • For p ∈ [ ∞ ) , a normed space F , and a measure space ( X , M , µ ) , we denote by L p ( X , M , µ ; F ) the space of all M µ -measurable functions u : X → F with the norm k u k L p ( X , M , µ ; F ) : = (cid:18)Z X k u ( x ) k pF µ ( d x ) (cid:19) / p < ∞ where M µ denotes the completion of M with respect to the measure µ . If there is no confusionfor the given measure and σ -algebra, we usually omit them. • We denote by |O | the Lebesgue measure of a measurable set
O ⊂ R d . • Let ( Ω , F , P ) be a probability space and let u ( ω , t , x ) and v ( ω , t , x ) be stochastic processes on Ω × ( ∞ ) × R d . We say that with probability one, for all t ∈ ( ∞ ) u ( ω , t , x ) = v ( ω , t , x ) ( x - a . e . ) if there exists Ω ′ ⊂ Ω such that P ( Ω ′ ) = ( ω ′ , t ) ∈ Ω ′ × ( ∞ ) , u ( ω ′ , t , x ) = v ( ω ′ , t , x ) holds for almost every x ∈ R d . For the notational convenience, the random parameter ω will beusually omitted. • We denote the d -dimensional Fourier transform of f by F [ f ]( ξ ) : = Z R d e − π i ξ · x f ( x ) d x and the d -dimensional inverse Fourier transform of f by F − [ f ]( x ) : = Z R d e π ix · ξ f ( ξ ) d ξ . • If we write C = C ( a , b , · · · ) , this means that the constant C depends only on a , b , · · · . • We shall write A ® B if there is a positive generic constant C such that | A | ≤ C | B | .Let ( Ω , F , P ) be a complete probability space, and {F t , t ≥ } be a filtration satisfying the usualcondition, i.e. {F t } is increasing , right continuous, and each F t contains all ( F , P ) -null sets. In otherwords, F t ⊂ F t if t ≤ t , T t < s F s = F t , and A ⊂ F t for all t ≥ B ∈ F such that A ⊂ B and P ( B ) =
0. We denote by P the predictable σ -field generated by {F t , t ≥ } . We assume that w kt areindependent one-dimensional Brownian motions (Wiener processes) on ( Ω , F , P ) for ( k =
1, 2, . . . ) andthey are relative to {F t , t ≥ } .We end this section by introducing inhomogeneous and homogeneous Sobolev spaces and relatedfunction spaces used in this article. Definition 1 (Inhomogeneous Sobolev spaces) . (1) For n ∈ R and p ∈ ( ∞ ) , define the spaceH np ( R ) = ( − ∆ ) − n / L p ( R ) (called the space of Bessel potentials or the Sobolev space with frac-tional derivatives) as the set of all tempered distributions u such that ( − ∆ ) n / u : = F − ”(cid:0) + | πξ | (cid:1) n / F ( u )( ξ ) — ∈ L p ( R ) ILDOO KIM AND MINSUK YANG with the norm k u k H np ( R ) : = k ( − ∆ ) n / u k L p ( R ) < ∞ . (2) The set of all u = ( u , u , u ) such that u , u , u ∈ H np ( R ) is denoted by H np ( R ; R ) and the normis given by k u k H np ( R ; R ) : = X i = (cid:13)(cid:13) ( − ∆ ) n / u i (cid:13)(cid:13) L p ( R ) < ∞ . H np , σ ( R ; R ) denotes the closure of C ∞ c , σ ( R ; R ) in H np ( R ; R ) . For a sequence a = ( a , a , . . . ) = ( a k ) k ∈ N ,we define k a k l : = P ∞ k = | a k | and denote by l the space of all sequences a so that k a k l < ∞ . For u =( u k ) k ∈ N ( u k ∈ H np ( R )) , we define the space H np ( R ; l ) as the set of all l -valued tempered distributionssuch that k u k H np ( R ; l ) : = (cid:13)(cid:13) | ( − ∆ ) n / u | l (cid:13)(cid:13) L p ( R ) < ∞ .The set of all u = ( u , u , u ) such that u , u , u ∈ H np ( R ; l ) is denoted by H np ( R ; R × l ) with the norm k u k H np ( R ; R × l ) : = X i = (cid:13)(cid:13) u i (cid:13)(cid:13) H np ( R ; l ) H np , σ ( R ; R × l ) denotes the subspace of H np ( R ; R × l ) in which every component is divergence-free,i.e. u = (cid:0) u ik ( i =
1, 2, 3, k ∈ N ) (cid:1) ∈ H np , σ ( R ; R × l ) if and only if u ik ∈ H np , σ ( R ; R ) for all i =
1, 2, 3 and k ∈ N . In particular, we put L p , σ : = H p , σ . For thenotational convenience, we set H np = H np ( R ; R ) , H np , σ = H np , σ ( R ; R ) , H np ( l ) = H np ( R ; R × l ) , H np , σ ( l ) = H np , σ ( R ; R × l ) .We write u ∈ H np ( T ) if u is an H np -valued ¯ P -measurable process satisfying k u k H np ( T ) : = ‚ E –Z T k u k pH np d t ™Œ / p < ∞ . TOCHASTIC MHD EQUATIONS 7
For the notational convenience, we set H np ( T ) : = L p ( Ω × ( T ) , ¯ P ; H np ) , H np , σ ( T ) : = L p ( Ω × ( T ) , ¯ P ; H np , σ ) H np ( T , l ) : = L p ( Ω × ( T ) , ¯ P ; H np ( l )) , H np , σ ( T , l ) : = L p ( Ω × ( T ) , ¯ P ; H np , σ ( l )) .Similarly, we write U np : = L p ( Ω , F ; H np ) , U np , σ : = L p ( Ω , F ; H np , σ ) ,and simply L p ( T ) : = H p ( T ) , L p , σ ( T ) : = H p , σ ( T ) , L p ( T , l ) : = H p ( T , l ) , L p , σ ( T , l ) : = H p , σ ( T , l ) . Definition 2 (Homogeneous Sobolev spaces) . (1) We denote by ˙ H np ( R ; R ) the space of all R -valuedtempered distribution u = ( u , u , u ) modulo by polynomials such that ( − ∆ ) n / u i : = F − (cid:2) | πξ | n F ( u i )( ξ ) (cid:3) ∈ L p ( R ) with the norm k u k ˙ H np ( R ; R ) : = X i = (cid:13)(cid:13) ( − ∆ ) n / u i (cid:13)(cid:13) L p ( R ) < ∞ . We denote by ˙ H np ( R ; R × l ) the space of all sequence u = ( u ik ) i = k ∈ N such that u k ∈ ˙ H np ( R ; R ) for all k ∈ N and k u k ˙ H np ( R ; R × l ) : = X i = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)‚ ∞ X k = (cid:12)(cid:12) ( − ∆ ) n / u ik (cid:12)(cid:12) Œ / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R ) < ∞ .˙ H n , σ p ( R ; R ) and ˙ H n , σ p ( R ; R × l ) denote the subspace of ˙ H np ( R ; R ) and ˙ H np ( R ; R × l ) whoseelements are divergence free, respectively, i.e. ˙ H n , σ p ( R ; R ) and ˙ H n , σ p ( R ; R × l ) are closures ofC ∞ c with respect the norms in ˙ H np ( R ; R ) and ˙ H np ( R ; R × l ) , respectively. ILDOO KIM AND MINSUK YANG (2) Similar to inhomogeneous function spaces, we set deterministic spaces ˙ H np = H np ( R ; R ) ˙ H np , σ = ˙ H np , σ ( R ; R ) ˙ H np ( l ) = H np ( R ; R × l ) ˙ H np , σ ( l ) = ˙ H np , σ ( R ; R × l ) , and stochastic spaces ˙ H np ( T ) : = L p ( Ω × ( T ) , ¯ P ; ˙ H np ) ˙ H np , σ ( T ) : = L p ( Ω × ( T ) , ¯ P ; ˙ H np , σ ) ˙ H np ( T , l ) : = L p ( Ω × ( T ) , ¯ P ; ˙ H np ( l )) ˙ H np , σ ( T , l ) : = L p ( Ω × ( T ) , ¯ P ; ˙ H np , σ ( l )) , and ˙ U np : = L p ( Ω , F ; ˙ H np ) ˙ U np : = L p ( Ω , F ; ˙ H np , σ ) . Remark 3.
It is well-known that two norms k · k H np and k · k ˙ H np + k · k L p are equivalent if < p < ∞ andn > (cf. [
4, Theorem 6.3.2 ] ). Thus, H np = ˙ H np ∩ L p and k · k ˙ H np ® k · k H np if < p < ∞ and n > .
3. L
INEAR THEORIES FOR STOCHASTIC HEAT EQUATIONS
In this section, we introduce linear theories for stochastic PDEs. Recently, analytic regularity theoriesfor stochastic PDEs have been well developed. We refer the reader to [ ] , which is considered as one ofbibles in this area. However, most of the estimates and theories are handled in inhomogeneous Sobolevspace setting. To prove our main theorems, we need to obtain homogeneous type estimates for linearstochastic PDEs. Since we could not find an appropriate reference to show homogeneous type estimatesfor stochastic PDEs, we give detailed proofs. In addition, we note that all functions in the followingtheorem are R -valued, but the results in [ ] are for scalar-valued functions. Since our leading operatoris Laplacian, those are easily extended to R -valued functions without any difficulty. Proposition 1.
Let n ∈ R and T ∈ ( ∞ ) . If u ∈ U n + − / pp , f ∈ H np ( T ) , and g ∈ H n + p ( T , l ) , then thereexists a unique solution u ∈ H n + p ( T ) tou t ( t , x ) = ∆ u ( t , x ) + f ( t , x ) + g k ( t , x ) dw k d tu ( x ) = u ( x ) (6) TOCHASTIC MHD EQUATIONS 9 in ( T ) × R in the sense that for any φ ∈ C ∞ c , with probability one, for all t ∈ ( T ) , ( u ( t , · ) , φ ) = ( u , φ ) + Z t [( u ( s , · ) , ∆ φ ) + ( f ( s , · ) , φ )] ds + Z t (cid:0) g k ( s , · ) , φ (cid:1) dw ks . Moreover, there exist positive constants C ( T , p ) , C ( p ) , C ( T , p ) , and C ( p ) such that k u k H n + p ( T ) ≤ C € k u k U n + − / pp + k f k H np ( T ) + k g k H n + p ( T , l ) Š (7) k u k ˙ H n + p ( T ) ≤ C € k u k ˙ U n + − / pp + k f k ˙ H np ( T ) + k g k ˙ H n + p ( T , l ) Š (8) and E sup ≤ t ≤ T k u ( t , · ) k pH n + − / pp ≤ C (cid:129) k u k p ˙ U n + − / pp + k f k p H n + − / pp ( T ) + k g k p H n + − / pp ( T , l ) ‹ (9) E sup ≤ t ≤ T k u ( t , · ) k p ˙ H n + − / pp ≤ C (cid:129) k u k p ˙ U n + − / pp + k f k p ˙ H n + − / pp ( T ) + k g k p ˙ H n + − / pp ( T , l ) ‹ . (10) Proof.
This proposition with the estimates (7) and (9) was proved in [
5, Lemma 4.1 and Theorem 4.10 ] .We only prove (8) since the proof of (10) is similar. We may prove it with assuming T = w t : = p T w
T t .is also a Brownian motion. If the proposition is proved for T =
1, then we define˜ u ( t , x ) = u ( T t , p T x ) ˜ u ( x ) = u ( p T x ) ˜ f ( t , x ) = T f ( T t , p T x ) ˜ g ( t , x ) = p T g ( T t , p x ) ,where u , u , f , g satisfy (6) so that ˜ u is a solution to˜ u t ( t , x ) = ∆ ˜ u ( t , x ) + ˜ f ( t , x ) + ˜ g k ( t , x ) d ˜ w k d t ˜ u ( x ) = ˜ u ( x ) in (
0, 1 ) × R . Due to the estimate for ˜ u , we have k u k ˙ H n + p ( T ) = T / p T − ( n + ) / T / ( p ) k ˜ u k ˙ H n + p ( ) ≤ T / p T − ( n + ) / T / ( p ) C € k ˜ u k ˙ U n + − / pp + k ˜ f k ˙ H np ( ) + k ˜ g k ˙ H n + p ( l ) Š = C € k u k ˙ U n + − / pp + k f k ˙ H np ( T ) + k g k ˙ H n + p ( T , l ) Š .Now, we assume T = • Case 1 . ( f = g = ( − ∆ ) ( n + ) / − / p to both sides of (6) and applying (7)with n = − + / p , we have k u k ˙ H n + p ® k ∆ ( n + ) / − / p u k H / pp ® k ∆ ( n + ) / − / p u k U p = k u k ˙ U n + − / pp . (11) • Case 2 . ( u = g = ( − ∆ ) n / to both sides of (6) and applying (7) with n =
0, we have k u k ˙ H n + p ® k ∆ n / u k H p ® k ∆ n / f k L p = k f k ˙ H np . (12) • Case 3 . ( u = f = ( − ∆ ) ( n + ) / to both sides of (6) and applying (7) with n = −
1, we have k u k ˙ H n + p ® k ∆ ( n + ) / u k H p ® k ∆ ( n + ) / g k L p ( T , l ) = k g k ˙ H n + p ( T , l ) . (13) • Case 4 .(General Case). Suppose u is a unique solution to u t ( t , x ) = ∆ u ( t , x ) u ( x ) = u ( x ) , u is a unique solution to u t ( t , x ) = ∆ u ( t , x ) + f ( t , x ) u ( x ) = u is a unique solution to u t ( t , x ) = ∆ u ( t , x ) + g k ( t , x ) dw k d tu ( x ) = u = u + u + u for the general case. Combining the estimates (11),(12), and (13), we obtain that (8) with T = (cid:3) Remark 4.
The constants C and C depend on T . However, one can take C and C uniformly for allT ∈ [
0, ¯ T ] , i.e. for all T ∈ [
0, ¯ T ] , k u k H n + p ( T ) ≤ C € k u k U n + − / pp + k f k H np ( T ) + k g k H n + p ( T , l ) Š , and E sup ≤ t ≤ T k u ( t , · ) k pH n + − / pp ≤ C (cid:129) k u k pU n + − / pp + k f k p H n + − / pp ( T ) + k g k p H n + − / pp ( T , l ) ‹ , where constants C and C depend only on ¯ T and p.
We denote by H ∞ c ( T , l ) the space of stochastic processes g = ( g , g , . . . ) such that g k = k and each g k is of the type g k ( t , x ) = j ( k ) X i = ( τ i − , τ i ] ( t ) g ik ( x ) , TOCHASTIC MHD EQUATIONS 11 where j ( k ) ∈ N , g ik ∈ C ∞ c , and τ i are stopping times with τ i ≤ T . Similarly, we denote by H ∞ c ( T ) thespace of the processes g such that g ( t , x ) = j X i = ( τ i − , τ i ] ( t ) g i ( x ) ,where j ∈ N , g i ∈ C ∞ c , and τ i are stopping times with τ i ≤ T . Lastly, we denote by H ∞ c ( R ) the spaceof the random variables g of the type g ( ω , x ) = A ( ω ) g ( x ) where g ∈ C ∞ c , and A ∈ F . Remark 5. (i) It is known that H ∞ c ( T , l ) is dense in H np ( T , l ) for all p ∈ ( ∞ ) and n ∈ R (for instance,see [
5, Theorem 3.10 ] ). In particular, H ∞ c ( T ) is dense in H np ( T ) for all p ∈ ( ∞ ) and n ∈ R .Following the idea of [
5, Theorem 3.10 ] , one can also easily check that H ∞ c ( R ) is dense in U np for allp ∈ ( ∞ ) and n ∈ R .(ii) If g ∈ H n + p ( T , l ) , then the stochastic integral Z t (cid:0) g k ( s , · ) , φ (cid:1) dw ks is well-defined in the classical Itô-sense (cf. [
5, Remark 3.2 ] ). Moreover, the H n + p ( l ) -valued stochasticintegral Z t g ( s , x ) dw s is defined by recently developed UMD-space valued stochastic integral theory (cf. [ ] ).(iii) The dual space of H np is H − nq , where q = pp − , and one can find a countable subset of C ∞ c which is densein H − nq . Thus, (7) can be interpreted in strong sense. i.e. (6) holds if and only if with probability one,for all t ∈ ( T ) , u ( t , x ) = u + Z t ( ∆ u ( s , x ) + f ( s , x )) ds + Z t g k ( s , x ) dw ks where the equality holds as an element of H np , R t ( ∆ u ( s , x ) + f ( s , x )) ds is H np -valued classical Bochner’sintegral, and R t g ( s , x ) dw s is H np ( l ) -valued stochastic integral.(iv) If u ∈ H ∞ c ( R ) , f ∈ H ∞ c ( T ) , and g ∈ H ∞ c ( T , l ) , then the solution u is given by (cf. [
5, proof ofTheorem 4.2 ] )u ( t , x ) = S ( t ) u ( x ) + Z t S ( t − s ) f ( s , · )( x ) ds + Z t S ( t − s ) g ( s , · ) dw ks , (14) where p ( t , x ) = ( π t ) − / exp (cid:0) −| x | / ( t ) (cid:1) S ( t ) u ( x ) = Z R p ( t , x − y ) u ( y ) d y . Due to (7) , the standard approximation, and UMD space-valued stochastic integration theories, (14) holds even for general u ∈ H n + p ( R ) , f ∈ H np ( T ) , and g ∈ H np ( T , l ) .(v) In Proposition 1, we assumed that u ∈ U n + − / pp , f ∈ H np ( T ) , g ∈ H n + p ( T , l ) . However by usingapproximations in u N ∈ ˙ U n + − / pp ∩ U n + − / pp , f N ∈ ˙ H np ( T ) ∩ H np ( T ) , g N ∈ ˙ H n + p ( T , l ) ∩ H n + p ( T , l )( N =
1, 2, . . . ) or applying a UMD-space valued stochastic maximal L p -inequality ( [
7, Theorem 1.1 ] ),one can easily check that (8) and (10) hold for all u ∈ ˙ U n + − / pp , f ∈ ˙ H np ( T ) , g ∈ ˙ H n + p ( T , l ) , and udefined as in (14) . We state a few corollaries, which is easily deduced by Proposition 1 and the representation (14).
Corollary 1.
There exists a positive constant C such that for all T ∈ ( ∞ ) , u ∈ ˙ U / p , and g ∈ ˙ H / ( T , l ) , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) S ( t ) u ( x ) + Z t S ( t − s ) g ( s , · ) dw ks (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˙ H / ( T ) + E sup ≤ t ≤ T (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) S ( t ) u ( x ) + Z t S ( t − s ) g ( s , · ) dw ks (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p ˙ H / ≤ C € k u k ˙ U / + k g k ˙ H / ( T , l ) Š . Corollary 2.
There exists a positive constant C such that for all T ∈ ( ∞ ) , u ∈ ˙ U − / , and g ∈ ˙ H − ( T , l ) , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) S ( t ) u ( x ) + Z t S ( t − s ) g ( s , · ) dw ks (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ˙ L ( T ) ≤ C € k u k ˙ U − / + k g k ˙ H − ( T , l ) Š . Corollary 3. (i) There exists a positive constants C such that for all T ∈ ( ∞ ) and u ∈ ˙ H / , Z T k S ( t ) u k H / d t ≤ C k u k ˙ H / . (ii) There exists a positive constant C such that for all T ∈ ( ∞ ) and f ∈ L (( T ) ; ˙ H − / ) , sup ≤ t ≤ T (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z t S ( t − s ) f ( s , · ) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H / + Z T (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z t S ( t − s ) f ( s , · ) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H / d t ≤ C Z T k f ( t , · ) k H − / d t . (iii) There exists a positive constant C such that for all T ∈ ( ∞ ) and u ∈ H − / , k S ( t ) u k L (( T ) × R ) ≤ C k u k H − / .4. B ILINEAR E STIMATES
The goal of this section is to prove bilinear estimates, Proposition 2 and Proposition 3, which play keyroles to obtain our existence results for stochastic MHD equations.
TOCHASTIC MHD EQUATIONS 13
Before going further, we recall that if v ∈ C ([ T ] ; ˙ H / ) ∩ L (( T ) ; ˙ H / ) ,then from an interpolation inequality in the Sobolev space, k v k L (( T ) ; ˙ H ) ® k v k / C ([ T ] ; ˙ H / ) k v k / L (( T ) ; ˙ H / ) (15)and hence C ([ T ] ; ˙ H / ) ∩ L (( T ) ; ˙ H / ) ⊂ L (( T ) ; ˙ H ) .For the notational convenience, we set X = C ([ T ] ; ˙ H / ) ∩ L (( T ) ; ˙ H / ) with the norm k · k X = k · k C ([ T ] ; ˙ H / ) + k · k L (( T ) ; ˙ H / ) .For the corresponding stochastic function spaces, we set X T : = L € Ω , F ; C ([ T ] ; ˙ H / ( R )) Š ∩ ˙ H / ( T ) , X T : = L € Ω , F ; L ([ T ] ; ˙ H / ( R )) Š with the norms k · k X T = k · k L € Ω , F ; C ([ T ] : ˙ H / ( R )) Š + k · k ˙ H / ( T ) , k · k X T = ‚ E Z T k·k H d t Œ / .Due to (15), we have X T ⊂ X T . Proposition 2.
There exists a constant C > such that for all T > and v , w ∈ X T , k B ( w , v ) k X T ≤ C k v k X T k w k X T . Proof.
We begin by recalling B ( u , v )( t ) = Z t S ( t − s ) P ∇ · ( u ⊗ v )( s ) ds .Since the embedding L / ⊂ ˙ H − / is continuous and the projection operator P is continuous on L / , wehave k P ∇ · ( v ⊗ w ) k ˙ H − / ® k P ∇ · ( v ⊗ w ) k L / ® k∇ · ( v ⊗ w ) k L / .Since ∇ · v =
0, we use the Hölder inequality and the Sobolev inequality to get k∇ · ( v ⊗ w ) k L / ® k v · ∇ w k L / ® k v k L k∇ w k L . ® k v k ˙ H k w k ˙ H Thus, we use the Cauchy–Schwarz inequality inequality to get Z T k P ∇ · ( v ⊗ w ) k H − / d t ® Z T k v k H k w k H d t ® ‚Z T k v k H d t Œ / ‚Z T k w k H d t Œ / . (16)If we put f : = P ∇ · ( v ⊗ w ) ∈ L (( T ) : ˙ H − / ( R )) ,then from Corollary 3(ii),sup ≤ t ≤ T (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z t S ( t − s ) f ( s , · ) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H / + Z T (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z t S ( t − s ) f ( s , · ) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H / d t ® Z T k f ( t , · ) k H − / d t . (17)Combining (16) and (17), we obtain for all T > v , w ∈ L (( T ) ; ˙ H ) , k B ( v , w ) k X ® ‚Z T k v k H d t Œ / ‚Z T k w k H d t Œ / .Thus, we use the Cauchy–Schwarz inequality inequality to obtain that k B ( w , v ) k X T ® E ‚Z T k w k H d t Œ / ‚Z T k v k H d t Œ / ® k v k X T k w k X T . (cid:3) We now prove that the bilinear operator is jointly continuous on L ( t ) × L ( t ) . We divide its proofinto a few steps. Proposition 3.
There exists a constant C > such that for all T > and v , w ∈ L ( t ) , k B ( v , w ) k L ( T ) ≤ C k v k L ( T ) k w k L ( T ) . Proof.
Step 1)
To prove this proposition, it is useful to have an integral representation B ( v , w )( t ) = Z t S ( t − s ) P ∇ · ( v ⊗ w )( s ) ds = Z t K ( t − s ) ∗ ( v ⊗ w )( s ) ds with a kernel K . TOCHASTIC MHD EQUATIONS 15
Let h be a L p ( R ) -valued function defined on ( T ) with T > p ∈ ( ∞ ) . For each s ∈ ( T ) , h ( s ) ∈ L p ( R ) . That is, fixing s , we can regard h ( s ) as a function defined on R andby h ( s , y ) we denote the value of this function at y ∈ R . The kernel K should satisfy K ( t − s , · ) ∗ h ( s , · ) = S ( t − s ) P ∇ h ( s ) = S ( t − s )( I + R ⊗ R ) ∇ h ( s ) so that we can write its j -th component of K as an inverse Fourier transform of the multiplier ( K ( t , x )) j = − i X k , m ∈{ } F − ” e − t | ξ | | ξ | − ξ j ξ k ξ m — ( x ) .The key points of this kernel representation are the following scaling property K ( t , x ) = t − K ( t − / x ) (18)and the pointwise decay propertysup x ∈ R ( + | x | ) | K ( x ) | < ∞ . (19)Ossen studied this kind of kernel and now its properties are regarded as well-known facts. Werefer the reader to [ ] for its generalization and other fine properties. For reader’s conveniencewe provide a proof of these two basic facts at Step 3. Step 2)
We use the Young inequality and the scaling property (18) to obtain that k B ( v , w )( t ) k L ( R ) ® Z t k K ( t − s , · ) ∗ ( v ⊗ w )( s ) k L ( R ) ds ® Z t k K ( t − s , · ) k L / ( R ) k ( v ⊗ w )( s ) k L / ( R ) ds ® Z t ( t − s ) − / k K ( · ) k L / ( R ) k ( v ⊗ w )( s ) k L / ( R ) ds .Since k K ( · ) k L / ( R ) < ∞ from the pointwise decay property (19), we use the Hardy–Littlewood–Sobolev theorem on fractional integration (cf. [
3, Theorem 6.1.3 ] ) to obtain that Z k B ( v , w )( t ) k L ( R ) d t ® Z T (cid:18)Z t ( t − s ) − / k ( v ⊗ w )( s ) k L / ( R ) ds (cid:19) d t ® ‚Z T k ( v ⊗ w )( t ) k / L / ( R ) d t Œ .Since k v ⊗ w k L / ( T ) ≤ k v k L ( T ) k w k L ( T ) , we use the Cauchy–Schwarz inequality inequality toget the result. Step 3)
Finally, we prove (18) and (19). We denote K j , k , m ( t , x ) = F − ” e − t | ξ | | ξ | − ξ j ξ k ξ m — . Since we have ( K ( t , x )) j = − i X k , m K j , k , m ( t , x ) ,it suffices to prove that K j , k , m ( t , x ) satisfies the same properties as (18) and (19). We notice that ∂ t K j , k , m ( t , x ) = − Z R e ix · ξ e − t | ξ | ξ j ξ k ξ m d ξ = iD jx D kx D m Z R e ix · ξ e − t | ξ | d ξ = iD jx D kx D mx ( π/ t ) / e − π | x | / t and K j , k , m ( t , x ) goes to 0 as t → ∞ . Thus, by the Fundamental theorem, of calculus, we have K j , k , m ( t , x ) = i Z ∞ t D jx D kx D mx ( π/ s ) / e − π | x | / s ds = i π / x j x k x m Z ∞ t s − / e − π | x | / s ds .By the change of variables π | x | / s = a , we get K j , k , m ( t , x ) = i π / x j x k x m Z ∞ t s − / e − π | x | / s dss = i π / x j x k x m | x | − Z π | x | / t a / e − a da .From this identity it is easy to check the scaling K j , k , m ( t , x ) = t − K j , k , m ( t − / x ) ,and the decay | K j , k , m ( x ) | ® ( + | x | ) − . (cid:3)
5. P
ROOF OF MAIN THEOREMS
The proofs of Theorem 2 and Theorem 1 go along the same series of steps. In order to write theprocedures in a tidy manner, we shall prove the following abstract fixed point lemma.We recall that if X is a Banach space with the norm k · k X , then the product space X × X is a Banachspace with the norm k ( v , w ) k X × X = k v k X + k w k X .We shall use the Banach fixed point theorem to the product space X × X . Lemma 1.
Let X be a Banach space with the norm k · k X . Assume that B ( v , w ) be a jointly continuousbilinear operator on X × X , that is, there exists a positive constant C X such that forall v , w ∈ X , k B ( v , w ) k X ≤ C X k v k X k w k X . (20) TOCHASTIC MHD EQUATIONS 17 If ǫ < ( C X ) − , k v k X + k w k X ≤ ǫ , (21) and ( v n , w n ) is a sequence in X × X satisfying v n + = v − B ( w n , v n ) w n + = w − B ( v n , w n ) , (22) then there exists a unique ( v , w ) ∈ X × X satisfying k v k X + k w k X ≤ ǫ . and v = v − B ( w , v ) w = w − B ( v , w ) . (23) Proof.
We note first that for all n ∈ N , k v n k X + k w n k X ≤ ǫ . (24)Indeed, mathematical induction yields k v n + k X + k w n + k X ≤ k v k X + k w k X + k B ( w n , v n ) k X + k B ( v n , w n ) k X ≤ ǫ + C X k v n k X k w n k X ≤ ǫ + C X ǫ ≤ ǫ .Since B is bilinear, we may write for n ≥ v n + − v n = − B ( w n , v n ) + B ( w n − , v n − )= − B ( w n , v n − v n − ) − B ( w n − w n − , v n − ) and w n + − w n = − B ( v n , w n ) + B ( v n − , w n − )= − B ( v n , w n − w n − ) − B ( v n − v n − , w n − ) .Thus, we have k ( v n + , w n + ) − ( v n , w n ) k X × X = k v n + − v n k X + k w n + − w n k X ≤ ǫ C X ( k v n − v n − k X + k w n − w n − k X ) < δ k ( v n , w n ) − ( v n − , w n − ) k X × X by (24), where δ is a constant satisfying 4 ǫ C X < δ < Therefore, the Banach fixed-point theorem implies that ( v n , w n ) converges to a unique ( v , w ) ∈ X × X satisfying (23), which follows by taking the limit to (22) and (24). (cid:3) Now, we are ready to prove Theorem 1.
Proof of Theorem 1.
Recall X T : = L € Ω , F ; C ([ T ] : ˙ H / ( R )) Š ∩ ˙ H / ( T ) , X T : = L (cid:0) Ω , F ; L ([ T ] : ˙ H ( R )) (cid:1) .From (5), Corollary 1, and Proposition 2, we have k ( v , w ) k X T × X T ® k ( v , w ) k ˙ U / p × ˙ U / p + k ( G , G ) k ˙ H / ( T , l ) × ˙ H / ( T , l ) + k v k X T k w k X T .Thus, it suffices to find a solution ( v , w ) ∈ X T × X T .We set v ( t , x ) = S ( t ) v ( x ) + Z t S ( t − s ) G ( s , · )( x ) dw ks w ( t , x ) = S ( t ) w ( x ) + Z t S ( t − s ) G ( s , · )( x ) dw ks and inductively define ( v n , w n ) for n ≥ v n ( t , x ) = S ( t ) w ( x ) − B ( w n − , v n − ) + Z t S ( t − s ) G ( s , · ) dw ks w n ( t , x ) = S ( t ) v ( x ) − B ( v n − , w n − ) + Z t S ( t − s ) G ( s , · ) dw ks .By mathematical induction, we have for all n ∈ N , v n , w n ∈ X T . (25)Indeed, due to (15) and Corollary 1, we have v , w ∈ X T . If we assume that (25) holds with n = k ,then by (15), Corollary 1, and Proposition 2, k ( v k + , w k + ) k X T × X T ® k ( v , w ) k ˙ U / p × ˙ U / p + k ( G , G ) k ˙ H / ( T , l ) × ˙ H / ( T , l ) + k v k k X T k w k k X T < ∞ .This proves (25).Finally, we can apply Lemma 1 due to (15), Corollary 1 and Proposition 2 so that we get the secondpart, Theorem 1(ii). TOCHASTIC MHD EQUATIONS 19
To obtain the first part, Theorem 1(i), it suffices to show that there exists a positive constant T ∈ ( ∞ ) such that k v k X T + k w k X T < C ,which is an immediate consequence of the fact thatlim T → k v k X T + k w k X T = (cid:3) The proof of Theorem 2 is almost identical to the proof of Theorem 1 except that Corollary 2 andProposition 3 are used in place of Corollary 1 and Proposition 2.A
CKNOWLEDGEMENTS
I. Kim’s work was supported by the National Research Foundation of Korea(NRF) grant funded bythe Korea government(MSIT) (No. 2017R1C1B1002830). M. Yang’s work was partially supported bythe National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No.2015R1A5A1009350) and by the Yonsei University Research Fund (No. 2018-22-0046).R
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