Well-posedness for Stochastic Generalized Fractional Benjamin-Ono Equation
aa r X i v : . [ m a t h . A P ] A p r Well-posedness for Stochastic GeneralizedFractional Benjamin-Ono Equation ∗ Wei Yan
Department of Mathematics and information science, Henan Normal University,Xinxiang, Henan 453007, P.R.ChinaEmail:[email protected]
Jianhua Huang † College of Science, National University of Defense and Technology,Changsha,P. R. China 410073Email: [email protected] andBoling Guo
Institute of Applied Physics and Computational Mathematics, Beijing, 100088Email: [email protected]
Abstract . This paper is devoted to the Cauchy problem for thestochastic generalized Benjamin-Ono equation. By using the Bourgainspaces and Fourier restriction method and the assumption that u is F -measurable, we prove that the Cauchy problem for the stochasticgeneralized Benjamin-Ono equation is locally well-posed for the initial data u ( x, w ) ∈ L (Ω; H s ( R )) with s ≥ − α , where 0 < α ≤ . In particular,when u ∈ L (Ω; H α +12 ( R )) ∩ L α ) α (Ω; L ( R )), we prove that there existsa unique global solution u ∈ L (Ω; H α +12 ( R )) with 0 < α ≤ . Keywords:
Cauchy problem; Stochastic fractional Benjamin-Ono equa-tion, bilinear estimate
Mathematics Subject Classification(2000) : 35E15, 35Q53, 60H15 ∗ Supported by the NSF of China (No.11371367) and Fundamental program of NUDT(JC12-02-03) † Corresponding author Introduction
In this paper, we consider the following stochastic generalized fractional Benjamin-Ono type equa-tion ( du ( t ) = [ −| D x | α +1 ∂ x u ( t ) + u ( t ) k u x ( t )] dt + Φ dW ( t ) ,u (0) = u (1.1)where 0 < α ≤ | D x | is the Fourier multiplier operator with symbol | ξ | . We recall thatthe Benjamin-Ono equation is a nonlinear partial integro-differential equation that describes one-dimensional internal waves in deep water, which was introduced by Benjamin [1] and Ono[34].In fact, equation (1.1) is equivalent to the following equations: ( du ( t ) dt = [ −| D x | α ∂ x u ( t ) + u ( t ) k u x ( t )] + Φ dW ( t ) dt ,u (0) = u . (1.2)Equation (1.2) is considered as the Benjamin-Ono type equation ( dv ( t ) dt = [ −| D x | α ∂ x v ( t ) + v k ( t ) v x ( t )] ,u (0) = u . (1.3)forced by a random term. (1.3) which contains Benjamin-Ono equation and KdV equation ariseas mathematical models for the weakly nonlinear propagation of long waves in shallow channels.When α = 1 and k = 1, (1.1) reduces to the stochastic KdV equation which has been studiedby some people, we refer the readers to [2, 3, 4].When α = 1 and k = 1 , (1.3) reduces to the KdV equation which has been investigated bymany authors, we refer the readers to [5, 20, 21, 7, 8, 9, 10, 14, 15, 22, 23, 24, 26]. The resultof [23] and [24] implies that s = − is the critical well-posedness indices for the Cauchy problemfor the KdV equation. Guo[15] and [26] almost proved that the KdV equation is globally well-posed in H − / with the aid of I -method and the dyadic bilinear estimates at the same time.When α = 1 and k = 2, (1.3) reduces to the mKdV equation which has been studied by somepeople, we refer the readers to [10, 15, 22, 26, 35]. Recently, Chen et.al [6] studied the Cauchyproblem for the stochastic Camassa-Holm equation. When α = 0 and k = 1 , (1.3) reduces tothe Benjamin-Ono equation which has been studied by many people, we refer the readers to[27, 28, 29, 30, 31, 32, 33, 37]. By using the gauge transformation introduced by [37] and a newbilinear estimate, Ionescu and Kenig [25] proved that the Benjamin-Ono equation is globally well-posed in H s ( R ) with s ≥ . When 0 < α ≤ k = 1 , (1.3) has been investigated by somepeople, we refer the readers to [11, 13, 17, 18, 20]. In [18], the author proved that (1.3) is locallywell-posed in H ( s,w ) and globally well-posed in H (0 ,w ) . Recently, by using a frequency dependentrenormalization method, Herr [19] proved that (1.3) is globally well-posed in L if 0 < α < H s with s ≥ − α if 0 ≤ α ≤ k = 1 and in H s with s ≥ − α . Richards prove a global well-posedness result for stochasticKDV-Burgers equation under an additional smoothing of the noise, we refer to [36] for the details.In this paper, we focus the case 0 < α ≤ k = 2 of (1.1). In this paper, we consider theCauchy problem for the stochastic generalized Benjamin-Ono equation. By using the Bourgainspaces and Sobolev spaces and the assumption that u is F - measurable and Φ ∈ L , α +12 ., weprove that the Cauchy problem for the stochastic generalized Benjamin-Ono equation is locallywell-posed for the initial data u ( x, w ) ∈ L (Ω; H s ( R )) with s ≥ − α , where 0 < α ≤ . Inparticular, when u ∈ L (Ω; H α +12 ( R )) ∩ L α ) α (Ω; L ( R )) and Φ ∈ L , α +12 , we prove that thereexists a unique global solution u ∈ L (Ω; H α +12 ( R )) if 0 < α ≤ X ∼ Y by A | X | ≤ | Y | ≤ A | X | , where A j > j = 1 ,
2) and denote X ≫ Y by | X | > C | Y | , where C is some positive numberwhich is larger than 2. h ξ i s = (1 + ξ ) s for any ξ ∈ R , and F u denotes the Fourier transformationof u with respect to its all variables. F − u denotes the Fourier inverse transformation of u withrespect to its all variables. F x u denotes the Fourier transformation of u with respect to itsspace variable. F − x u denotes the Fourier inverse transformation of u with respect to its spacevariable. is the Schwartz space and is its dual space. H s ( R ) is the Sobolev space with norm k f k H s ( R ) kh ξ i s F x f k L ξ ( R ) . For any s, b ∈ R , X s, b ( R ) is the Bourgain space with phase function φ ( ξ ) = ξ | ξ | α . That is, a functions u ( x, t ) in belongs to X s,b ( R ) iff k u k X s, b ( R ) = (cid:13)(cid:13) h ξ i s h τ − ξ | ξ | α +1 i b F u ( ξ, τ ) (cid:13)(cid:13) L τ ( R ) L ξ ( R ) < ∞ . For any given interval L , X s, b ( R × L ) is the space of the restriction of all functions in X s, b ( R )on R × L , and for u ∈ X s, b ( R × L ) its norm is k u k X s, b ( R × L ) = inf {k U k X s, b ( R ) ; U | R × L = u } . When L = [0 , T ], X s, b ( R × L ) is abbreviated as X Ts,b . Throughout this paper, we always assumethat w ( ξ ) = ξ | ξ | α +1 , ψ is a smooth function, ψ δ ( t ) = ψ ( tδ ) , satisfying 0 ≤ ψ ≤ , ψ = 1 when t ∈ [0 , , supp ψ ⊂ [ − ,
2] and σ = τ − ξ | ξ | α +1 , σ k = τ k − ξ k | ξ k | α +1 ( k = 1 , ,U ( t ) u = Z R e i ( xξ − ξ | ξ | α +1 ) F x u ( ξ ) dξ, k f k L qt L px = Z R (cid:18)Z R | f ( x, t ) | p dx (cid:19) qp dt ! q , k f k L pt L px = k f k L pxt . We assume that B ( x, t ), t ≥ , x ∈ R , is a zero mean gaussian process whose covariance functionis given by E ( B ( t, x ) B ( s, y )) = ( t ∧ s )( x ∧ y )3or t, s ≥ , x, y ∈ R and W ( t ) = ∂B∂x = ∞ P i =1 β j e j , where ( e i ) i ∈ N is an orthonormal basis of L ( R )and ( β i ) is a sequence of mutually independent real brownian motions in a fixed probability space,is a cylindrical Wiener process on L ( R ) . ( ., . ) denotes the L space duality product, i.e., ( f, g ) = R R f ( x ) g ( x ) dx. (Ω , F , P ) is a probability space endowed with a filtration ( F t ) t ≥ . E f = R Ω f d P .W ( t ) is a cylindrical Wiener process ( W ( t )) t ≥ on L ( R ) associated with the filtration ( F ) t ≥ . Forany orthonormal basis ( e k ) k ∈ N of L ( R ), W = P ∞ k =0 β k e k for a sequence ( β k ) k ∈ N of real, mutuallyindependent brownian motions on (Ω , F , P , F t ) t ≥ ) . Let H be a Hilbert space, L ( L ( R ) , H ) thespace of Hilbert-Schmidt operators from L ( R ) into H . Its norm is given by k Φ k L ( L ( R ) ,H ) = X j ∈ N | Φ e j | H . When H = H s ( R ), L ( L ( R ) , H s ( R )) = L ,s .The main results of this paper are as follows: Theorem A
Let 0 < α ≤ , u ( x ) ∈ L (Ω; H s ( R )) be F -measurable with s ≥ − α andΦ ∈ L , α +12 . Then the Cauchy problem for (1.1) locally well-posed with k = 2. Theorem B
Let 0 < α ≤ , u ∈ L (Ω; H α +12 ( R )) ∩ L α ) α (Ω; L ( R )) be F -measurableand Φ ∈ L , α +12 . Then the Cauchy problem for (1.1) possesses a unique global solution u ∈ L (Ω; H α +12 ( R )) with k = 2.The rest of the paper is organized as follows. In section 2, some key interpolate inequalities andpreliminary estimates are established. The section 3 is devoted to bilinear estimate by Fourierrestriction norm method. We will show the trilinear estimate and local well-posedness of Cauchyproblem in section 4. The global well-posedness of Cauchy problem is established in section 5. Lemma 2.1.
Let θ ∈ [0 , and W γ ( t ) u ( x ) = R R e i ( tφ ( ξ )+ xξ ) | φ ′′ ( ξ ) | γ F x u ( ξ ) dξ . Then k W θ ( t ) u k L qt L px ≤ C k u k L x , where ( p, q ) = ( − θ , θ ) . For the proof of Lemma 2.1, we refer the readers to Theorem 2.1 of [21].
Lemma 2.2.
Let b = + ǫ , then k u k L xt ≤ C k u k X , α +32( α +2) ( 12 + ǫ ) (2.1) and (cid:13)(cid:13)(cid:13) D α x u (cid:13)(cid:13)(cid:13) L xt ≤ C k u k X , b . (2.2)4 roof. Let θ = , it follows from lemma 1 that (cid:13)(cid:13)(cid:13)(cid:13)Z R e itφ ( ξ )+ ixξ | φ ′′ ( ξ ) | F x u ( ξ ) dξ (cid:13)(cid:13)(cid:13)(cid:13) L xt ≤ C k u k L ξ . where | φ | = | ξ | α +1 , | φ ′′ | = c | ξ | α , then (cid:13)(cid:13)(cid:13)(cid:13)Z R e itφ ( ξ )+ ixξ | ξ | α F x u ( ξ ) dξ (cid:13)(cid:13)(cid:13)(cid:13) L xt ≤ C k u k L ξ . Due to k f k L α +6 xt ≤ C k D γx D γt f k L xt where γ = α α +3) . Then k W ( t ) u ( x ) k L α +4 xt = C (cid:13)(cid:13)(cid:13)(cid:13)Z R e i ( tφ + xξ ) F x u ( ξ ) dξ (cid:13)(cid:13)(cid:13)(cid:13) L α +4 xt ≤ C (cid:13)(cid:13)(cid:13)(cid:13) D γx D γt Z R e i ( tφ + xξ ) F x u ( ξ ) dξ (cid:13)(cid:13)(cid:13)(cid:13) L xt = C (cid:13)(cid:13)(cid:13)(cid:13)Z R e i ( tφ + xξ ) | ξ | α F x u ( ξ ) dξ (cid:13)(cid:13)(cid:13)(cid:13) L xt ≤ C k u k L x . (2.3)By a standard argument, it follows from k W ( t ) u ( x ) k L α +6 xt ≤ C k u k L x that k u ( x ) k L α +6 xt ≤ C k u k X , + ǫ . (2.4)By using the Plancherel identity, we have that k u k L xt = C k u k X , . (2.5)Interpolating (2.4) with (2.5) yields k u k L xt ≤ C k u k X , α +32( α +2) ( 12 + ǫ ) . (2.6)From (2.3), by using a standard proof, we have that k D α x u k L xt ≤ C k u k X ,b . (2.7)Interpolating (2.7) with (2.5) yields k D α x u k L xt ≤ C k u k X , b . (2.8)Hence, the proof of Lemma 2.2 is completed. Lemma 2.3.
Let b = + ǫ. Then, for ≤ s ≤ , we have that k I s ( u , u ) k L xt ≤ C Y j =1 k u j k X , α +3+2( α +1) s α +2) b , (2.9) where F I s ( u , u )( ξ, τ ) = Z ξ = ξ + ξ τ = τ + τ || ξ | α +1 − | ξ | α +1 | s F u ( ξ , τ ) F u ( ξ , τ ) dξ dτ . roof. Let F j ( ξ j , τ j ) = h σ j i α +3+2( α +1) s α +4 b F u j ( ξ j , τ j )( j = 1 , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z ξ = ξ + ξ τ = τ + τ || ξ | α +1 − | ξ | α +1 | s F h σ i α +3+2( α +1) s α +2 b F h σ i α +3+2( α +1) s α +2 b dξ dτ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ξτ ≤ C Y j =1 k F j k L ξτ . (2.10)Assume that b = α +3+2( α +1) s α +4 b . By using the Young inequality, since 0 < s < , we have that || ξ | α +1 − | ξ | α +1 | s h σ i − b h σ i − b = || ξ | α +1 − | ξ | α +1 | s h σ i − bs h σ i − bs h σ i − ( b − bs ) h σ i − ( b − bs ) ≤ s || ξ | α +1 − | ξ | α +1 | / h σ i − b h σ i − b + (1 − s ) h σ i − α +32 α +4 b h σ i − α +32 α +4 b ≤ || ξ | α +1 − | ξ | α +1 | / h σ i − b h σ i − b + h σ i − α +32 α +4 b h σ i − α +32 α +4 b . (2.11)By using (2.11), Plancherel identity, Lemma 3.1 in [18], we have that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z ξ = ξ + ξ τ = τ + τ || ξ | α +1 − | ξ | α +1 | s F h σ i α +3+2( α +1) s α +4 b F h σ i α +3+2( α +1) s α +2 b dξ dτ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ξτ ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z ξ = ξ + ξ τ = τ + τ || ξ | α +1 − | ξ | α +1 | / Y j =1 F j h σ j i b dξ dτ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ξτ + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z ξ = ξ + ξ τ = τ + τ Y j =1 F h σ j i α +32 α +4 b dξ dτ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ξτ ≤ C Y j =1 (cid:13)(cid:13)(cid:13)(cid:13) F − (cid:18) F j h σ j i b (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) X ,b + C Y j =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) F − F j h σ j i α +32 α +4 b !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X , α +32 α +4 b ≤ C Y j =1 k F j k L ξτ . (2.12)The proof of Lemma 2.3 is completed. Lemma 2.4.
Let u ∈ H s ( R ) , c > / , < b < / . Then for t ∈ [0 , T ] , W ( t ) u ∈ X Ts, c and thereis a constant k > such that k U ( t ) u k X Ts,c ≤ k k u k H s . (2.13) There is a constant c > such that for t ∈ [0 , and f ∈ X Ts, b , (cid:13)(cid:13)(cid:13)(cid:13)Z T U ( t − s ) f ( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) X Ts,b ≤ CT − b k f k X − b,s . (2.14)For the proof of Lemma 2.4, we refer the readers to Lemma 3.1 of [3].6 emma 2.5. Let u = Z t U ( t − s )Φ dW ( s ) and Φ ∈ L ,s , for t ∈ [0 , T ] , we have E ( sup t ∈ [0 ,T ] k u k H s ) ≤ T k Φ k L ,s . (2.15)Lemma 2.5 can be proved similarly to Proposition 3.1 of [2]. Lemma 2.6.
Let u = Z t U ( t − s )Φ dW ( s ) and Φ ∈ L ,s , for t ∈ [0 , T ] , we have Ψ u ∈ L (Ω; X s,b ) and E ( sup t ∈ [0 ,T ] k Ψ u k X s,b ) ≤ M ( b, Ψ) k Φ k L ,s . (2.16)Lemma 2.6 can be proved similarly to Proposition 2.1 of [3]. Theorem 3.1.
For all u, v on R × R and < α ≤ , < ǫ ≤ α α +8) and b = − ǫ , we have k u u k L ≤ C k u k X − ,b k u k X − α ,b . (3.1) Proof.
Define F ( ξ , τ ) = h ξ i − / h σ i b F u ( ξ , τ ) , F ( ξ , τ ) = h ξ i − α h σ i b F u ( ξ , τ ) ,σ j = τ j − | ξ j | α +1 ξ j , j = 1 , . To obtain (4.4), it suffices to prove that Z R Z ξ = ξ + ξ τ = τ + τ K ( ξ , τ , ξ, τ ) | F | Y j =1 | F j | dξ dτ dξdτ ≤ C k F k L ξτ Y j =1 k F j k L ξτ , (3.2)where K ( ξ , τ , ξ, τ ) = h ξ i / h ξ i α − h σ i b h σ i b . F ≥ , F j ≥ j = 1 , . Obviously, { ( ξ , τ , ξ, τ ) ∈ R , ξ = X j =1 ξ j , τ = X j =1 τ j } ⊂ X j =1 Ω j where Ω = { ( ξ , τ , ξ, τ ) ∈ R , ξ = X j =1 ξ j , τ = X j =1 τ j , | ξ | ≤ | ξ | ≤ } , Ω = { ( ξ , τ , ξ, τ ) ∈ R , ξ = X j =1 ξ j , τ = X j =1 τ j , | ξ | ≥ , | ξ | ≫ | ξ |} , Ω = { ( ξ , τ , ξ, τ ) ∈ R , ξ = X j =1 ξ j , τ = X j =1 τ j , | ξ | ≥ , | ξ | ∼ | ξ |} , Ω = { ( ξ , τ , ξ, τ ) ∈ R , ξ = X j =1 ξ j , τ = X j =1 τ j , | ξ | ≤ | ξ | ≤ } , Ω = { ( ξ , τ , ξ, τ ) ∈ R , ξ = X j =1 ξ j , τ = X j =1 τ j , | ξ | ≥ , | ξ | ≫ | ξ |} , Ω = { ( ξ , τ , ξ, τ ) ∈ R , ξ = X j =1 ξ j , τ = X j =1 τ j , | ξ | ≥ , | ξ | ≥ | ξ | , | ξ | ∼ | ξ |} , We define f j = F − F j h σ j i b , j = 1 , . The integrals corresponding to Ω j (1 ≤ j ≤ , j ∈ N + ) will be denoted by J k (1 ≤ k ≤ , k ∈ N + )in (3.2), respectively.(1). Ω = { ( ξ , τ , ξ, τ ) ∈ R , ξ = P j =1 ξ j , τ = P j =1 τ j , | ξ | ≤ | ξ | ≤ } . In this subregion, we havethat K ( ξ , τ , ξ, τ ) ≤ C Q j =1 h σ i b .
8y using the Plancherel identity and the H¨older inequality and α +32( α +2) ( + ǫ ) < − ǫ , we have that J ≤ C Z R Z ξ = ξ + ξ τ = τ + τ F Q j =1 F j Q j =1 h σ j i b dξ dτ dξdτ ≤ C Z R F − ( F ) f f dxdt ≤ C kF − ( F ) k L xt Y j =1 k f j k L xt ≤ C k F k L ξτ Y j =1 k f j k X , α +32( α +2) ( 12 + ǫ ) ≤ C k F k L ξτ Y j =1 k F j k L ξτ . (3.3)(2).Ω = { ( ξ , τ , ξ, τ ) ∈ R , ξ = P j =1 ξ j , τ = P j =1 τ j , | ξ | ≥ , | ξ | ≫ | ξ |} . If | ξ | ≤ , we have that K ( ξ , τ , ξ, τ ) ≤ C Q j =1 h σ j i b This case can be proved similarly to Ω . If | ξ | ≥ , we have that K ( ξ , τ , ξ, τ ) ≤ C | ξ | α Q j =1 h σ j i b ≤ C || ξ | α +1 − | ξ | α +1 | α α +1) Q j =1 h σ j i b . By using the Cauchy-Schwartz inequality and Lemma 2.3 as well as 0 < ǫ < , we have that J ≤ C Z R Z ξ = ξ + ξ τ = τ + τ || ξ | α +1 − | ξ | α +1 | α α +1) F Q j =1 F j Q j =1 h σ j i b dξ dτ dξdτ ≤ C k F k L ξτ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z ξ = ξ + ξ τ = τ + τ || ξ | α +1 − | ξ | α +1 | α α +1) F Q j =1 F j Q j =1 h σ j i b dξ dτ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ξτ ≤ C k F k L ξτ Y j =1 k F j k L ξτ . (3). Ω = { ( ξ , τ , ξ, τ ) ∈ R , ξ = P j =1 ξ j , τ = P j =1 τ j , | ξ | ≥ , | ξ | ∼ | ξ |} . In this subregion, wehave that K ( ξ , τ , ξ, τ ) ≤ C | ξ | α Q j =1 h σ j i b ≤ C Q j =1 | ξ j | α Q j =1 h σ j i b
9y using the Plancherel identity and the Cauchy-Schwartz inequality as well as ( + ǫ ) < − ǫ ,we have that J ≤ C Z R Z ξ = ξ + ξ τ = τ + τ Q j =1 | ξ j | α F Q j =1 F j Q j =1 h σ j i b dξ dτ dξdτ ≤ C k F k L ξτ Y j =1 (cid:13)(cid:13)(cid:13)(cid:13) D α x F − (cid:18) F j h σ j i b (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L xt ≤ C k F k L ξτ Y j =1 k F j k L ξτ . (3.4)(4). Ω = { ( ξ , τ , ξ, τ ) ∈ R , ξ = P j =1 ξ j , τ = P j =1 τ j , | ξ | ≤ | ξ | ≤ } . In this subregion, we havethat K ( ξ , τ , ξ, τ ) ≤ C Q j =1 h σ j i b . Thus subregion can be proved similarly to Ω . (5). Ω = { ( ξ , τ , ξ, τ ) ∈ R , ξ = P j =1 ξ j , τ = P j =1 τ j , | ξ | ≥ , | ξ | ≫ | ξ |} . In this subregion, wehave that K ( ξ , τ , ξ, τ ) ≤ C || ξ | α +1 − | ξ | α +1 | α +1) Q j =1 h σ j i b . By using the Cauchy-Schwartz inequality and using Lemma 2.3 as well as α +42( α +2) ( + ǫ ) ≤ − ǫ ,we have that J ≤ C Z R Z ξ = ξ + ξ τ = τ + τ || ξ | α +1 − | ξ | α +1 | α +1) F Q j =1 F j Q j =1 h σ j i b dξ dτ dξdτ ≤ C k F k L ξτ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z ξ = ξ + ξ τ = τ + τ || ξ | α +1 − | ξ | α +1 | α +1) F Q j =1 F j Q j =1 h σ j i b dξ dτ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ξτ ≤ C k F k L ξτ Y j =1 k F j k L ξτ . (6). Ω = { ( ξ , τ , ξ, τ ) ∈ R , ξ = P j =1 ξ j , τ = P j =1 τ j , | ξ | ≥ , | ξ | ≥ | ξ | , | ξ | ∼ | ξ |} . In thissubregion, | ξ | ∼ | ξ | . This subregion can be proved similarly to Ω . Thus, We have completed the proof of Theorem 3.1.10
Local well-posedness
In this section, we will establish two new trilinear estimates which play a crucial role in establishingthe local well-posedness of solution, and then we will show the local well-posedness of the Cauchyproblem (1.1) by Banach Fixed Point theorem.We will establish the Lemma 4.1 with the aid of the idea in [38]. Let Z = R and Γ k ( Z ) denotethe hyperplane in R k Γ k ( Z ) := (cid:8) ( ξ , · · · , ξ k ) ∈ Z k , ξ + · · · + ξ k = 0 (cid:9) endowed with the induced measure Z Γ k ( Z ) f := Z Z k − f ( ξ , · · · , ξ k − , − ξ − · · · − ξ k − ) dξ · · · dξ k . A function m : Γ k ( Z ) → C is said to be a [ k ; Z ]-multiplier, and we define the norm k m k [ k ; Z ] to bethe best constant such that the inequality (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Γ k ( Z ) m ( ξ ) k Y j =1 f j ( ξ j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k m k [ k ; Z ] k Y j =1 k f j k L . holds for all test function f j on Z. Lemma 4.1.
Let s = − α , b = − ǫ . Then k ∂ x ( u u u ) k X s , − b ≤ C Y j =1 k u j k X s ,b . (4.1) Proof.
By duality, Plancherel identity and the definition, to obtain (4.4), it suffices to prove that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ( P j =1 ξ j ) h ξ i − α Q j =1 h τ j − w ( ξ j ) i − ǫ Q j =1 h ξ j i − α (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) [4; R × R ] ≤ C. (4.2)By using the symmetry and h ξ i − α ≤ C h ξ i " X j =1 h ξ j i − α resulting from | ξ + ξ + ξ | ≤ h ξ i , to obtain (4.2), it suffices to prove (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h ξ i / h ξ i / h ξ i − α h ξ i − α Q j =1 h τ j − w ( ξ j ) i − ǫ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) [4; R × R ] ≤ C. (4.3)(4.3) follows from T T ⋆ identity in Lemma 3.7 of [38] and Theorem 3.1.The proof of lemma 4.1 has been completed.11 emma 4.2. Let s ≥ s = − α , b = − ǫ . Then k ∂ x ( u u u ) k X s, − b ≤ C Y j =1 k u j k X s,b . (4.4) Proof. (4.4) is equivalent to the following inequality Z R Z ξ = ξ + ξ + ξ τ = τ + τ + τ | ξ |h ξ i s F Q j =1 F j h σ i b Q j =1 h ξ j i s h σ j i b dξ dτ dξ dτ dξdτ ≤ C k F k L ξτ Y j =1 k F j k L ξτ . (4.5)Since h ξ i s − s ≤ C Y j =1 h ξ j i s − s , (4.6)(4.5) is equivalent to the following inequality Z R Z ξ = ξ + ξ + ξ τ = τ + τ + τ | ξ |h ξ i s F Q j =1 F j h σ i b Q j =1 h ξ j i s h σ j i b dξ dτ dξ dτ dξdτ ≤ C k F k L ξτ Y j =1 k F j k L ξτ , (4.7)which is just the lemma 4.1. Hence, the proof of lemma 4.2 is completed.Following the method of [3], combining Lemma 4.2, Lemmas 2.4, 2.5 with the Banach FixedPoint Theorem, we have the following local well-posedness of Cauchy problem (1.1). Theorem 4.1.
Let < α ≤ , u ( x ) ∈ L (Ω; H s ( R )) be F -measurable with s ≥ − α and Φ ∈ L , α +12 . Then the Cauchy problem for (1.1) locally well-posed with k = 2 . In this section, we always assume that 0 < α ≤ , u ∈ L (Ω; H α +12 ( R )) ∩ L α ) α (Ω; L ( R )) be F -measurable and Φ ∈ L , α +12 . In order to obtain the global well-posedness, we follow the argumentgiven by Bouard and Debussche in [3], in which, they established the global well-posedness forstochastic Korteweg-de Vries equation driven by white noise, we also refer [36] to the global well-posedness for stochastic KVD-Burgers equation.Notice that the deterministic equation (1.3) possesses two important conservation laws:12 Z R u dx, Z R ( D α +12 x u ) dx − k + 1 Z R u k +1 dx. Let (Φ m ) m ∈ N be a sequence in L , such that Φ m −→ Φ in L , α +12 and let ( u m ) m ∈ N be asequence in H ( R ) such that u m −→ u in L (Ω; H α +12 ( R )) ∩ L α ) α (Ω; L ( R )) and in H α +12 ( R )a.s.. 12 emma 5.1. For sufficiently large m , then there exists a unique solution u m P -a.s in L ∞ (0 , T ; H α +12 ( R )) of du m + ( D α +1 x u mx − u m u mx ) = Φ m dW, (5.1) u m (0) = u m (5.2) For any
T > .Proof. Define v m = u m − u m , then v m satisfies v mt + D α +1 x v mx − ( v m + u m ) ∂ x ( v m + u m ) = 0 , (5.3) v m (0) = u m . (5.4)By using Lemmas 2.4-2.6 and Lemma 4.2 as well as the fixed point point, we know that (5.3)-(5.4)have a local solution u m ∈ L ∞ (0 , T ( ω ) , H α +12 ( R )) a.s. for a.s.u m ∈ H α +12 , where T ( ω ) is thelifespan of the local solution. It follows from Lemma 2.5 that u m = Z t U ( t − s )Φ m dW ( s ) −→ Z t U ( t − s )Φ dW ( s ) (5.5)in L (Ω; H ( R )) and u m is in L ∞ (0 , T ; H ) a.s. From (5.5), we have that there exists a subsequence,still denoted by u m , such that u m −→ u (5.6)in H ( R ) a.s.We claim that the sequence u m is bounded in L (Ω; L ∞ (0 , T ; H α +12 ( R )) when u ∈ L (Ω; H α +12 ( R )) ∩ L α ) α (Ω; L ( R )) for any T > I ( u m ) = 12 Z R ( D α +12 x u m ) dx − Z R u m dx. (5.7)Applying the Itˆ o formula to I ( u m ) yields I ( u m ) = I ( u m ) − Z t ( D α +1 x u m − u m , Φ dW ( s )) + 12 Z t T r ( I ′′ ( u m )Φ m Φ ⋆m ) ds. (5.8)with I ′′ ( u m ) φ = D α +1 x φ − u m φ. e i ) i ∈ N be an orthonormal basis of L ( R ), by using H α +12 ֒ → L ∞ , we have that T r ( I ′′ ( u m )Φ m Φ ⋆m ) = − X j ∈ N Z R (cid:2) D α +1 x (Φ m e j )Φ m e j + 3 u m (Φ m e j ) (cid:3) dx ≤ X j ∈ N (cid:18)(cid:13)(cid:13)(cid:13) D α +12 x (Φ m e j ) (cid:13)(cid:13)(cid:13) L + 3 k u m k L k Φ m e j k L ∞ (cid:19) ≤ k Φ m k L , α +122 (cid:2) k u m k L + 1 (cid:3) . (5.9)We derive from (5.8) thatE sup t ∈ [0 ,T ] Z t T r ( I ′′ ( u m )Φ m Φ ⋆m ) ds ! ≤ C E sup t ∈ [0 ,T ] k u m k L ! + CT k Φ m k L , α +122 + C. (5.10)Applying the martingale inequality ( Theorem 3.14 of [12]) yieldsE sup t ∈ [0 ,T ] − Z t ( D α +12 x u m − u m , Φ dW ( s )) ! ≤ (cid:18)Z T (cid:12)(cid:12)(cid:12) Φ ⋆m (cid:16) D α +12 x u m − u m (cid:17)(cid:12)(cid:12)(cid:12) ds (cid:19) / ! . (5.11)By using the Sobolev embedding H α +12 ֒ → L ∞ and interpolation Theorem, we obtain (cid:12)(cid:12) Φ ⋆m (cid:0) D α +1 x u m − u m (cid:1)(cid:12)(cid:12) = X j ∈ N (cid:2) ( D α +1 x u m , Φ m e j ) + ( u m , Φ m e j ) (cid:3) ≤ C X j ∈ N h k u m k H α +12 k Φ m e j k H α +12 + k u m k L k Φ m e j k L ∞ i ≤ C (cid:20) k u m k H α +12 + k u m k α ) α +1 L k D α +12 x u m k α +1 L (cid:21) k Φ m k L , α +122 . (5.12)Substituting (5.12) into (5.11), using Cauchy-Schwartz inequality and Young inequality, we deduceE sup t ∈ [0 ,T ] − Z t ( D α +1 x u m − u m , Φ dW ( s )) ! ≤
18 E sup t ∈ [0 ,T ] k D α +12 x u m k L ! + C E sup t ∈ [0 ,T ] k u m k α ) α L ! + CT k Φ m k L , α +122 . (5.13)Applying Itˆ o formula to F ( u m ) = k u m k α ) α L yields k u m k α ) α L = k u m k α ) α L + 2(2 + 3 α ) α Z t k u m k α ) α L ( u m , Φ m dW ) + 12 Z t T r ( F ′′ ( u m )Φ m Φ ⋆m ) ds, (5.14)where F ′′ ( u m ) φ = 4(2 + 3 α )(2 + 2 α ) α k u m k α ) α L ( u m , φ ) u m + 2(2 + 3 α ) α k u m k α ) α L φ.
14y using a martingale inequality ( Theorem 3.14 in [12]) and Young inequality, we have thatE sup t ∈ [0 ,T ] α ) α Z t k u m k α ) α L ( u m , Φ m dW ) ! ≤ (cid:18)Z T k u m k α ) α L k Φ ⋆m u m k L (cid:19) / ! ≤
116 E sup t ∈ [0 ,T ] k u m k α ) α L ! + CT k Φ m k α +6 L , . (5.15)Direct computation implies T r ( F ′′ ( u m )Φ m Φ ⋆m )= X j ∈ N α )(2 + 2 α ) α k u m k α ) α L ( u m , Φ m e j ) + X j ∈ N α ) α k u k α ) α L k Φ m e j k L ≤ α )(4 + 5 α ) α k u m k α ) α L k Φ m k L , ≤ k u m k α ) α L + C k Φ m k αα L , . (5.16)Combining (5.15), (5.16) with (5.14), we have thatE sup t ∈ [0 ,T ] k u m ( t ) k α ) α L ! ≤ C E (cid:18) k u m k α ) α L (cid:19) + C k Φ m k αα L , + CT k Φ m k α +6 L , . (5.17)By using interpolation Theorem, we have that k u m k L ≤ C k u m k α +1) α +1 L k D α +12 x u m k α +1 L ≤ C k u m k α +1) α L + 18 k D α +12 x u m k L . (5.18)Similarly, we derive thatE sup t ∈ [0 ,T ] k u m ( t ) k α ) α L ! ≤ C E (cid:18) k u m k α ) α L (cid:19) + CT k Φ m k C ( α ) L , (5.19)and E sup t ∈ [0 ,T ] k u m ( t ) k L ! ≤ C E (cid:0) k u m k L (cid:1) + CT k Φ m k L , , (5.20)E sup t ∈ [0 ,T ] k u m ( t ) k L ! ≤ C E (cid:0) k u m k L (cid:1) + CT k Φ m k L , , (5.21)where C ( α ) is a constant relative to α . Combining (5.18) with (5.19), we getE sup t ∈ [0 ,T ] k u m k L ! ≤ C E (cid:18) k u m k α ) α L (cid:19) + CT k Φ m k C ( α ) L , + 18 E sup t ∈ [0 ,T ] k D α +12 x u m k L ! . (5.22)15ombining (5.10), (5.13), (5.17), (5.20)-(5.22) with (5.8), we obtain12 E sup t ∈ [0 ,T ] Z R h u m + ( D α +12 x u m ) i dx ! = 12 E sup t ∈ [0 ,T ] Z R u m dx ! + E sup t ∈ [0 ,T ] I ( u m ) ! + 14 E Z R u m dx ≤ C E Z R h u m + ( D α +12 x u m ) i dx + C E (cid:18) k u m k α ) α L (cid:19) + C E (cid:18) k u m k α ) α L (cid:19) + 14 E sup t ∈ [0 ,T ] Z R h u m + ( D α +12 x u m ) i dx ! + C E (cid:0) k u m k L (cid:1) + CT k Φ m k C ( α ) L , α +122 + CT k Φ m k D ( α ) L , α +122 + CT k Φ m k E ( α ) L , α +122 + CT k Φ m k L , α +122 + C, (5.23)where C ( α ) , D ( α ) , E ( α ) are constants relative to α. From (5.22), by using the Young inequality, we have thatE sup t ∈ [0 ,T ] Z R h u m + ( D α +12 x u m ) i dx ! ≤ C E Z R h u m + ( D α +12 x u m ) i dx + C E (cid:18) k u m k α ) α L (cid:19) + C E (cid:18) k u m k α ) α L (cid:19) + CT k Φ m k C ( α ) L , α +122 + CT k Φ m k D ( α ) L , α +122 + CT k Φ m k E ( α ) L , α +122 + C + C E (cid:0) k u m k L (cid:1) ≤ C E Z R h u m + ( D α +12 x u m ) i dx + C E (cid:18) k u m k α ) α L (cid:19) + CT k Φ m k F ( α ) L , α +122 (1 + T ) + C + CT , (5.24)where F ( α ) = max { , C ( α ) , D ( α ) , E ( α ) } . From u m −→ u in L (Ω; L ∞ (0 , T ; H α +12 ( R )) ∩ L α ) α (Ω; L ( R )) and Φ m −→ Φ in L , α +12 , weknow that ∀ ǫ > , there exists sufficiently large m ∈ N + such thatE k u m k H α +12 + E (cid:18) k u m k α ) α L (cid:19) + k Φ m k F ( α ) L , α +122 ≤ E k u k H α +12 + E (cid:18) k u k α ) α L (cid:19) + k Φ k F ( α ) L , α +122 + C + CT + ǫ. (5.25)16ombining (5.24) with (5.25), we have thatE sup t ∈ [0 ,T ] Z R h u m + ( D α +12 x u m ) i dx ! ≤ C E k u k H α +12 + C E (cid:18) k u k α ) α L (cid:19) + CT k Φ k F ( α ) L , α +122 (1 + T ) + Cǫ. (5.26)Combining (5.26) with (5.5), we have thatE sup t ∈ [0 ,T ] Z R h v m + ( D α +12 x v m ) i dx ! ≤ C (cid:18) E k u k H α +12 , E (cid:18) k u k α ) α L (cid:19) , T, k Φ k F ( α ) L , α +122 (cid:19) . (5.27)Thus, combining the the local existence of solution with (5.28), we obtain that (5.3)-(5.4) possessesa unique global solution. Consequently, (5.1)-(5.2) possesses a unique global solution u m P-a.s in L ∞ (0 , T ; H α +12 ( R )) for any T >
Theorem 5.1.
Let < α ≤ , u ∈ L (Ω; H α +12 ( R )) ∩ L α ) α (Ω; L ( R )) be F -measurable.Then the Cauchy problem for (1.1) possesses a unique global solution u ∈ L (Ω; H α +12 ( R )) with k = 2 .Proof. From (5.25), we know that after extraction of a subsequence, we can find a function˜ u ∈ L (Ω; L ∞ (0 , T ; H α +12 ( R )))such that u m ⇀ ˜ u (5.28)in L (Ω; L ∞ (0 , T ; H α +12 ( R ))) weak star. Moreover, we have thatE sup t ∈ [0 ,T ] k ˜ u k H α +12 ! ≤ C E (cid:16) k u k H α +12 (cid:17) + CT k Φ k F ( α ) L , α +122 + CT + C. (5.29)Now we define the mapping G m v = U ( t ) u m + Z t U ( t − τ ) (cid:26) ∂ x ( v ) (cid:27) dτ + u m , (5.30)it is easily checked that G m is a strict contraction uniformly on B t w r w , where r w ≥ C (cid:20)(cid:18) sup m ∈ N k Ψ u m k X α +12 ,b (cid:19) + K k ˜ u k L ∞ (0 ,T ; H α +12 ) (cid:21) Ct − bw (cid:16) r w + K k ˜ u k L ∞ (0 ,T ; H α +12 ) (cid:17) ≤ . It is easily checked that G m has a unique fixed point u m for sufficiently large m ∈ N + . Now weprove that u m → u in X t wα +12 , b . Let z m = U ( t ) u m , v m = u m − z m − u m . Then v m ( t ) = − Z t U ( t − s ) ∂ x (cid:2) ( v m + z m ( t ) + u m ) (cid:3) ds. (5.31)Combining lemma 4.2 with (5.31), by using Lemmas 2.4-2.6, we have that k v m − v k X twα +12 ,b ≤ Ct − bw (cid:20) k v k X twα +12 ,b + k v m k X twα +12 ,b + k z k X twα +12 ,b + k z m k X twα +12 ,b + k u k X twα +12 ,b + k u m k X twα +12 ,b (cid:21) × (cid:20) k v m − v k X twα +12 ,b + k z m − z k X twα +12 ,b + k u − u m k X twα +12 ,b (cid:21) ≤ Ct − bw (cid:20) k u k X twα +12 ,b + k u m k X twα +12 ,b + k z k X twα +12 ,b + k z m k X twα +12 ,b + k u k X twα +12 ,b + k u m k X twα +12 ,b (cid:21) × (cid:20) k v m − v k X twα +12 ,b + k z m − z k X twα +12 ,b + k u − u m k X twα +12 ,b (cid:21) ≤ Ct − bw h k u k H α +12 + k u m k H α +12 + r w i × (cid:20) k v m − v k X twα +12 ,b + k z m − z k X twα +12 ,b + k u − u m k X twα +12 ,b (cid:21) ≤ (cid:20) k v m − v k X twα +12 ,b + k z m − z k X tw ,b + k u − u m k X twα +12 ,b (cid:21) . (5.32)From (5.32), we have that k v m − v k X twα +12 ,b ≤ k z m − z k X twα +12 ,b + k u − u m k X twα +12 ,b ≤ C k u m − u k H α +12 + k u − u m k X twα +12 ,b . (5.33)From Lemma 2.6, we have that E ( sup t ∈ [0 ,T ] k Ψ u − Ψ u m k X α +12 ,b ) ≤ M ( b, Ψ) k Φ k L , α +122 . (5.34)Combining (5.33), (5.34) with the fact that u m −→ u . (5.35)18n H α +12 a.s., we have that u m → u (5.36)in X t wα +12 , b . From (5.36), we have that u = e u (5.37)on [0 , t w ] and k u ( t w ) k H α +12 ≤ k e u k L ∞ (0 ,T ; H α +12 ) . (5.38)Combining (5.38) with Theorem 4.1, we can construct a solution on [ t w , t w ]; starting from u (2 t w ),we obtain a solution on [0 , T ] by reiterating this argument. Thus, the proof of Theorem 5.1 hasbeen completed. Acknowledgment
We would like to thank an anonymous referee for the valuable suggestion for improving thequality of this paper.