Wellposedness and regularity estimate for stochastic Cahn--Hilliard equation with unbounded noise diffusion
aa r X i v : . [ m a t h . P R ] J un WELLPOSEDNESS AND REGULARITY ESTIMATE FORSTOCHASTIC CAHN–HILLIARD EQUATION WITH UNBOUNDEDNOISE DIFFUSION ∗ JIANBO CUI † AND
JIALIN HONG ‡ Abstract.
In this article, we consider the one dimensional stochastic Cahn–Hilliard equationdriven by multiplicative space-time white noise with diffusion coefficient of sublinear growth. Byintroducing the spectral Galerkin method, we first obtain the well-posedness of the approximatedequation in finite dimension. Then with the help of the semigroup theory and the factorizationmethod, the approximation processes is shown to possess many desirable properties. Further, we showthat the approximation process is strongly convergent in certain Banach space via the interpolationinequality and variational approach. Finally, the global existence and regularity estimate of theunique solution process are proven by means of the strong convergence of the approximation process.
Key words. stochastic Cahn–Hilliard equation, multiplicative space-time white noise, spectralGalerkin method, global existence, regularity estimate
AMS subject classifications.
1. Introduction.
In this article, we consider the following stochastic Cahn–Hilliard equation with multiplicative space-time white noise dX ( t ) + A ( AX ( t ) + F ( X ( t ))) dt = G ( X ( t )) dW ( t ) , t ∈ (0 , T ] , (1) X (0) = X . Here 0 < T < ∞ , H := L ( O ) with O = (0 , L ) , L > − A : D ( A ) ⊂ H → H is the Laplacian operator under homogenous Dirichlet or Neumman boundarycondition, and { W ( t ) } t ≥ is a generalized Wiener process on a filtered probabilityspace (Ω , F , {F t } t ≥ , P ). The nonlinearity F is assumed to be the Nemytskii operatorof f ′ , where f is a polynomial of degree 4, i.e., c ξ + c ξ + c ξ + c ξ + c with c i ∈ R , i = 0 , · · · , c >
0. A typical example is the double well potential f = ( ξ − .For more general drift nonlinearities, we refer to [14] and references therein. Thediffusion coefficient G is assumed to be the Nemytskii operator of g , where g is a globalLipschitz function with the sublinear growth condition | g ( ξ ) | ≤ C (1 + | ξ | α ) , α < G = I , Eq. (1) corresponds to the stochastic Cahn–Hilliard–Cook equation.This equation is used to describe the complicated phase separation and coarseningphenomena in a melted alloy that is quenched to a temperature at which only twodifferent concentration phases can exist stably (see e.g. [1, 3, 17]). The physicalimportance of the Dirichlet problem was pointed out to us by M. E. Gurtin: it governsthe propagation of a solidification front into an ambient medium which is at restrelative to the front (see e.g. [15]).The existence and uniqueness of the solution to Eq. (1) have already been provenby [12] in the case of G = I for the space dimension d = 1. Moreover, if G = I but d ≥
2, the driving noise should be more regular than the space-time white noise. When ∗ This work was funded by National Natural Science Foundation of China (No. 91630312, No.91530118, No.11021101 and No. 11290142). † School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA([email protected] (corresponding author)) ‡ LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences,Beijing, 100190, China School of Mathematical Science, University of Chinese Academy of Sci-ences, Beijing, 100049, China ([email protected]) 1
This manuscript is for review purposes only.
JIANBO CUI AND JIALIN HONG G is a bounded diffusion coefficient, the authors in [4] obtain the global existence andpath regularity of the solution in d = 1, and the local existence of the solution inhigher dimension d = 2 ,
3. Recently, the authors in [2] extend the results on thelocal existence and uniqueness of the solution in the case that | g ( ξ ) | ≤ C (1 + | ξ | α ), α ∈ (0 , d ≤
3. Meanwhile, the global existence of the solution is achieved underthe restriction that α < , d = 1. However, for the global existence of the solution, itis still unknown whether the sublinear growth condition α < could be extended tothe general sublinear growth condition, i.e., | g ( ξ ) | ≤ C (1 + | ξ | α ) , α ∈ (0 , dX N ( t ) + A ( AX N ( t ) + P N F ( X N ( t ))) dt = P N G ( X N ( t )) dW ( t ) , t ∈ (0 , T ](2) X N (0) = P N X , where N ∈ N + . Then by making use of the factorization formula and the equivalentrandom form of the semi-discrete equation, we show the well-posedness of the semi-discrete equation (2), as well as its uniform a priori estimate and regularity estimate.Furthermore, we show that the limit of the solution of the spectral Galerkin methodexists globally and is the unique mild solution of Eq. (1). As a consequence, the expo-nential integrability property, the optimal temporal and spatial regularity estimatesof the exact solution are proven. Meanwhile, with the help of the Sobolev interpo-lation equality and the smoothing effect of the semigroup S ( t ) := e − A t , the sharpspatial strong convergence rate of the spectral Galerkin method is established underhomogenous Dirichlet boundary condition. To the best of our knowledge, this is notonly the new result on the global existence and regularity estimate of the solution,but also the first result on the strong convergence rate of numerical approximationfor the stochastic Cahn–Hilliard equation driven by multiplicative space-time whitenoise.The rest of this article is organized as follows. In Section 2 the setting andassumptions used are formulated. In Section 3, we prove several uniform a prioriestimates and regularity estimates of the spatial spectral Galerkin method. The strongconvergence analysis of the spatial spectral Galerkin method is presented in Section4. Our main result which states existence, uniqueness and regularity of solutions ofEq. (1) with nonlinear multiplicative noise is presented in Section 5.
2. Preliminaries.
In this section, we present some preliminaries and notations,as well as the assumptions on Eq. (1).Given two separable Hilbert spaces ( H , k · k H ) and ( e H, k · k e H ), L ( H , e H ) and L ( H , e H ) are the Banach spaces of all linear bounded operators and the nuclearoperators from H to e H , respectively. The trace of an operator T ∈ L ( H ) is tr [ T ] = P k ∈ N hT f k , f k i H , where { f k } k ∈ N ( N = { , , , · · · } ) is any orthonormal basisof H . In particular, if T ≥ tr [ T ] = kT k L . Denote by L ( H , e H ) the space ofHilbert–Schmidt operators from H into e H , equipped with the usual norm given by This manuscript is for review purposes only. k · k L ( H , e H ) = ( P k ∈ N k · f k k e H ) . The following useful property and inequality hold kST k L ( H , e H ) ≤ kSk L ( H , e H ) kT k L ( H ) , T ∈ L ( H ) , S ∈ L ( H , e H ) , (3) tr [ Q ] = kQ k L ( H ) = kT k L ( e H, H ) , Q = T T ∗ , T ∈ L ( e H, H ) , where T ∗ is the adjoint operator of T .Given a Banach space ( E , k · k E ) and T ∈ L ( H , E ), we denote by γ ( H , E ) the spaceof γ -radonifying operators endowed with the norm kT k γ ( H , E ) = ( e E k P k ∈ N γ k T f k k E ) ,where ( γ k ) k ∈ N is a Rademacher sequence on a probability space ( e Ω , f F , e P ). For conve-nience, let L q = L q ( O ), 2 ≤ q < ∞ equipped with the usual inner product and norm.We also need the following Burkerholder inequality (see e.g. [20]), (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup t ∈ [0 ,T ] (cid:13)(cid:13)(cid:13) Z t φ ( r ) d f W ( r ) (cid:13)(cid:13)(cid:13) L q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) ≤ C p,q k φ k L p (Ω; L ([0 ,T ]; γ ( H ; L q )) (4) ≤ C p,q (cid:16) E (cid:16) Z T (cid:13)(cid:13)(cid:13) X k ∈ N ( φ ( t ) e k ) (cid:13)(cid:13)(cid:13) L q dt (cid:17) p (cid:17) p , where f W is the H -valued cylindrical Wiener process and { e k } k ∈ N is any orthonormalbasis of H .Next, we introduce some assumptions and spaces associated with A . We denoteby H k := H k ( O ) the standard Sobolev space and E := C ( O ). For convenience,we mainly focus on the well-posedness and numerical approximation for Eq. (1)under homogenous Dirichlet boundary condition. We would like to mention that theapproach for proving the global existence of the unique solution is also available forEq. (1) under homogenous Neumman boundary condition. Denote A = − ∆ theDirichlet Laplacian operator with D ( A ) = (cid:8) v ∈ H ( O ) : v = 0 on ∂ O (cid:9) . It is known that A is a positive definite, self-adjoint and unbounded linear operator on H . Thus there exists an orthonormal eigensystem { ( λ j , e j ) } j ∈ N such that 0 < λ ≤· · · ≤ λ j ≤ · · · with λ j ∼ j and sup j ∈ N + k e j k E < ∞ . We define H α , α ∈ R as thespace of the series v := P ∞ j =1 v j e j , v j ∈ R , such that k v k H α := ( P ∞ j =1 λ αj v j ) < ∞ .Equipped with the norm k · k H α and corresponding inner product, the Hilbert space H α equals D ( A α ). It is obvious that H = H . We denote k · k = k · k H . The followingsmoothing effect of the analytical semigroup S ( t ) = e − tA , t > k A β S ( t ) v k ≤ Ct − β k v k , β > , v ∈ H (5)and the contractivity property of S ( t ) (see e.g. [18, Appendix B]), k S ( t ) v k L q ≤ Ct − ( p − q ) k v k L p , ≤ p ≤ q < ∞ , v ∈ L p , (6) k S ( t ) v k E ≤ Ct − p k v k L p , v ∈ L p , will be used frequently. Throughout this article, the Wiener process W is assumedto be the H -valued cylindrical Wiener process, which implies that for any γ ∈ (0 , ), k A γ − Q k L ( H ) < ∞ . We denote by C a generic constant which may depend onseveral parameters but never on the projection parameter N and may change fromoccurrence to occurrence. We also remark that the approach for proving the globalexistence of the unique solution is available for the cases of higher dimension and moreregular Q -Wiener process. This manuscript is for review purposes only.
JIANBO CUI AND JIALIN HONG
3. A priori estimate and regularity estimate of the spectral Galerkinmethod.
In this section, we give the a priori estimate and regularity estimate of thesolution of Eq. (2). Notice that Eq. (2) is equivalent to the following random PDEand the equation of the discrete stochastic convolution Z N , dY N ( t ) + A ( AY N ( t ) + P N F ( Y N ( t ) + Z N ( T ))) dt = 0 , Y N (0) = P N X , (7) dZ N ( t ) + A Z N ( t ) dt = P N G ( Y N ( t ) + Z N ( t )) dW ( t ) , Z N (0) = 0 . (8)The above decomposition is inspired by [5] where the authors use similar decom-position to show the well-posedness of stochastic reaction-diffusion systems. In thefollowing, we present the a priori and regularity estimates of Z N and Y N . Lemma
Let X ∈ H , T > and q ≥ . There exists a unique solution X N of Eq. (2) satisfying sup t ∈ [0 ,T ] E h(cid:13)(cid:13) X N ( t ) (cid:13)(cid:13) q H − i ≤ C ( X , T, q ) , (9) where C ( X , T, q ) is a positive constant.Proof. Thanks to the fact all the norms in finite dimensional normed linear spacesare equivalent, the norm k · k := k · k H and k · k H − in P N ( H ) are equivalent up toconstants depending on N . The existence of a unique strong solution for Eq. (2)in H − can be obtained by the arguments in [19, Chapter 3]. However, the momentbound of the exact solution will depend on N by this procedure. To prove (9), we needto find a proper Lyapunov functional and to derive the a priori estimate independentof N . According to Eq. (7), by using the chain rule and integration by parts, we havefor any t ≤ T , k Y N ( t ) k H − = k Y N (0) k H − − Z t h∇ Y N ( s ) , ∇ Y N ( s ) i ds − Z t h F ( Y N ( s ) + Z N ( s )) , Y N ( s ) i ds = k Y N (0) k H − − Z t k∇ Y N ( s ) k ds − Z t h F ( Y N ( s ) + Z N ( s )) , Y N ( s ) i ds. The expression of F and Young inequality implies that k Y N ( t ) k H − + 2 Z t k∇ Y N ( s ) k ds + 8( c − ǫ ) Z t k Y N ( s ) k L ds (10) ≤ k Y N (0) k H − + C ( ǫ ) Z t (1 + k Z N ( s ) k L ) ds. Thus it suffices to deduce the a priori estimate of R t k Z N ( s ) k L ds . From the mildform of Z N , the H¨older inequality, the Burkholder inequality and the contractivity of This manuscript is for review purposes only. S ( · ) (6), it follows that for p ≥ q ≥ E [ k Z N ( s ) k qL p ]= E h(cid:13)(cid:13)(cid:13) Z s S ( s − r ) P N G ( Y N ( r ) + Z N ( r )) dW ( r ) (cid:13)(cid:13)(cid:13) qL p i ≤ C E h(cid:16) Z s (cid:13)(cid:13)(cid:13) S ( s − r ) P N G ( Y N ( r ) + Z N ( r )) (cid:13)(cid:13)(cid:13) γ ( H ,L p ) dr (cid:17) q i ≤ C E h(cid:16) Z s ∞ X k =1 (cid:13)(cid:13)(cid:13) S ( s − r ) P N ( G ( Y N ( r ) + Z N ( r )) e k ) (cid:13)(cid:13)(cid:13) L p dr (cid:17) q i ≤ C E h(cid:16) Z s ( s − r ) − ( − p ) ∞ X k =1 (cid:13)(cid:13)(cid:13) S ( s − r P N ( G ( Y N ( r ) + Z N ( r )) e k ) (cid:13)(cid:13)(cid:13) dr (cid:17) q i . The Paserval equality and the sublinear growth of G yield that E [ k Z N ( s ) k qL p ] ≤ C E h(cid:16) Z s ( s − r ) − ( − p ) ∞ X j,k =1 D G ( Y N ( r ) + Z N ( r )) e k , e − λ j ( s − r ) e j E dr (cid:17) q i = C E h(cid:16) Z s ( s − r ) − ( − p ) ∞ X j =1 e − λ j ( s − r ) k G ( Y N ( r ) + Z N ( r )) e j k dr (cid:17) q i ≤ C E h(cid:16) Z s ( s − r ) − ( − p ) ∞ X j =1 e − λ j ( s − r ) k G ( Y N ( r ) + Z N ( r )) k dr (cid:17) q i ≤ C E h(cid:16) Z s ( s − r ) − + p k G ( Y N ( r ) + Z N ( r )) k dr (cid:17) q i ≤ C E h(cid:16) Z s ( s − r ) − p +12 p (1 + k Y N ( r ) k α + k Z N ( r )) k α ) dr (cid:17) q i ≤ C ( Z s ( s − r ) − p +1 p dr ) q E h(cid:16) Z s ( k Y N ( r ) k α + k Z N ( r )) k α ) dr (cid:17) q i . Using the Young inequality, we obtain for 0 ≤ s ≤ t , E [ k Z N ( s ) k qL p ] ≤ Cs q p (cid:16) E [( Z s k Y N ( r ) k α dr ) q ] + Z s E [ k Z N ( r ) k q ] dr (cid:17) ≤ Cs q p (cid:16) E [( Z s k Y N ( r ) k αL p dr ) q ] + Z s E [ k Z N ( r ) k qL p ] dr (cid:17) . Since the moment bound of Z N and Y N are finite depending on N , we can apply theGronwall’s inequality and get that for 0 ≤ s ≤ T , E [ k Z N ( s ) k qL p ] ≤ C ( T ) (cid:16) E [( Z s k Y N ( r ) k αL p dr ) q ] (cid:17) . (11) This manuscript is for review purposes only.
JIANBO CUI AND JIALIN HONG
Now taking k th moment, k ∈ N + on (10) and letting p = 4, q = 4 k , we have E h ( Z t k Y N ( s ) k L ds ) k i ≤ C k Y N (0) k k H − + C ( ǫ ) Z t (1 + E [ k Z N ( s ) k kL ]) ds ≤ C k Y N (0) k k H − + C ( ǫ, T ) (cid:16) C ( ǫ ) + ǫ Z t E [( Z s k Y N ( r ) k L dr ) k ] ds (cid:17) , where ǫ > C ( ǫ, T ) ǫ T < . The above estimationleads to E h ( Z t k Y N ( s ) k L ds ) k i ≤ C k Y N (0) k k H − + C ( k, ǫ, ǫ , T ) , which in turns yields that for k ∈ N + , E h k Y N ( t ) k k H − i + E h ( Z t k∇ Y N ( s ) k ds ) k i + E h ( Z t k Y N ( s ) k L ds ) k i (12) ≤ C ( X , T, k ) . Based on the a priori estimates of Z N and Y N in L p and H − , respectively, wecomplete the proof via the H¨older inequality. Lemma
Let X ∈ H , T > and q ≥ . There exists a positive constant C ( X , T, q ) such that E h sup t ∈ [0 ,T ] (cid:13)(cid:13) Z N ( t ) (cid:13)(cid:13) qE i ≤ C ( X , T, q ) . (13) Proof.
By using the factorization formula in [13, Chapter 5], we have that for α > p + γ , p > γ = , E h sup s ∈ [0 ,T ] k Z N ( s ) k qE i ≤ C ( q, T ) E h k Y α ,N k qL p (0 ,T ; H ) i , where Y α ,N ( s ) = R s ( s − r ) − α S ( s − r ) P N G ( Y N ( r ) + Z N ( r )) dW ( r ). Thus it sufficesto estimate E h k Y α ,N k qL p (0 ,T ; H ) i . From the H¨older and Burkholder inequalities, itfollows that for q ≥ max( p, E h k Y α ,N k qL p (0 ,T ; H ) i = E h(cid:16) Z T (cid:13)(cid:13)(cid:13) Z s ( s − r ) − α S ( s − r ) P N G ( Y N ( r ) + Z N ( r )) dW ( r ) (cid:13)(cid:13)(cid:13) p ds (cid:17) qp i ≤ C ( T, q ) Z T E h(cid:13)(cid:13)(cid:13) Z s ( s − r ) − α S ( s − r ) P N G ( Y N ( r ) + Z N ( r )) dW ( r ) (cid:13)(cid:13)(cid:13) q i ds ≤ C ( T, q ) Z T E h(cid:16) Z s ( s − r ) − α X i ∈ N + (cid:13)(cid:13)(cid:13) S ( s − r ) P N G ( Y N ( r ) + Z N ( r )) e i (cid:13)(cid:13)(cid:13) dr (cid:17) q ds i ≤ C ( T, q ) Z T E h(cid:16) Z s ( s − r ) − (2 α + ) (1 + k Y N ( r ) k α + k Z N ( r ) k α ) dr (cid:17) q ds i . This manuscript is for review purposes only.
Since α <
1, one can choose a positive number l > p such that 2 αl < α + ) ll − <
1. Then by using a priori estimates (11) and(12), we obtain E h k Y α,N k qL p (0 ,T ; H ) i ≤ C ( T, q, α ) Z T ( Z s ( s − r ) − (2 α + ) ll − dr ) q ( l − l × E h(cid:16) Z s (1 + k Y N ( r ) k αl + k Z N ( r ) k αl ) dr (cid:17) q l i ds ≤ C ( T, q, α, X ) , which implies that E h sup s ∈ [0 ,T ] k Z N ( s ) k qE i ≤ C ( T, q, α, X ) . Corollary
Let X ∈ H , T > and q ≥ . Then the solution X N of Eq. (2) satisfies E h sup t ∈ [0 ,T ] (cid:13)(cid:13) X N ( t ) (cid:13)(cid:13) q H − i ≤ C ( X , T, q ) , (14) where C ( X , T, q ) is a positive constant.Proof. Similar arguments in the proof of (12) yield that for any k ≥ E h sup t ∈ [0 ,T ] k Y N ( t ) k k H − i ≤ C ( X , T, k ) . Combining this estimate with Lemma 3.2, we complete the proof.Thanks to the above a priori estimates of Y N and Z N , we are now in a position todeduce the a priori estimate of X N in H . Lemma
Let X ∈ H , T > and q ≥ . There exists a positive constant C ( X , T, q ) such that E h sup t ∈ [0 ,T ] (cid:13)(cid:13) X N ( t ) (cid:13)(cid:13) q H i ≤ C ( X , T, q ) . (15) Proof.
By applying the integration by parts and the dissipativity of − F , we obtain k Y N ( t ) k + (2 − ǫ ) Z t k ( − A ) Y N ( s ) k ds ≤ k X N k + C Z t ( k Z N ( s ) k E + 1) k Y N ( s ) k L ds + C Z t (1 + k∇ Y N ( s ) k + k Z N ( s ) k L ) ds. Taking the p th moment and using the a priori estimates (12) and (13), we have that This manuscript is for review purposes only.
JIANBO CUI AND JIALIN HONG for p ≥ E h sup t ∈ [0 ,T ] k Y N ( t ) k p i + E h Z T k ( − A ) Y N ( s ) k p ds i ≤ C ( p, T ) (cid:16) E h k X N k p i + E h (1 + sup s ∈ [0 ,T ] k Z N ( s ) k pE )( Z T k Y N ( s ) k L ds ) p i + E h(cid:16) Z T (1 + k∇ Y N ( s ) k + k Z N ( s ) k L ) ds (cid:17) p i(cid:17) ≤ C ( T, X N , p ) , which, together with (13) and the H¨older inequality, completes the proof.Based on the a priori estimate of k X N k , we are in a position to deduce theregularity estimate of X N . Before that, we first give the regularity estimate of Z N . Lemma
Let X ∈ H , q ≥ and γ ∈ (0 , ) . Then the discrete stochasticconvolution Z N satisfies E h sup t ∈ [0 ,T ] (cid:13)(cid:13) Z N ( t ) (cid:13)(cid:13) q H γ i ≤ C ( X , T, q )(16) for a positive constant C ( X , T, q ) .Proof. By the factorization method, we have for α > p + β , p > β = γ , E h sup s ∈ [0 ,T ] k Z N ( s ) k q H γ i ≤ C ( T, q ) E h k Y α ,N k qL p (0 ,T ; H ) i , where Y α ,N ( s ) = R s ( s − r ) − α S ( s − r ) P N G ( Y N ( r )+ Z N ( r )) dW ( r ). From the H¨olderand Burkholder inequalities, the estimates (12) and (15), it follows that for q ≥ max( p, E h k Y α ,N k qL p (0 ,T ; H ) i ≤ C ( T, q ) Z T E h(cid:16) Z s ( s − r ) − α X i ∈ N + (cid:13)(cid:13)(cid:13) S ( s − r ) P N G ( Y N ( r ) + Z N ( r )) e i (cid:13)(cid:13)(cid:13) dr (cid:17) q ds i ≤ C ( T, q ) Z T E h(cid:16) Z s ( s − r ) − (2 α + ) (1 + k Y N ( r ) k α + k Z N ( r ) k α ) dr (cid:17) q ds i ≤ C ( T, q ) (cid:16) E h sup r ∈ [0 ,T ] k Y N ( r ) k αq i + E h sup r ∈ [0 ,T ] k Z N ( r ) k αq i(cid:17) × Z T ( Z s ( s − r ) − (2 α + ) dr ) q ds ≤ C ( X , T, q, α ) Z T ( Z s ( s − r ) − (2 α + ) dr ) q ds. Since γ < , one can choose a positive a large enough number p such that p + γ + < α + <
1. Thus we obtain E h k Y α ,N k qL p (0 ,T ; H ) i ≤ C ( X , T, q, γ ) , This manuscript is for review purposes only. which implies that for any q ∈ N + , E h sup s ∈ [0 ,T ] k Z N ( s ) k q H γ i ≤ C ( X , T, q, γ ) . Next, we deduce the following uniform regularity estimate of X N . Proposition
Let X ∈ H γ , γ ∈ [1 , ) , T > , q ≥ and N ∈ N + . Then theunique mild solution X N of Eq. (2) satisfies E h sup t ∈ [0 ,T ] (cid:13)(cid:13) X N ( t ) (cid:13)(cid:13) q H γ i ≤ C ( X , T, q )(17) for a positive constant C ( X , T, q ) .Proof. Due to (16), it suffices to give the regularity estimate for Y N . Before that,we give the following estimate of k Y N ( t ) k L . The Sobolev embedding theorem, thecontractivity (6) from L to L , the smoothing effect (5), and the Gagliardo–Nirenberginequality yield that k Y N ( t ) k L ≤ k S ( t ) X N k L + Z t k S ( t − s S ( t − s A ) P N F ( Y N ( s ) + Z N ( s )) k L ds ≤ C k X N k L + C Z t ( t − s ) − k S ( t − s A kk F ( Y N ( s ) + Z N ( s )) k ds ≤ C k X N k L + C Z t ( t − s ) − (cid:16) k Z N ( s ) k L + k Y N ( s ) k L (cid:17) ds ≤ C k X N k H + C Z t ( t − s ) − (cid:16) k Z N ( s ) k L + k AY N ( s ) k k Y N ( s ) k (cid:17) ds. From the H¨older inequality, the estimates (12), (15) and (16), it follows that for any q ≥ E h sup t ∈ [0 ,T ] k Y N ( t ) k qL i ≤ C ( p ) k X N k q H + C ( p ) E h(cid:16) Z T k AY N ( s ) k ds (cid:17) q i + C ( p )( Z T ( t − s ) − ds ) q E h sup s ∈ [0 ,T ] k Y N ( s ) k q i + C ( p )( Z T ( t − s ) − ds ) q (cid:16) E h sup s ∈ [0 ,T ] k Z N ( s ) k qE i(cid:17) ≤ C ( X , T, q ) . The mild form of Y N ( t ) and (5) lead to k Y N ( t ) k H γ ≤ k e − A t X N k H γ + Z t (cid:13)(cid:13) e − A ( t − s ) AF ( Y N ( s ) + Z N ( s )) (cid:13)(cid:13) H γ ds ≤ C k X N k H γ + C Z t ( t − s ) − (cid:13)(cid:13) e − A ( t − s ) F ( Y N ( s ) + Z N ( s )) (cid:13)(cid:13) H γ ds ≤ C k X k H γ + C Z t ( t − s ) − − γ (cid:0) k Y N ( s ) k L + k Z N ( s ) k L (cid:1) ds. By taking q th moment and making use of the a priori estimates of k Y N k L and k Z N k H γ , we finish the proof. This manuscript is for review purposes only. JIANBO CUI AND JIALIN HONG
Remark If X ∈ H γ , γ ∈ (0 , , the estimate (17) also holds for γ ∈ (0 , .The key ingredient of the proof is use of the contractivity of S ( t ) to deal with the term k S ( t ) X N k L . Indeed, (6) yields that k S ( t ) X N k L ≤ Ct − k X k . From the H¨older inequality, it follows that there exist p , q satisfying p + q = 1 , ( + γ ) p < and q < , such that Z t ( t − s ) − − γ k Y N ( s ) k L ds ≤ ( Z t ( t − s ) − ( + γ ) p ds ) p ( Z t k Y N ( s ) k q L ds ) q . Based on the above estimate and similar arguments in the proof of Proposition 3.1,we obtain the desired result.
After these preparations, we are able to answer the well-posedness problem ofEq. (1). Before that, we give the useful lemma whose proof is similar to that of [10,Lemma 4.3].
Lemma
Let g : L → H be the Nemytskii operator of a polynomial of seconddegree. Then for any β ∈ (0 , , it holds that k g ( x ) y k H − ≤ C (cid:0) k x k E + k x k H β (cid:1) k y k H − β , where x ∈ E, x ∈ H β and y ∈ H . Proposition
Let sup N ∈ N + k X N k E ≤ C ( X ) , T > and q ≥ . Then theunique solution X N of Eq. (2) satisfies E h sup t ∈ [0 ,T ] (cid:13)(cid:13) X N ( t ) (cid:13)(cid:13) qE i ≤ C ( X , T, q )(18) for a positive constant C ( X , T, q ) .Proof. Due to Corollary 3.2, it remains to bound E h sup t ∈ [0 ,T ] (cid:13)(cid:13) Y N ( t ) (cid:13)(cid:13) qE i . The mildform of Y N , combined with (6), (5) and the estimation of k Y N k L , yields that E h sup t ∈ [0 ,T ] k Y N ( t ) k qE i ≤ E h sup t ∈ [0 ,T ] k S ( t ) X N k qE i + C E h(cid:16) Z T ( t − s ) − k F ( Y N + Z N ) k ds (cid:17) q i ≤ C ( X , T, q ) , which completes the proof.
4. Strong convergence analysis of the spectral Galerkin method.
Themain idea of our approach to proving the global existence of the solution is to showthe uniform convergence of the sequence { ( Y N , Z N ) } N ∈ N + and then to prove thelimit process is the unique mild solution of Eq. (1). In the following, we first presentthe strong convergence analysis of the spectral Galerkin approximation in H − . Wewould like to mention that there already exists some convergence result of finitedimensional approximation for Eq. (1) driving by additive space-time white noise This manuscript is for review purposes only.
Proposition
Let X ∈ H γ , γ ∈ (0 , ) , T > , p ≥ and sup N ∈ N + k X N k E < ∞ . Assume that X N and X M are the spectral Galerkin approximations with differentparameters N, M ∈ N + , N < M . Then it holds that sup t ∈ [0 ,T ] E h k X N ( t ) − X M ( t ) k p H − i ≤ C ( T, X , p ) λ − γpN , (19) where C ( T, X , p ) is a positive constant.Proof. Due to Proposition 3.1, we obtain that for t ∈ [0 , T ], p ≥ γ ∈ (0 , ), E h k ( I − P N ) X M ( t ) k p H − i ≤ E h k ( I − P N ) A − − γ A γ X M ( t ) k p i ≤ C ( X , T, p, γ ) λ − p − γp N . Thus it remains to estimate k X N − P N X M k H − . From the Taylor expansion and Itˆoformula, it follows that for p ≥ k X N ( t ) − P N X M ( t ) k p H − = − p Z t k X N ( s ) − P N X M ( s ) k p − H − k X N ( s ) − P N X M ( s ) k H ds − p Z t k X N ( s ) − P N X M ( s ) k p − H − D Z F ′ ( θX N ( s ) + (1 − θ ) X M ( s )) dθ ( X N ( s ) − P N X M ( s ) − ( I − P N ) X M ( s )) , X N ( s ) − P N X M ( s ) E ds + 2 p Z t k X N ( s ) − P N X M ( s ) k p − H − h X N ( s ) − P N X M ( s ) , ( G ( X N ( s )) − G ( X M ( s ))) dW ( s ) i H − + 2 p Z t X i ∈ N + k X N ( s ) − P N X M ( s ) k p − H − k P N (( G ( X N ( s )) − G ( X M ( s ))) e i ) k H − ds + 2 p (2 p − Z t k X N ( s ) − P N X M ( s ) k p − H − X i ∈ N + |h X N ( s ) − P N X M ( s ) ,P N ( G ( X N ( s )) − G ( X M ( s )) e i ) i H − | ds =: − p Z t k X N ( s ) − P N X M ( s ) k p − H − k X N ( s ) − P N X M ( s ) k H ds + I ( t ) + I ( t ) + I ( t ) + I ( t ) . The monotonicity of − F and the Young inequality yield that I ≤ C Z t k X N ( s ) − P N X M ( s ) k p − H − k X N ( s ) − P N X M ( s ) k ds + 2 Z t k X N ( s ) − P N X M ( s ) k p − H − D A − Z F ′ ( θX N ( s ) + (1 − θ ) X M ( s )) dθ ( I − P N ) X M ( s )) , A ( X N ( s ) − P N X M ( s )) E ds. This manuscript is for review purposes only. JIANBO CUI AND JIALIN HONG
From Lemma 3.5, it follows that for β ∈ (0 ,
1) and small ǫ > I ≤ ǫ Z t k X N ( s ) − P N X M ( s ) k p − H − k X N ( s ) − P N X M ( s ) k H ds + C ( ǫ ) Z t k X N ( s ) − P N X M ( s ) k p H − ds + C ( ǫ ) Z t (cid:13)(cid:13)(cid:13) X N ( s ) − P N X M ( s ) k p − H − k A − Z F ′ ( θX N ( s ) + (1 − θ ) X M ( s )) dθ ( I − P N ) X M ( s )) k ds ≤ ǫ Z t k X N ( s ) − P N X M ( s ) k p − H − k X N ( s ) − P N X M ( s ) k H ds + C ( ǫ ) Z t k X N ( s ) − P N X M ( s ) k p H − ds + C ( ǫ ) λ − γ − βN Z t k X N ( s ) − P N X M ( s ) k p − H − (cid:16) k X N k H β + k X M k H β + k X N k E + k X M k E (cid:17) k X M ( s ) k H γ ds. The uniform boundedness of { e j } j ∈ N + and the Young inequality yield that for small ǫ > E h I + I i ≤ C E h Z t k X N ( s ) − P N X M ( s ) k p − H − X i ∈ N + k P N (( G ( X N ( s )) − G ( X M ( s ))) e i ) k H − ds i ≤ C E h Z t k X N ( s ) − P N X M ( s ) k p − H − X j ∈ N + k ( G ( X N ( s )) − G ( X M ( s ))) e j ) k λ − j ds i ≤ C ( ǫ ) E h Z t k X N ( s ) − P N X M ( s ) k p H − ds i + ǫ E h Z t k X N ( s ) − P N X M ( s ) k p − H − k X N ( s ) − P N X M ( s ) k H ds i + C E h Z t k X N ( s ) − P N X M ( s ) k p − H − k ( I − P N ) X M ( s ) k ds i . The above estimations, combined with the Young inequality and the martingale prop-
This manuscript is for review purposes only. I , yield that for β ∈ (0 ,
1) and small ǫ > E h k X N ( t ) − P N X M ( t ) k p H − i ≤ − p Z t E h k X N ( s ) − P N X M ( s ) k p − H − k X N ( s ) − P N X M ( s ) k H i ds + E h I ( t ) + I ( t ) + I ( t ) + I ( t ) i ≤ C ( ǫ ) Z t E h k X N ( s ) − P N X M ( s ) k p H − i ds + C ( ǫ ) λ − γ − βN Z t E h k X N ( s ) − P N X M ( s ) k p − H − (cid:16) k X N k H β + k X M k H β + k X N k E + k X M k E (cid:17) k X M ( s ) k H γ i ds + C Z t E h k X N ( s ) − P N X M ( s ) k p − H − k ( I − P N ) X M ( s ) k i ds ≤ C ( ǫ ) Z t E h k X N ( s ) − P N X M ( s ) k p H − i ds + C ( ǫ ) λ − γp − βpN Z t E h(cid:16) k X N k H β + k X M k H β + k X N k E + k X M k E (cid:17) p k X M ( s ) k p H γ i ds + Cλ − γpN Z t E h k X M ( s ) k p H γ i ds. Combining the regularity estimates of X N and X M in Proposition 3.1, we completethe proof by using the Gronwall inequality.Now, we are in the position to deduce the error estimate in H , which impliesthat { X N } N ∈ N + is a Cauchy sequence in L p (Ω; C ([0 , T ]; H )). The following strongconvergence rate of the spectral Galerkin approximation is also applied for analyzingthe strong convergence of the full discretization and its density function in [7]. Theorem
Let X ∈ H γ , γ ∈ (0 , ) , T > , p ≥ and sup N ∈ N + k X N k E < ∞ .Assume that X N and X M are the spectral Galerkin approximations with differentparameters N, M ∈ N + , N < M . Then for τ ∈ (0 , γ ) , it holds that E h sup t ∈ [0 ,T ] k X N ( t ) − X M ( t ) k p i ≤ C ( T, X , p ) λ − τpN . (20) for a positive constant C ( T, X , p ) .Proof. From the mild form of X N and P N X M , the smoothing effect (5) of S ( t ),Lemma 3.5, Proposition 4.1, the interpolation inequality and the Burkholder inequal- This manuscript is for review purposes only. JIANBO CUI AND JIALIN HONG ity, it follows that for p ≥ max( l , l > β ∈ (0 , E h k X N ( t ) − P N X M ( t ) k p i ≤ C ( p, T ) (cid:0) Z T ( t − s ) − l l − ds (cid:1) p ( l − l E h(cid:16) Z T (cid:0) k X N k lE + k X M k lE + k X N k l H β + k X M k l H β (cid:1)(cid:13)(cid:13) X N ( s ) − X M ( s ) (cid:13)(cid:13) βl H − (cid:13)(cid:13) X N ( s ) − X M ( s ) (cid:13)(cid:13) (1 − β ) l ds (cid:17) pl i + C ( p, T ) E h(cid:13)(cid:13) Z t S ( t − s )( G ( X N ( s )) − G ( X M ( s ))) dW ( s ) (cid:13)(cid:13) p i ≤ C ( p, T, X , β ) λ − βpγN + C ( p, T ) E h(cid:16) Z t X i ∈ N + k S ( t − s )( G ( X N ( s )) − G ( X M ( s ))) e i k ds (cid:17) p i ≤ C ( p, T, X , β ) λ − βpγN + C ( p, T ) E h(cid:16) Z t k X N ( s ) − P N X M ( s ) k ds (cid:17) p i ≤ C ( p, T, X , β ) λ − βpγN + C ( p, T ) Z t E h k X N ( s ) − P N X M ( s ) k p i ds. From the Gronwall inequality, it follows thatsup t ∈ [0 ,T ] E h k X N ( t ) − P N X M ( t ) k p i ≤ C ( X , T, p, γ ) λ − βpγN . (21)Furthermore, taking supreme over t ∈ [0 , T ], similar arguments yield that for p ≥ max( l , l > β ∈ (0 , E h sup t ∈ [0 ,T ] k X N ( t ) − P N X M ( t ) k p i ≤ C ( p, T, X , β ) λ − βpγN + C ( p ) E h sup t ∈ [0 ,T ] (cid:13)(cid:13) Z t S ( t − s )( G ( X N ( s )) − G ( X M ( s ))) dW ( s ) (cid:13)(cid:13) p i . The factorization method yields that for α > q , q > E h sup t ∈ [0 ,T ] (cid:13)(cid:13) Z t S ( t − s )( G ( X N ( s )) − G ( X M ( s ))) dW ( s ) (cid:13)(cid:13) p i ≤ C ( p, q, T ) E h k Z α ,N,M k pL q ([0 ,T ]; H ) i , where Z α ,N,M ( s ) = R s ( s − r ) − α S ( s − r ) P N ( G ( X N ( r )) − G ( X M ( r ))) dW ( r ). Thusit suffices to estimate E h k Z α ,N,M k pL q ([0 ,T ]; H ) i . From the H¨older and Burkholder This manuscript is for review purposes only. p ≥ q , E h k Z α ,N,M k pL q ([0 ,T ]; H ) i = E h(cid:16) Z T (cid:13)(cid:13)(cid:13) Z s ( s − r ) − α S ( s − r ) P N ( G ( X N ( s )) − G ( X M ( s ))) dW ( r ) (cid:13)(cid:13)(cid:13) q ds (cid:17) pq i ≤ C ( T, p ) Z T E h(cid:13)(cid:13)(cid:13) Z s ( s − r ) − α S ( s − r ) P N ( G ( X N ( s )) − G ( X M ( s ))) dW ( r ) (cid:13)(cid:13)(cid:13) p i ds ≤ C Z T E h(cid:16) Z s ( s − r ) − α X i ∈ N + (cid:13)(cid:13)(cid:13) S ( s − r ) P N ( G ( X N ( s )) − G ( X M ( s ))) e i (cid:13)(cid:13)(cid:13) dr (cid:17) p i ds ≤ C Z T E h(cid:16) Z s ( s − r ) − (2 α + ) k X N ( s ) − P N X M ( s ) k dr (cid:17) p ds i + C ( T, p ) λ − γpN ≤ Cλ − γβpN . Combining the above estimates and E h k sup t ∈ [0 ,T ] ( I − P N ) X M ( t ) k p i ≤ C ( T, X , p, γ ) λ − γpN , we complete the proof. Remark If X ∈ H γ , γ > , then k X N k E ≤ C ( X ) holds for every N ∈ N + .If the bound of k X N k is not uniform, then by using (6) , we have that E (cid:2) k X N ( t ) k qE (cid:3) ≤ C ( X , T, q )(1 + t min( − − ǫ + γ , q ) . As a result, it’s is not hard to check that Proposition 4.1, Theorem 4.1 and Proposition5.1 still hold with p = 1 , which is helpful for establishing the wellposedness result undermild assumptions.
5. Global existence and regularity estimate.
Based on the convergence ofthe approximate process X N , we are in a position to show the global existence of theunique solution for Eq. (1) driven by multiplicative space-time white noise. Proposition
Let
T > , X ∈ H γ , γ ∈ (0 , ) , p ≥ and sup N ∈ N + k X N k E ≤ C ( X ) . Then Eq. (1) possesses a unique mild solution X in L p (Ω; C (0 , T ; H )) .Proof. We first show the local uniqueness of the mild solution for Eq. (1). Let τ R := inf { t ∈ [0 , T ] (cid:12)(cid:12) k X ( t ) k > R } . Then the uniqueness in [0 , τ R ] is obtained due tothe Lipschitz continuity of G and the local Lipschitz continuity of F . More precisely,assume that we have two different mild solutions X and X for Eq. (1) with the sameinitial datum X . Next, we prove the local uniqueness, i.e., in each [0 , τ R ], X = X ,a.s. Since the decompositions X = Y + Z and X = Y + Z , we have for t ∈ [0 , τ R ], d ( Y ( t ) − Y ( t )) = − A ( Y ( t ) − Y ( t )) dt − A ( F ( X ( t )) − F ( X ( t ))) dt,Y (0) − Y (0) = 0 , and d ( Z ( t ) − Z ( t )) = − A ( Z ( t ) − Z ( t )) dt + ( G ( X ( t )) − G ( X ( t ))) dW ( t ) ,Z (0) − Z (0) = 0 . This manuscript is for review purposes only. JIANBO CUI AND JIALIN HONG
From the mild form of Z − Z and the factorization method, it follows that E h sup t ∈ [0 ,τ R ] k Z ( t ) − Z ( t ) k pE i = E h sup t ∈ [0 ,τ R ] (cid:13)(cid:13)(cid:13) Z t S ( t − s )( G ( X ( s )) − G ( X ( s ))) dW ( s ) (cid:13)(cid:13)(cid:13) p i ≤ C ( q, T ) E h Z τ R (cid:16) Z t ( t − s ) − α − X i ∈ N + k S ( t − s )( G ( X ( s )) − G ( X ( s ))) e i k ds (cid:17) p dt i ≤ C ( q, T ) E h sup t ∈ [0 ,τ R ] (cid:13)(cid:13) X ( t ) − X ( t ) (cid:13)(cid:13) p i , where α > p + for large enough p >
1. Similar arguments, together with the Youngand Gagliardo–Nirenberg inequality, yield that for t ∈ [0 , τ R ] and for some ǫ < k Y ( t ) − Y ( t ) k ≤ − Z t k A ( Y ( s ) − Y ( s )) k ds + 2 Z t h− A ( F ( X ( s )) − F ( X ( s ))) , Y ( t ) − Y ( t ) i ds ≤ C ( ǫ ) Z t k F ( X ( s )) − F ( X ( s )) k ds ≤ C ( ǫ ) Z t k X ( s ) − X ( s ) k (1 + k X ( s ) k E + k X ( s ) k E ) ds ≤ C ( ǫ ) Z t k X ( s ) − X ( s ) k (1 + k AY ( s ) k + k AY ( s ) k + k Y ( s ) k + k Y ( s ) k + k Z ( s ) k E + k Z ( s ) k E ) ds. Notice that for i = 1 , t ∈ [0 , τ R ], Z t k AY i ( s ) k ds ≤ C k Y k + C Z t ( k Z i ( s ) k E + 1) k Y i ( s ) k L ds + C Z t (1 + k∇ Y i ( s ) k + k Z i ( s ) k L ) ds ≤ C ( R, T, Y ) < ∞ . By Gronwall’s inequality, we get k Y ( t ) − Y ( t ) k ≤ exp( C ( R, T, Y )) Z t k X ( s ) − X ( s ) k ds. From the previous estimates, we conclude that for 0 ≤ s ≤ t ≤ τ R and p ≥ k X ( s ) − X ( s ) k p ≤ C p k Y ( s ) − Y ( s ) k p + C p k Z ( s ) − Z ( s ) k p ≤ C p exp( C ( R, T, Y )) Z s k X ( s ) − X ( s ) k p ds + C p k Z ( s ) − Z ( s ) k p , This manuscript is for review purposes only. k X ( s ) − X ( s ) k p ≤ exp( C ( p, T ) exp( C ( R, T, Y ))) k Z ( s ) − Z ( s ) k p . Taking expectation and using the Burkholder inequality, we have for large enough q > E h sup s ∈ [0 ,t ] k X ( s ) − X ( s ) k p i ≤ exp( C ( p, T ) exp( C ( R, T, Y ))) E h sup s ∈ [0 ,t ] k Z ( s ) − Z ( s ) k p i ≤ C ( R, T, p, Y ) E h Z t (cid:16) Z s ( s − r ) − − q k G ( X ( r )) − G ( X ( r )) k dr (cid:17) p dt i ≤ C ( R, T, p, Y ) Z t E h sup r ∈ [0 ,s ] k X ( r )) − X ( r ) k p i ds. From Gronwall’s inequality, it follows that for any t ≤ τ R , E h sup s ∈ [0 ,t ] k X ( s ) − X ( s ) k p i = 0 . Thus the local uniqueness of the mild solution holds. Once the global existence of themild solution holds, we have E h sup s ∈ [0 ,T ] k X ( s ) − X ( s ) k p i ≤ lim R →∞ E h sup s ∈ [0 ,τ R ] k X ( s ) − X ( s ) k p i = 0 , since it holds that lim R →∞ τ R = T, a.s.
In the following, we show the existence of the global mild solution. According toTheorem 4.1, we have that { X N } N ∈ N + is a Cauchy sequence in L p (Ω; C ([0 , T ]; H )).Then we denote X the limit of X N in L p (Ω; C ([0 , T ]; H )). From lim N →∞ k X N − X k L p (Ω; C ([0 ,T ]; H )) = 0, it follows that for each i ∈ N + ,lim N →∞ k|h X N − X, e i i|k L p (Ω; C ([0 ,T ]; R )) = 0 , which implies that for a subsequence { X N k } k ∈ N + ,lim k →∞ sup t ∈ [0 ,T ] |h X N k ( t ) , e i i| λ γi = sup t ∈ [0 ,T ] |h X ( t ) , e i i| λ γi , a.s. , and lim N →∞ X N k = X in C ([0 , T ]; H γ ), a.s.The uniform boundedness of k X N k L p (Ω; C ([0 ,T ]; H γ )) , together with Fatou’s lemma,yields that k X k pL p (Ω; C ([0 ,T ]; H γ )) ≤ lim inf N →∞ k X N k k pL p (Ω; C ([0 ,T ]; H γ )) ≤ C ( T, X , p ) . Thus it suffices to prove that X is the mild solution of Eq. (1), i.e., X ( t ) = S ( t ) X + Z t S ( t − s ) F ( X ( s )) ds + Z t S ( t − s ) G ( X ( s )) dW ( s ) , a.s. This manuscript is for review purposes only. JIANBO CUI AND JIALIN HONG
The mild form of X N and (5) yield that Err := k S ( t )( I − P N ) X k L p (Ω; C ([0 ,T ; H ])) + (cid:13)(cid:13)(cid:13) Z t S ( t − s ) A ( F ( X ( s )) − P N F ( X N ( s ))) ds (cid:13)(cid:13)(cid:13) L p (Ω; C ([0 ,T ]; H )) + (cid:13)(cid:13)(cid:13) Z t S ( t − s )( G ( X ( s )) − P N G ( X N ( s ))) dW ( s ) (cid:13)(cid:13)(cid:13) L p (Ω; C ([0 ,T ]; H )) ≤ C ( T, X ) λ − γ N + (cid:13)(cid:13)(cid:13) Z t S ( t − s ) A ( I − P N ) F ( X N ( s )) ds (cid:13)(cid:13)(cid:13) L p (Ω; C ([0 ,T ]; H )) + (cid:13)(cid:13)(cid:13) Z t S ( t − s ) AP N ( F ( X ( s ) − F ( X N ( s ))) ds (cid:13)(cid:13)(cid:13) L p (Ω; C ([0 ,T ]; H )) + (cid:13)(cid:13)(cid:13) Z t S ( t − s )( I − P N ) G ( X ( s ) dW ( s ) (cid:13)(cid:13)(cid:13) L p (Ω; C ([0 ,T ]; H )) + (cid:13)(cid:13)(cid:13) Z t S ( t − s ) P N ( G ( X ( s )) − G ( X N ( s ))) dW ( s ) (cid:13)(cid:13)(cid:13) L p (Ω; C ([0 ,T ]; H )) ≤ C ( T, X , p ) λ − γ N + C ( T, p ) λ − γ N (cid:13)(cid:13)(cid:13) Z t ( t − s ) − − γ k F ( X N ( s )) k H ds (cid:13)(cid:13)(cid:13) L p (Ω; C ([0 ,T ]; R )) + C ( T, p ) (cid:13)(cid:13)(cid:13) Z t ( t − s ) − (1 + k X ( s ) k E + k X N ( s ) k E ) k X ( s ) − X N ( s ) k ds (cid:13)(cid:13)(cid:13) L p (Ω; C ([0 ,T ]; R )) + C ( T, p ) (cid:13)(cid:13)(cid:13) Z t ( t − s ) − α S ( t − s )( I − P N ) G ( X ( s ) dW ( s ) (cid:13)(cid:13)(cid:13) L p (Ω; L q ([0 ,T ]; H )) + C ( T, p ) (cid:13)(cid:13)(cid:13) Z t ( t − s ) − α S ( t − s ) P N ( G ( X ( s )) − G ( X N ( s ))) dW ( s ) (cid:13)(cid:13)(cid:13) L p (Ω; L q ([0 ,T ]; H )) According to the factorization method, the Burkholder inequality, (5) and the errorestimate (21), we have that for 2 p ≥ q , α > q , sufficient large q > β ∈ (0 , Err ≤ C ( T, X , p ) λ − βγ N + C ( T, p ) (cid:13)(cid:13)(cid:13) Z t ( t − s ) − α S ( t − s )( I − P N ) G ( X ( s ) dW ( s ) (cid:13)(cid:13)(cid:13) L p (Ω; L q ([0 ,T ]; H )) + C ( T, p ) (cid:13)(cid:13)(cid:13) Z t ( t − s ) − α S ( t − s ) P N ( G ( X ( s )) − G ( X N ( s ))) dW ( s ) (cid:13)(cid:13)(cid:13) L p (Ω; L q ([0 ,T ]; H )) ≤ C ( T, X , p ) λ − βγ N + C ( T, p ) λ − γ N (cid:13)(cid:13)(cid:13) Z t ( t − s ) − α − − γ (1 + k X ( s ) k α ) ds (cid:13)(cid:13)(cid:13) L p (Ω; L q ([0 ,T ]; R )) + C ( T, p ) (cid:13)(cid:13)(cid:13) Z t ( t − s ) − α − k X ( s ) − X N ( s ) k ds (cid:13)(cid:13)(cid:13) L p (Ω; L q ([0 ,T ]; R )) ≤ C ( T, X , p ) λ − βγ N . The above estimation implies that S ( t ) X + Z t S ( t − s ) F ( X ( s )) ds + Z t S ( t − s ) G ( X ( s )) dW ( s ) This manuscript is for review purposes only. X N in L p (Ω; C ([0 , T ]; H )).By the uniqueness of the limit in L p (Ω; C ([0 , T ]; H )), we conclude that X ( t ) = S ( t ) X + R t S ( t − s ) F ( X ( s )) ds + R t S ( t − s ) G ( X ( s )) dW ( s ), a.s.From the arguments in the above proof, we immediately get that the followingwell-posedness result under mild assumptions. As a cost, we can not obtain theoptimal convergence rate of this Cauchy sequence { X N } N ∈ N + . Theorem
Let
T > , X ∈ H γ , γ > , p ≥ . Then Eq. (1) possesses aunique mild solution X in L p (Ω; C ([0 , T ]; H )) .Proof. Since the strong convergence in Theorem 4.1 holds with p = 1 (see Remark4.1), we have that { X N } N ∈ N + is a Cauchy sequence in L (Ω; C ([0 , T ]; H )), whichimplies that there exists a subsequence { X N k } k ∈ N + converging to X in C ([0 , T ]; H )a.s. Notice that Lemma 3.3 implies that X N ∈ L p (Ω; C ([0 , T ]; H )) for any p ≥
1. Byusing the H¨older inequality and Fatou’s lemma, we obtain k X − X N k L p (Ω; C ([0 ,T ]; H )) ≤ k X − X N k L p − (Ω; C ([0 ,T ]; H )) k X − X N k L (Ω; C ([0 ,T ]; H )) ≤ k X − X N k p L (Ω; C ([0 ,T ]; H )) (cid:0) C ( X , T, p ) + lim k →∞ k X N k k L p − (Ω; C (0 ,T ; H )) (cid:1) p − p ≤ C ( X , T, p ) k X − X N k p L (Ω; C ([0 ,T ]; H )) , which implies that { X N } N ∈ N + is also a Cauchy sequence in L p (Ω; C ([0 , T ]; H )). Remark
Let
T > , X ∈ H , p ≥ . By the similar arguments in the proofof Proposition 5.3 and Theorem 5.1, one may prove that Eq. (1) possesses a uniquemild solution X in C ([0 , T ]; L p (Ω; H )) . After establishing the well-posedness of Eq. (1), we turn to giving the followingproperties of the exact solution X . Corollary
Let X ∈ H γ , γ ∈ (0 , ) , T > and p ≥ . The unique mildsolution X of Eq. (1) satisfies E h sup t ∈ [0 ,T ] (cid:13)(cid:13) X ( t ) (cid:13)(cid:13) p H γ i ≤ C ( X , T, p ) . (22) Proof.
By Proposition 3.1, Theorem 4.1 and Fatou’s Lemma, we completes theproof.
Proposition
Let X ∈ E , T > and p ≥ . The unique mild solution X of Eq. (1) satisfies sup t ∈ [0 ,T ] E h(cid:13)(cid:13) X ( t ) (cid:13)(cid:13) pE i ≤ C ( X , T, p ) . Proof.
The proof is similar to that of Proposition 3.2.
Remark
Under the condition of Proposition 5.2, one can prove that thesolution X has almost surely continuous trajectories in E . In addition we assumethat X is β -H¨older continuous with β ∈ (0 , . By using the fact that S ( · ) is ananalytical semigroup in E and similar arguments in the proof of Proposition 5.3, wehave that X is almost surely β -continuous in space and β -continuous in time.This manuscript is for review purposes only. JIANBO CUI AND JIALIN HONG
Proposition
Let X ∈ H γ , γ ∈ (0 , ) , p ≥ . Then the unique mild solution X of Eq. (1) satisfies k X ( t ) − X ( s ) k L p (Ω; H ) ≤ C ( X , T, p )( t − s ) γ (23) for a positive constant C ( X , T, p ) and ≤ s ≤ t ≤ T .Proof. From the mild form of X , it follows that k X ( t ) − X ( s )) k ≤ k ( S ( t ) − S ( s )) X k + Z s (cid:13)(cid:13)(cid:13) ( S ( t − r ) − S ( s − r )) AF ( X ( r )) (cid:13)(cid:13)(cid:13) dr + Z ts (cid:13)(cid:13)(cid:13) S ( t − r ) AF ( X ( r )) (cid:13)(cid:13)(cid:13) dr + (cid:13)(cid:13)(cid:13) Z s ( S ( t − r ) − S ( s − r )) G ( X ( r )) dW ( r ) (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) Z ts S ( t − r ) G ( X ( r )) dW ( r ) (cid:13)(cid:13)(cid:13) . By taking p th moment, using (22) and the smoothing effect of S ( t ), we get E h(cid:13)(cid:13) ( S ( t ) − S ( s )) X (cid:13)(cid:13) p i ≤ C ( T, X , p, γ )( t − s ) γp , E h Z s (cid:13)(cid:13) ( S ( t − r ) − S ( s − r )) AF ( X ( r )) (cid:13)(cid:13) p dr i ≤ C ( T, p ) E h(cid:16) Z s ( s − r ) − − γ (cid:13)(cid:13) ( S ( t − s ) − I ) A − γ (cid:13)(cid:13)(cid:13)(cid:13) F ( X ( r )) (cid:13)(cid:13) dr (cid:17) p i ≤ C ( T, X , p, γ )( t − s ) γp , and E h(cid:16) Z ts (cid:13)(cid:13)(cid:13) S ( t − r ) AF ( X ( r )) (cid:13)(cid:13)(cid:13) dr (cid:17) p i ≤ C ( T, p ) E h(cid:16) Z ts ( t − r ) − k F ( X ( r )) k dr (cid:17) p i ≤ C ( T, X , p )( t − s ) p . The Burkholder inequality and (22) yield that E h(cid:13)(cid:13)(cid:13) Z s ( S ( t − r ) − S ( s − r )) G ( X ( r )) dW ( r ) (cid:13)(cid:13)(cid:13) p i ≤ C ( T, p ) E h(cid:16) Z s X i ∈ N + k S ( s − r )( S ( t − s ) − I ) G ( X ( r )) e i k ds (cid:17) p i ≤ C ( T, X , p, γ )( t − s ) γp . and E h(cid:13)(cid:13)(cid:13) Z ts S ( t − r ) G ( X ( r )) dW ( r ) (cid:13)(cid:13)(cid:13) p i ≤ C ( T, p ) E h(cid:16) Z ts X i ∈ N + k S ( t − r ) G ( X ( r )) e i k ds (cid:17) p i ≤ C ( T, p, X )( t − s ) p . Combining all the above estimates, we complete the proof.
This manuscript is for review purposes only. Remark
Under the same condition as in Proposition 5.3, the solution of thespectral Galerkin method X N satisfies k X N ( t ) − X N ( s ) k L p (Ω; H ) ≤ C ( X , T, p )( t − s ) γ , where C ( X , T, p ) > and ≤ s ≤ t ≤ T . As a result of Proposition 5.1, we have the following strong convergence rate ofthe spectral Galerkin method.
Corollary
Let X ∈ H γ , γ ∈ (0 , ) , T > , p ≥ and sup N ∈ N + k X N k E ≤ C ( X ) . Then for τ ∈ (0 , γ ) , there exists C ( X , T, p ) > such that (cid:13)(cid:13) X N − X (cid:13)(cid:13) L p (Ω; C ([0 ,T ]; H ) ≤ C ( X , T, p ) λ − τ N . (24)As a consequence of the strong convergence of the spectral Galerkin method, thefollowing exponential integrability property of the mild solution holds. We wouldlike to mention that the exponential integrability property has many applications innon-global SPDE and its numerical approximation(see e.g. [6, 9, 11]). Corollary
Let X ∈ H . There exist β > , c > such that for t ∈ [0 , T ] , E h exp (cid:16) e − βt k X ( t ) k H − + c Z t e − βs k X ( s ) k L ds + c Z t e − βs k∇ X ( s ) k ds (cid:17)i ≤ C ( X , T ) . Proof.
From the Gagliardo–Nirenberg and Young inequalities, it follows that Z t k X N − X k L ds ≤ C Z t k∇ ( X N − X ) k ds + C Z t k X N − X k ds. The similar arguments in Proposition 4.1 yield that for t ∈ [0 , T ],lim N →∞ k X N − X k L (Ω; L ([0 ,t ]; H )) = 0 , which together with the strong convergence of X N in C ([0 , t ]; L (Ω; H )) implies thatlim N →∞ k X N − X k L (Ω; L ([0 ,t ]; L )) = 0 . Thus by Fatou’s lemma, it suffices to showthe uniform boundedness of the exponential moment for X N .Denote µ ( x ) = − A x − AP N F ( x ) and σ ( x ) = P N G ( x ) I H and U ( x ) = k x k H − ,where x ∈ P N ( H ). By direct calculations and the interpolation inequality, we get h DU ( x ) , µ ( x ) i + 12 tr[ D U ( x ) σ ( x ) σ ∗ ( x )] + 12 k σ ( x ) ∗ DU ( x ) k = h x, − A x + AF ( x ) i H − + 12 X i ∈ N + k P N ( G ( x ) e i ) k H − + 12 X i ∈ N + h x, G ( x ) e i i H − ≤ − (1 − ǫ ) k∇ x k − (4 c − ǫ ) k x k L + ǫ k x k H − + C ( ǫ ) . Using the exponential integrability lemma in [8] and taking β = ǫ , we have E h exp (cid:16) e − βt k X N ( t ) k H − + (4 c − ǫ ) Z t e − βs k X N ( s ) k L ds + (1 − ǫ ) Z t e − βs k∇ X N ( s ) k ds (cid:17)i ≤ C ( X , T, ǫ ) , which, combined with Fatou’s lemma, completes the proof. This manuscript is for review purposes only. JIANBO CUI AND JIALIN HONG
6. Conclusion.
In this paper, we introduce a new approach to studying theglobal existence and regularity estimate of the solution process for stochastic Cahn–Hilliard equation driven by multiplicative space-time white noise. Compared to theexisting work, we use the spectral Galerkin method, instead of the cut-off equation,to approximate the original equation. Then by proving the well-posedness and apriori estimates of the approximated equation, we show that the solution { X N } N ∈ N + possesses the sharp strong convergence rate and thus is a Cauchy sequence in certainBanach space. As a consequence, the limit process of X N is shown to be the globalsolution of stochastic Cahn–Hilliard equation and to possess the optimal regularityestimates.
7. Acknowledgement.
The authors are very grateful to Professor YaozhongHu(University of Alberta) for his helpful discussions and suggestions.
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