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SMOOTH SOLUTIONS OF NON-LINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS XICHENG ZHANG Department of Mathematics, Huazhong University of Science and Technology Wuhan, Hubei 430074, P.R.China 8 Email: [email protected] 0 0 2 Abstract. In this paper, we study the regularities of solutions of nonlinear stochas- n tic partial differential equations in the framework of Hilbert scales. Then we apply a our general result to several typical nonlinear SPDEs such as stochastic Burgers and J Ginzburg-Landau’s equations on the real line, stochastic 2D Navier-Stokes equations in 5 the whole space and a stochastic tamed 3D Navier-Stokes equation in the whole space, 2 and obtain the existence of their respectively smooth solutions. ] R P . h Contents t a m 1. Introduction 1 [ 2. General Settings and Main Result 3 1 3. Proof of Main Theorem 5 v 3.1. Regularized Stochastic Differential Equations 5 3 3.2. Convergence of uǫ(t) 8 8 8 4. Strong Solutions of Semilinear SPDEs in Euclidean space 12 3 5. Stochastic Burgers and Ginzburg-Landau’s equations on the real line 15 . 1 6. Stochastic tamed 3D Navier-Stokes equation in R3 18 0 7. Stochastic 2D Navier-Stokes equation in R2 23 8 0 References 26 : v i X r a 1. Introduction Consider the following stochastic Burgers and Ginzburg-Landau equation on the real line: du(t,x) = ν∂2u(t,x)+c ∂ u(t,x)2 +c u(t,x) c u(t,x)3 dt x 0 · x 1 · − 2 ·  h+ σ (t,x,u(t,x))dWk(t), i (1.1) k k  u(0,x) = u (x), x R, 0 P ∈ wherec ,c R and ν,c > 0, Wk(t),k N is a sequence of independent Brownian 0 1 2 motions,the∈coefficients σ ,k {N satisfy∈some}smoothnessconditions. Uptonow, there k { ∈ } are many papers devoted to the studies of stochastic Burgers’ equation and stochastic Ginzburg-Landau’s equation (cf. [1, 6, 19, 7] and references therein). In [6], using heat kernel estimates, Gyöngy and Nualart proved the existence and uniqueness of L2(R)- solutiontostochasticBurgers’equationontherealline. Bysolving aninfinitedimensional Kolmogorov’s equation, Röckner and Sobol [19] developed a new method to solve the generalized stochastic Burgers and reaction diffusion equations. More recently, Kim [7] 1 studied the stochastic Burgers type equation with the first order term having polynomial growth, as well as the existence of associated invariant measures. Since allof these works areconcerned with stochastic Burgers equation driven by space- time white noises, they had to consider weak or mild solutions rather than strong or classical solutions. Anaturalquestionisthat: doesthereexist smoothsolutionorclassical solution in x to the equation (1.1) if allthe datas aresmooth? Of course, for this question, we can only consider the equation (1.1) driven by time white and space correlated noises. We remark that for the deterministic Burgers equation, i.e., σ = c = c = 0 and k 1 2 c = 1,ν > 0, it is well known that there exists a unique smooth solution if the initial 0 data is smooth (cf. [8]). Let us also consider the following stochastic 2D Navier-Stokes equation in R2: ∂ u = ν∆u u ∂ u u ∂ u ∂ p+f t 1 1 − 1 x1 1 − 2 x2 1 − x1 1 ∂ u = ν∆u u ∂ u u ∂ u ∂ p+f (1.2)  t 2 2 − 1 x1 2 − 2 x2 2 − x2 2 ∂ u +∂ u = 0,  x1 1 x2 2 where ν is the viscosity constant, u(t,x) = (u ,u ) is the velocity field, p is the pressure  1 2 function, and f = (f ,f ) is the white in time and additive stochastic forcing. In [16], 1 2 Mikulevicius and Rozovskii studied the existence of martingale solutions for any dimensional stochastic Navier-Stokes equations in the whole space. In particular, they obtained the existence of a unique weak solution for the above two dimensional equation. In the periodic boundary case, using Galerkin’s approximation and Fourier’s transformation, Mattingly [15] proved the spatial analyticity for the solution to the above stochastically forced 2D Navier-Stokes equation. However, using his method, it seems to be hard to consider the multiplicative noise force. As for the stochastic 3D Navier-Stokes equation, Röckner and the author [20, 21] recently studied the following tamed or modified scheme in the whole space R3 or periodic boundary case: ∂ u = ν∆u 3 u ∂ u ∂ p t j j − i=1 i xi j − xj g 3 u 2 u +f , j = 1,2,3, (1.3)  − N Pi=1| i| j j  3 ∂ u =(cid:16) 0, (cid:17) i=1 xi i P where the tamingfunPction gN : R+ 7→ R+ is smooth and satisfies that g (r) = 0 on r 6 N and g (r) = (r N)/ν on r > N +2. N N − In [21], we proved the existence of a unique strong solution and the ergodicity of associated invariant measure in the case of periodic boundary conditions. For the existence, the method is mainly based on the Galerkin approximation, and the smooth solution of Eq.(1.3) is not obtained. Our main purpose in this paper is to present a unified settings for proving the existence of smooth solutions to the above three typical nonlinear stochastic partial differential equations. That is, we shall consider an abstract semilinear stochastic evolution equation in the scope of Hilbert scales determined by a sectorial operator. Here, the analytic semigroup generated by the sectorial operator plays a mollifying role, and will be used to construct an approximating sequence of regularized stochastic ordinary differential equations in Hilbert spaces. After obtaining some uniform estimates of the approximating solutions in the spaces of Hilbert scale, we can prove that the solutions of approximating equations strongly converge to a smooth solution. Our approach is much inspired by the energy method used in the deterministic case (cf. [14]), and is different from Galerkin’s approximation and semigroup methods which were extensively used in the well known 2 studies of stochastic partial differential equations(abbrev. SPDEs) (cf. [11, 4], etc.). We remark that the regularity of solution will be decreasing when we use the semigroup method to deal with SPDEs (cf. [2, Sections 5,8]). The main advantage of our method is that we can obtain better regularities unlike the semigroup method. In [26], using the semigroup method and nonlinear interpolations, we have already proved the existence of smooth solutions to a large class of semilinear SPDEs when the coefficients are smooth and have all bounded derivatives. However, the result in [26] can not be applied to the above mentioned equations. It should be emphasized that the existence of smooth solutions for nonlinear partial differential equations, fox exam- ple, Navier-Stokes equations, usually depends on the spatial dimension. Thus, it is not expected to use our general result(see Theorem 2.2 below) to treat high dimensional nonlinear SPDEs for obtaining smooth solutions. Nevertheless, we may still apply our general result to achieving the existence of strong solutions for a class of semilinear SPDEs with Lipschitz nonlinear coefficients in Euclidean space (cf. [9, 10, 16]). We also want to say that although our main attention concentrates on the above three types nonlinear SPDEs, our result can also be applied to dealing with the stochastic Kuramoto-Sivashinsky equation and stochastic Cahn-Hilliard equation (cf. [23]), as well as the stochastic partial differential equation in the abstract Wiener space (cf. [25]), which are not discussed here. This paper is organized as follows: in Section 2, we shall give the general framework and state our main result. In Section 3, we devote to the proof of our main result. In Section 4, we investigate a class of semilinear SPDEs in the whole space and in bounded smooth domains of Euclidean space, and obtain the existence of unique strong solutions. In Section 5, we study stochastic Burgers and Ginzburg-Landau’s equations on the real line, and get the existence of smooth solutions. In Section 6, we prove the existence of smoothsolutions to stochastic tamed3D Navier-Stokes equations. InSection 7, stochastic 2D Navier-Stokes equation in the whole space and with multiplicative noise is considered. Convention: The letter C with or without subscripts will denote a positive constant, which is unimportant and may change from one line to another line. 2. General Settings and Main Result Let (H, H) be a separable Hilbert space, a symmetric and non-positive sectorial operator inkH·kthat generates a symmetric analyLtic semigroup ( ) in H (cf. [18]). For ǫ ǫ>0 α > 0, we define the Sobolev space Hα by T Hα := D((I )α/2) −L together with the norm u α := (I )α/2u H. k k k −L k The inner product in Hα is denoted by , . The dual space of Hα is denoted by H−α h· ·iα with the norm u −α := (I )−α/2u H. k k k −L k Then (Hα)α∈R forms a Hilbert scale (cf. [12, 22]), i.e.: (i) for any α < β, Hβ Hα; (ii) for any α < γ < β ⊂and u Hβ, ∈ β−γ γ−α u 6 C u β−α u β−α. (2.1) k kγ k kα ·k kβ Set H∞ := m∈NHm. Then (cf. [18]) ∩ Proposition 2.1. For all integer m Z, we have ∈ 3 (i) H∞ is dense in Hm, and for every ǫ > 0 and u Hm, u H∞; ǫ (ii) for every ǫ > 0 and u Hm ∈ T ∈ ∈ (I )m/2 u = (I )m/2u; ǫ ǫ −L T T −L (iii) for every ǫ > 0 and u Hm, k = 1,2, ∈ ··· u u 6 C ǫk/2 u ; ǫ m m m+k kT − k k k (iv) for every ǫ > 0 and u Hm, k = 0,1,2, ∈ ··· C u 6 mk u . kTǫ km+k ǫk/2 k km Let l2 be the usual Hilbert space of square summable sequences of real numbers. Let (Ω, ,( ) ,P) be a complete filtration probability space. A family of independent one t t>0 dimFensiFonal -adapted Brownian motions Wk(t);t > 0,k = 1,2, on (Ω, ,P) are t F { ···} F given. Then W(t),t > 0 can be regarded as a cylindrical Brownian motion in l2 (cf. { } [4]). Consider the following type stochastic evolution equation du(t) = [ u(t)+F(t,u(t))]dt+ B (t,u(t))dWk(t), u(0) = u , (2.2) k 0 L k X where the stochastic integral is understood as Itô’s integral, and for some N N the ∈ coefficients F(t,ω,u) : R Ω HN H−1, + × × → B(t,ω,u) : R Ω H0 H0 l2 + × × → ⊗ are two measurable functions, and for every t > 0 and u HN, ∈ F(t, ,u) /B(H−1), B(t, ,u) /B(H0 l2). t t · ∈ F · ∈ F ⊗ We also require that F(t,ω,u) H0 for u HN+1 and B (t,ω,u) Hm for any m N k and u Hm+1. ∈ ∈ ∈ ∈ ∈ We make the following assumptions on F and B: (H1 ) There exist q ,q > 1 and constants λ ,λ ,λ ,λ ,λ > 0 such that for all (t,ω) N 1 2 0 1 2 3 4 R Ω and u,v HN ∈ + × ∈ F(t,ω,u) F(t,ω,v) 6 λ ( u q1 + v q1 +1) u v , (2.3) k − k−1 0 k kN k kN ·k − k0 B (t,ω,u) B (t,ω,v) 2 6 λ u v 2, (2.4) k k − k k0 1k − k0 k X and for u HN+1 ∈ 1 u,F(t,ω,u) 6 u 2 +λ ( u 2 +1), (2.5) h i0 2k k1 2 k k0 F(t,ω,u) 6 λ ( u + u q2 +1), (2.6) k k0 3 k kN+1 k kN B (t,ω,u) 2 6 λ ( u 2+1). (2.7) k k k0 4 k k0 k X (H2 ) For some integer > N, and each m = 1, , , any δ (0,1), there exist N α ,β > 1 and cNonstants λ ,λ > 0 such··t·haNt for all u∈ H∞ and (t,ω) m m 1m 2m R Ω ∈ ∈ + × u,F(t,ω,u) = (I )m+21u,(I )−21F(t,ω,u) h im h −L −L i0 1 6 u 2 +λ ( u αm +1), (2.8) 2k km+1 1m k km−1 4 B (t,ω,u) 2 6 δ u 2 +λ ( u βm +1). (2.9) k k km k km+1 2m k km−1 k X Our main result is that Theorem 2.2. Under (H1 ) and (H2 ) with > N > 1, for any u HN, there exists N N 0 a unique process u(t) HN such that N ∈ ∈ (i) The process t u(t) H0 is -adapted and continuous, and for any T > 0 and t 7→ ∈ F p > 2 T E sup u(s) p + E u(s) 2 ds < + . k kN k kN+1 ∞ s∈[0,T] ! Z0 (ii) u(t) satisfies the following equation in H0: for all t > 0 t t u(t) = u + [ u(s)+F(s,u(s))]ds+ B (s,u(s))dWk(s), P a.s.. 0 k L − Z0 k Z0 X Remark 2.3. By (2.6), (2.7) and (i), one knows that all the integrals appearing in (ii) are well defined. Moreover, if for some C > 0, p > 1 and any u HN+1 ∈ F(s,u) 6 C( u + u p +1), k kN−1 k kN+1 k kN then we can find an HN-valued continuous version of u (cf. [22]). Remark 2.4. The solution u(t) also satisfies the following integral equation written in terms of the semigroup : t T t t u(t) = u + F(s,u(s))ds+ B (s,u(s))dWk(s). t 0 t−s t−s k T T T Z0 k Z0 X Using this representation, we can further prove the Hölder continuity of u(t) in t (cf. [25]). 3. Proof of Main Theorem 3.1. Regularized Stochastic Differential Equations. For any m = 0, , , let us consider the following regularized stochastic differential equation in Hm: ··· N duǫ(t) = Aǫ(t,uǫ(t))dt+ Bǫ(t,uǫ(t))dWk(t), uǫ(0) = u , (3.1) k 0 k X where the regularized operators are given by: Aǫ(t,ω,u) := u+ F(t,ω, u) ǫ ǫ ǫ ǫ T LT T T Bǫ(t,ω,u) := B (t,ω, u). k Tǫ k Tǫ The following two lemmas are direct from (H1 ) and (H2 ). We omit the proof. N N Lemma 3.1. There exists a constant C > 0 such that for any ǫ > 0 and all (t,ω) R Ω, u H0 ∈ + × ∈ 1 u,Aǫ(t,ω,u) 6 u 2 +C( u 2+1), h i0 −2kTǫ k1 k k0 Bǫ(t,ω,u) 2 6 C( u 2 +1). k k k0 k k0 k X 5 Lemma 3.2. Foranym = 1, , and δ (0,1), there existtwo constants C ,C > 0 m m,δ such that for any ǫ > 0 and a·ll··(t,Nω) R ∈ Ω and u Hm, we have + ∈ × ∈ 1 u,Aǫ(t,ω,u) 6 u 2 +C ( u αm +1) h im −2kTǫ km+1 m k km−1 Bǫ(t,ω,u) 2 6 δ u 2 +C ( u βm +1), k k km kTǫ km+1 m,δ k km−1 k X where α and β are same as in (H2 ). m m N We also have Lemma 3.3. For any m = 0, , , the functions Aǫ and Bǫ are locally Lipschitz continuous in Hm with respect to u··.·MNore precisely, for any R > 0 there are C ,C′ > 0 R,ǫ R,ǫ such that for any (t,ω) R Ω and u,v Hm with u , v 6 R + m m ∈ × ∈ k k k k Aǫ(t,ω,u) Aǫ(t,ω,v) 6 C u v m R,ǫ m k − k k − k Bǫ(t,ω,u) Bǫ(t,ω,v) 2 6 C′ u v 2 . k k − k km R,ǫk − km k X Proof. By (iv) of Proposition 2.1, we have ( u) ( v) = 2(u v) 6 C u v , kTǫL Tǫ −TǫL Tǫ km kLTǫ − km ǫk − km and by (H1 ) N F(t, u) F(t, v) 2 + B (t, u) B (t, v) 2 kTǫ Tǫ −Tǫ Tǫ km kTǫ k Tǫ −Tǫ k Tǫ km k X 6 C F(t, u) F(t, v) 2 +C B (t, u) B (t, v) 2 ǫk Tǫ − Tǫ k−1 ǫ k k Tǫ − k Tǫ k0 k X 6 C u v 2 6 C u v 2 . R,ǫkTǫ −Tǫ k0 R,ǫk − km The proof is complete. (cid:3) We now prove the following key estimate about the solution of regularized stochastic differential equation (3.1). Theorem 3.4. For any u HN, there exists a unique continuous -adapted solution 0 t uǫ to Eq.(3.1) in HN such th∈at for any p > 1 and T > 0 F T sup E sup uǫ(t) 2p + sup E uǫ(s) 2 ds 6 C . (3.2) k kN kTǫ kN+1 p,T ǫ∈(0,1) t∈[0,T] ! ǫ∈(0,1)Z0 Proof. First of all, by Lemma 3.3, there exists a unique continuous -adapted local t solution uǫ(t) in Hm for any m = 0, , . We now use induction methFod to prove that ··· N uǫ(t) is non-explosive in Hm and for any p > 1 and T > 0 (P ) : m sup E sup uǫ(t) 2p 6 C . (cid:26) ǫ∈(0,1) t∈[0,T]k km m,p,T By the standard stopping t(cid:0)imes technique, it(cid:1)suffices to prove the estimate in (Pm). In the following, we fix T > 0. By Itô’s formula, we have for any p > 1 uǫ(t) 2p = u 2p +I (t)+I (t)+I (t)+I (t), (3.3) k km k 0km m1 m2 m3 m4 where t I (t) := 2p uǫ(s) 2(p−1) uǫ(s),Aǫ(uǫ(s)) ds m1 k km h im Z0 6 ∞ t I (t) := 2p uǫ(s) 2(p−1) uǫ(s),Bǫ(s,uǫ(s)) dWk m2 k km h k im s k=1Z0 X ∞ t I (t) := p uǫ(s) 2(p−1) Bǫ(s,uǫ(s)) 2 ds m3 k km k k km k=1Z0 X ∞ t I (t) := 2p(p 1) uǫ(s) 2(p−2) uǫ(s),Bǫ(s,uǫ(s)) 2ds. m4 − k km |h k im| k=1Z0 X Let us first look at the case of m = 0. By Lemma 3.1 and Young’s inequality, we have t I (t)+I (t)+I (t) 6 C ( uǫ(s) 2p +1)ds. (3.4) 01 03 04 p k k0 Z0 Taking expectations for (3.3) gives that t E uǫ(t) 2p 6 u 2p +C (E uǫ(s) 2p +1)ds. k k0 k 0k0 p k k0 Z0 By Gronwall’s inequality, we obtain for any p > 1 sup E uǫ(t) 2p 6 C ( u 2p +1). (3.5) k k0 p,T k 0k0 t∈[0,T] On the other hand, by BDG’s inequality and Lemma 3.1, we have T 1/2 E sup I (s) 6 C E uǫ(s) 4(p−1) uǫ(s),Bǫ(s,uǫ(s)) 2ds | 02 | p k k0 kh · i0kl2 s∈[0,T] ! (cid:18)Z0 (cid:19) 1/2 T 6 C E uǫ(s) 4p−2 Bǫ(s,uǫ(s)) 2 ds p k k0 · k k k0 Z0 (cid:16)Xk (cid:17) ! T 1/2 6 C E ( uǫ(s) 4p +1)ds p k k0 (cid:18)Z0 (cid:19) T 1/2 6 C (E uǫ(s) 4p +1)ds p k k0 (cid:18)Z0 (cid:19) 6 C ( u 2p +1). p,T k 0k0 Thus, from (3.3)-(3.5), one knows that (P ) holds. 0 Suppose now that (P ) holds. By Lemma 3.2 and Young’s inequality, we have m−1 t I (t) 6 p uǫ(s) 2(p−1) uǫ(s) 2 m1 −k km kTǫ km+1 Z0 h +C uǫ(s) 2(p−1) ( uǫ(s) αm +1) ds mk km · k km−1 t i t 6 p uǫ(s) 2(p−1) uǫ(s) 2 ds+C uǫ(s) 2pds − k km kTǫ km+1 m,p k km Z0 Z0 t +C ( uǫ(s) p·αm +1)ds, (3.6) m,p k km−1 Z0 and for any δ (0,1) ∈ t I (t)+I (t) 6 p(2p 1) uǫ(s) 2(p−1) Bǫ(s,uǫ(s)) 2 ds m3 m4 − k km · k k km Z0 7 (cid:16)Xk (cid:17) t 6 p(2p 1)δ uǫ(s) 2(p−1) uǫ(s) 2 ds − k km kTǫ km+1 Z0 t t +C uǫ(s) 2pds+C ( uǫ(s) p·βm +1)ds. (3.7) m,p k km m,p k km−1 Z0 Z0 Choosing δ = 1 and taking expectations for (3.3) gives that 2(2p−1) p t E uǫ(t) 2p + E uǫ(s) 2(p−1) uǫ(s) 2 ds k km 2 k km ·kTǫ km+1 Z0 (cid:16) (cid:17) t t 6 u 2p +C E uǫ(s) 2pds+C E( uǫ(s) pm +1)ds, k 0km m,p k km m,p k km−1 Z0 Z0 where p := p (α β ). m m m By Gronwall·’s ineq∨uality again and (P ), we get for any p > 1 m−1 T sup E uǫ(t) 2p + E uǫ(s) 2(p−1) uǫ(s) 2 ds 6 C . (3.8) k km k km kTǫ km+1 m,p,T t∈[0,T] Z0 (cid:16) (cid:17) Furthermore, by BDG’s inequality and (3.8), we have T 1/2 E sup I (s) 6 C E uǫ(s) 4(p−1) uǫ(s),Bǫ(s,uǫ(s)) 2 ds | m2 | p k km ·kh · imkl2 s∈[0,T] ! (cid:18)Z0 (cid:19) 1/2 T 6 C E uǫ(s) 4p−2 Bǫ(s,uǫ(s)) 2 ds p k km k k km Z0 (cid:16)Xk (cid:17) ! T 6 C E sup uǫ(s) p uǫ(s) 2(p−1) p k km · k km × s∈[0,T] "Z0 1/2 δ uǫ(s) 2 +C ( uǫ(s) βm +1) ds kTǫ km+1 m,δ k km−1 # ! (cid:16) (cid:17) T 6 C δ E uǫ(s) 2(p−1) uǫ(s) 2 ds p k km kTǫ km+1 Z0 (cid:16) (cid:17) T +C E uǫ(s) 2(p−1)( uǫ(s) βm +1) ds m,p k km k km−1 Z0 (cid:16) (cid:17) 1 + E sup uǫ(t) 2p 2 k km t∈[0,T] ! 1 6 C + E sup uǫ(t) 2p , m,p,T 2 k km t∈[0,T] ! which together with (3.3) and (3.6)-(3.8) yields (P ). The proof is complete. (cid:3) m 3.2. Convergence of uǫ(t). Lemma 3.5. For any R > 0, there exists a constant C > 0 such that for any 0 < ǫ′ < R ǫ < 1, (t,ω) R Ω and u,v HN+1 with u , v 6 R + N N ∈ × ∈ k k k k hu−v,Aǫ(t,ω,u)−Aǫ′(t,ω,v)i0 6 CR ·√ǫ·(1+kTǫ′vkN+1)+CR ·ku−vk20, kBkǫ(t,ω,u)−Bkǫ′(t,ω,v)k20 6 CR ·√ǫ·(1+kTǫ′vk22)+CR ·ku−vk20. k X 8 Proof. We only prove the first estimate. The second one is similar. Above of all, by (ii) and (iii) of Proposition 2.1, we have hu−v,TǫL(Tǫu)−Tǫ′L(Tǫ′v)i0 = hTǫ(u−v),LTǫ(u−v)i0 +hu−v,(T2ǫ −T2ǫ′)Lvi0 6 −kTǫ(u−v)k21 +kTǫ(u−v)k20 +CR ·k(T2ǫ −T2ǫ′)vk1 6 −kTǫ(u−v)k21 +Cku−vk20 +CR ·√ǫ·kTǫ′vk2. (3.9) Secondly, we decompose the term involving F in Aǫ as: hu−v,TǫF(t,Tǫu)−Tǫ′F(t,Tǫ′v)i0 = hTǫ(u−v),F(t,Tǫu)−F(t,Tǫ′v)i0 +h(Tǫ −Tǫ′)(u−v),F(t,Tǫ′v)i0 =: I +I . 1 2 By (2.3) and (iii) of Proposition 2.1, we have for I 1 1 I1 6 4kTǫ(u−v)k21 +kF(t,Tǫu)−F(t,Tǫ′v)k2−1 1 6 4kTǫ(u−v)k21 +CR ·kTǫu−Tǫ′vk20 1 6 (u v) 2 +C √ǫ+C u v 2, 4kTǫ − k1 R · R ·k − k0 and for I , by (2.6) 2 I2 6 ( ǫ ǫ′)(u v) 0 F(t, ǫ′v) 0 k T −T − k ·k T k 6 CR √ǫ ( ǫ′v N+1 +1). · · kT k Combining the above calculations yields the first one. (cid:3) We now prove that Lemma 3.6. For any T > 0, we have lim E sup uǫ(t) uǫ′(t) 2 = 0. ǫ,ǫ′↓0 t∈[0,T]k − k0! Proof. For any R > 0 and 1 > ǫ > ǫ′ > 0, define the stopping time τǫ,ǫ′ := inf t > 0 : uǫ(t) uǫ′(t) > R . R { k kN ∧k kN } Then, by Theorem 3.4 we have E sup ( uǫ(t) 2 uǫ′(t) 2 ) P(τǫ,ǫ′ < T) 6 t∈[0,T] k kN ∧k kN 6 CT. (3.10) R (cid:16) R2 (cid:17) R2 Set v(t) := uǫ(t) uǫ′(t). − By Itô’s formula, we have v(t) 2 = J (t)+J (t)+J (t), (3.11) k k0 1 2 3 where t J (t) := 2 v(s),Aǫ(s,uǫ(s)) Aǫ′(s,uǫ′(s)) ds 1 h − i0 Z0 9 t J (t) := Bǫ(s,uǫ(s)) Bǫ′(s,uǫ′(s)) 2ds 2 k k − k k0 k Z0 X t J (t) := 2 v(s),Bǫ(s,uǫ(s)) Bǫ′(s,uǫ′(s)) dWk. 3 h k − k i0 s k Z0 X By Lemma 3.5, we have J (t τǫ,ǫ′)+J (t τǫ,ǫ′) 1 ∧ R 2 ∧ R ′ T t∧τǫ,ǫ 6 CR ·√ǫ· 1+ kTǫ′uǫ′(s)k2N+1ds +CR R kv(s)k20ds (cid:18) Z0 (cid:19) Z0 T t 6 CR ·√ǫ· 1+ kTǫ′uǫ′(s)k2N+1ds +CR kv(s∧τRǫ,ǫ′)k20ds. (3.12) (cid:18) Z0 (cid:19) Z0 Hence, by Theorem 3.4 and taking expectations for (3.11), we obtain t E v(t τǫ,ǫ′) 2 6 C √ǫ+C E v(s τǫ,ǫ′) 2ds. k ∧ R k0 R,T · R k ∧ R k0 Z0 By Gronwall’s inequality, we get sup E v(t τǫ,ǫ′) 2 6 C √ǫ. (3.13) k ∧ R k0 R,T · t∈[0,T] On the other hand, setting (s,ǫ,ǫ′) := Bǫ(s,uǫ(s)) Bǫ′(s,uǫ′(s)) 2, B k k − k k0 k X by BDG’s inequality and Young’s inequality, we have E sup J (s) 3 s∈[0,T∧τǫ,ǫ′]| |! R T∧τǫ,ǫ′ 1/2 R 6 CE v(s) 2 (s,ǫ,ǫ′)ds k k0 ·B Z0 ! ′ 1 T∧τRǫ,ǫ 6 E sup v(s) 2 +CE (s,ǫ,ǫ′)ds . 2 s∈[0,T∧τǫ,ǫ′]k k0! Z0 B ! R Thus, by (3.11)-(3.13) and Lemma 3.5 we obtain E sup v(s) 2 6 C √ǫ. s∈[0,T∧τǫ,ǫ′]k k0! R,T · R Therefore, by Theorem 3.4 and (3.10) we get E sup v(s) 2 k k0 s∈[0,T] ! = E sup kv(s)k20 ·1{τǫ,ǫ′>T} +E sup kv(s)k20·1{τǫ,ǫ′<T} s∈[0,T] R ! s∈[0,T] R ! 10