Stochastic logarithmic Schrodinger equations: energy regularized approach
aa r X i v : . [ m a t h . P R ] M a r Stochastic logarithmic Schr¨odinger equations: energy regularizedapproach
Jianbo Cui a , Liying Sun b, ∗ a School of Mathematics, Georgia Tech, Atlanta, GA 30332, USA b
1. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing,100190, China2. School of Mathematical Science, University of Chinese Academy of Sciences, Beijing, 100049, China
Abstract
In this paper, we prove the global existence and uniqueness of the solution of the stochasticlogarithmic Schr¨odinger (SlogS) equation driven by additive noise or multiplicative noise. Thekey ingredient lies on the regularized stochastic logarithmic Schr¨odinger (RSlogS) equation withregularized energy and the strong convergence analysis of the solutions of (RSlogS) equations.In addition, temporal H¨older regularity estimates and uniform estimates in energy space H ( O )and weighted Sobolev space L α ( O ) of the solutions for both SlogS equation and RSlogS equationare also obtained. Keywords: stochastic Schr¨odinger equation, logarithmic nonlinearity, energy regularizedapproximation, strong convergence
1. Introduction
The deterministic logarithmic Schr¨odinger equation has wide applications in quantum me-chanics, quantum optics, nuclear physics, transport and diffusion phenomena, open quantumsystem, Bose–Einstein condensations and so on (see e.g. [1, 6, 14, 16, 19, 20]). It takes the formof ∂ t u ( t, x ) = i ∆ u ( t, x ) + i λu ( t, x ) log( | u ( t, x ) | ) + i V ( t, x, | u | ) u ( t, x ) , x ∈ R d , t > ,u (0 , x ) = u ( x ) , x ∈ R d , where ∆ is the Laplacian operator on O ⊂ R d with O being either R d or a bounded domainwith homogeneous Dirichlet or periodic boundary condition, t is time, x is spatial coordinate, λ ∈ R / { } characterizes the force of nonlinear interaction, and V is a real-valued function. Whileretaining many of the known features of the linear Schr¨odinger equation, Bialynicki–Birula andMycielski show that only such a logarithmic nonlinearity satisfies the condition of separability ofnoninteracting systems (see [6]). The logarithmic nonlinearity makes the logarithmic Schr¨odingerequation unique among nonlinear wave equations. For instance, the longtime dynamics of thelogarithmic Schr¨odinger equation is essentially different from the Schr¨odinger equation. There is ⋆ This work was funded by National Natural Science Foundation of China (No. 91630312, No. 91530118,No.11021101 and No. 11290142). ∗ Corresponding author.
Email addresses: [email protected] (Jianbo Cui), [email protected] (Liying Sun)
Preprint submitted to Elsevier March 2, 2021 faster dispersive phenomenon when λ < λ > du ( t ) = i ∆ u ( t ) dt + i λu ( t ) log( | u ( t ) | ) dt + e g ( u ) ⋆ dW ( t ) , t > u (0) = u , where W ( t ) = P k ∈ N + Q e k β k ( t ), { e k } k ∈ N + is an orthonormal basis of L ( O ; C ) with { β k } k ∈ N + being a sequence of independent Brownian motions on a probability space (Ω , F , ( F t ) t ≥ , P ) . Here e g is a continuous function and e g ( u ) ⋆ dW ( t ) is defined by e g ( u ) ⋆ dW ( t ) = − X k ∈ N + | Q e k | (cid:16) | g ( | u | ) | u (cid:17) dt − i X k ∈ N + g ( | u | ) g ′ ( | u | ) | u | uIm ( Q e k ) Q e k dt + i g ( | u | ) udW ( t )if e g ( x ) = i g ( | x | ) x , and by e g ( u ) ⋆ dW ( t ) = dW ( t )if e g = 1. We would like to remark that when W is L ( O ; R )-valued and e g ( x ) = i g ( | x | ) x , e g ( u ) ⋆ dW ( t ) is just the classical Stratonovich integral.The SlogS equation (1) could be derived from the deterministic model by using Nelson’smechanics [18]. Applying the Madlung transformation u ( t, x ) = p ρ ( t, x ) e i S ( t,x ) , [17] obtains afluid expression of the solution as follows, ∂ t S ( t, x ) = −|∇ S ( t, x ) | − δIδρ ( ρ ( t, x ))+ λ log( ρ )+ V ( t, x, ρ ( t, x )) ,∂ t ρ ( t, x ) = − div ( ρ ( t, x ) ∇ S ( t, x )) , S (0 , x ) = S ( x ) , ρ (0) = ρ , where I ( ρ ) = R O |∇ log( ρ ) | ρdx is the Fisher information. If V is random and fluctuates rapidly,the term i V u can be approximated by some multiplicative Gaussian noise e g ( u ) ˙ W , which playsan important role in the theory of measurements continuous in time in open quantum systems(see e.g. [5]). Then we could use the inverse of Madlung transformation and formally obtain thestochastic logarithmic Schr¨odinger equation ∂ t u ( t, x ) = i ∆ u ( t, x ) + i λu ( t, x ) log( | u ( t, x ) | ) + e g ( u ( t, x )) ˙ W ( t, x ) , x ∈ O , t > u (0 , x ) = u ( x ) , x ∈ O . The main assumption on W and e g is stated as follows. Assumption 1.
The diffusion operator is the Nemystkii operator of e g . Wiener process W and e g satisfies one of the following condition, Case 1. { W ( t ) } t ≥ is L ( O ; C ) -valued and e g = 1;Case 2. { W ( t ) } t ≥ is L ( O ; C ) -valued, e g ( x ) = i g ( | x | ) x, g ∈ C b ( R ) and satisfies the growth condi-tion sup x ∈ [0 , ∞ ) | g ( x ) | + sup x ∈ [0 , ∞ ) | g ′ ( x ) x | + sup x ∈ [0 , ∞ ) | g ′′ ( x ) x | ≤ C g ;2ase 3. { W ( t ) } t ≥ is L ( O ; R ) -valued, e g ( x ) = i g ( | x | ) x, g ∈ C b ( R ) and satisfies the growth condi-tion sup x ∈ [0 , ∞ ) | g ( x ) | + sup x ∈ [0 , ∞ ) | g ′ ( x ) x | ≤ C g . Assumption 2.
Assume that g satisfies ( x + y )( g ( | x | ) − g ( | y | )) ≤ C g | x − y | , x, y ∈ [0 , ∞ ) . (2) When W ( t ) is L ( O ; C ) -valued, we in addition assume that g satisfies following one-side Lipschitzcontinuity | (¯ y − ¯ x )( g ′ ( | x | ) g ( | x | ) | x | x − g ′ ( | y | ) g ( | y | ) | y | y ) | ≤ C g | x − y | , x, y ∈ C . (3)A typical example is e g ( u ) = u , and then Eq. (1) becomes the SlogS equation driven by linearmultiplicative noise in [4].There are two main difficulties in proving the well-posedness of the SlogS equation. On onehand, the random perturbation in SlogS equation destroys a lot of physical conservation laws,like the mass and energy conservation laws in Case 1 and Case 2, and the energy conservation lawin Case 3. Similar phenomenon has been observed in stochastic nonlinear Schr¨odinger equationwith polynomial nonlinearity (see [13]). On the other hand, the logarithmic nonlinearity inSlogS equation is not locally Lipschitz continuous. The contraction mapping arguments viaStrichartz estimates (see e.g. [3, 13, 15]) for stochastic nonlinear Schr¨odinger equation withsmooth nonlinearity are not applicable here. We only realize that in Case 2, when the drivingnoise is a linear multiplicative noise ( e g ( u ) = i u ), [4] uses a rescaling technique, together withmaximal monotone operator theory to obtain a unique global mild solution in some Orlicz space.As far as we know, there has no results concerning the well-posedness of the SlogS equationdriven by additive noise or general multiplicative noise.To show the well-posedness of the considered model, we introduce an energy regularizedproblem inspired by [2] where the authors use the regularized problem to study error estimatesof numerical methods for deterministic logarithmic Schr¨odinger equation. The main idea isfirstly constructing a proper approximation of log( | x | ) denoted by f ǫ ( | x | ) . Then it induces theregularized entropy F ǫ which is an approximation of the entropy F ( ρ ) = R O ( ρ log( ρ ) − ρ ) dx, where ρ = | u | . The RSlogS equation is defined by du ǫ = i ∆ u ǫ dt + i u ǫ f ǫ ( | u ǫ | ) dt + e g ( u ǫ ) ⋆ dW ( t ) , (4)whose regularized energy is k∇ u k + λ F ǫ ( u ). Denoting H := L = L ( O ; C ) with the product h u, v i := R R d Re ( u ¯ v ) dx, for u, v ∈ H , we obtain the existence and uniqueness of the solution ofregularized SlogS equation by proving ǫ -independent estimate in H := W , and the weighted L -space L α := { v ∈ L | x (1 + | x | ) α v ( x ) ∈ L } with the norm k v k L α ( R d ) := k (1 + | x | ) α v ( x ) k L ( R d ) . Then we are able to prove that the limit of { u ǫ } ǫ> is convergent to a uniquestochastic process u which is shown to be the unique mild solution of (1). Meanwhile, the sharpconvergence rate of { u ǫ } ǫ> is given when O = R d , or O is a bounded domain in R d equippedwith homogenous Dirichlet or periodic boundary condition. Our main result is formulated asfollows. Theorem 1.
Let
T > , Assumptions 1 and 2 hold, u ∈ H ∩ L α , α ∈ (0 , , be F measurableand has any finite p th moment. Assume that P i ∈ N + k Q e i k L α + k Q e i k H < ∞ when e g = 1 andthat P i ∈ N + k Q e i k H + k Q e i k W , ∞ < ∞ when e g ( x ) = i g ( | x | ) x . Then there exists a unique ild solution u ∈ C ([0 , T ]; H ) of Eq. (1) . Moreover, for p ≥ , there exists C ( Q, T, λ, p, u ) > such that E h sup t ∈ [0 ,T ] k u ( t ) k p H i + E h sup t ∈ [0 ,T ] k u ( t ) k pL α i ≤ C ( Q, T, λ, p, u ) . When W ( t ) is L ( O ; R )-valued, the well-posedness of SlogS equation with a super-linearly grow-ing diffusion coefficient is also proven (see Theorem 2).The reminder of this article is organized as follows. In section 2, we introduce the RSlogSequation and show the local well-posedness of RSlogS equation driven by both additive andmultiplicative noise. Section 3 is devoted to ǫ -independent estimate of the mild solution in H and L α of the RSlogS equation. In section 4, we prove the main result by passing the limit ofthe sequence of the regularized mild solutions and providing the sharp strong convergence rate.Several technique details are postponed to the Appendix. Throughout this article, C denotesvarious constants which may change from line to line.
2. Regularized SLogS equation
In this section, we show the well-posedness of the solution for Eq. (4) (see Appendix forthe definition of the solution). We would like to remark that there are several choices of theregularization function f ǫ ( | x | ). For instance, one may take f ǫ ( | x | ) = log( | x | + ǫ ǫ | x | ) (see Lemma8 in Appendix for the necessary properties) or f ǫ ( | x | ) = log( | x | + ǫ ) (see e.g. [2] and referencestherein for more choices of regularization functions). If the regularization function f ǫ enjoys thesame properties of log( | x | + ǫ ǫ | x | ), then one can follow our approach to obtain the well-posednessof Eq. (4). In the following, we first present the local well-posedness of Eq. (4), and then deriveglobal existence and uniform estimate of its solution. For simplicity, we always assume that0 < ǫ ≪ In this part, we give the detailed estimates to get the local well-posedness in H of Eq. (4)if d ≤ f ǫ ( | x | ) = log( | x | + ǫ ) . In the case of d ≥
3, one coulduse the regularization function like f ǫ ( | x | ) = log( | x | + ǫ ǫ | x | ) to get the local well-posedness in H .Assume that u ∈ H when using the regularization log( | x | + ǫ ) and that u ∈ H when applyingthe regularization log( | x | + ǫ ǫ | x | ) . Denote by M p F (Ω; C ([0 , T ]; H )) with p ∈ [1 , ∞ ) the space of process v : [0 , T ] × Ω → H withcontinuous paths in H which is F t -adapted and satisfies k v k p M p F (Ω; C ([0 ,T ]; H )) := E h sup t ∈ [0 ,T ] k u ( t ) k p H i < ∞ . Let τ ≤ T be an F t -stopping time. And we call v ∈ M p F (Ω; C ([0 , τ ); H )), if there exists { τ n } n ∈ N + with τ n ր τ as n → ∞ a.s., such that v ∈ M p F (Ω; C ([0 , τ n ]; H )) for n ∈ N + . Next we show theexistence and uniqueness of the local mild solution (see Definition 1 in Appendix).For the sake of simplicity, let us ignore the dependence on ǫ and write u R := u ǫR , where u R is the solution of the truncated equation du R = i ∆ u R dt + i λ Θ R ( u R , t ) u R f ǫ ( | u R | ) dt (5) −
12 Θ R ( u R , t ) X k ∈ N + | Q e k | (cid:16) | g ( | u R | ) | u R (cid:17) dt i Θ R ( u R , t ) X k ∈ N + g ( | u R | ) g ′ ( | u R | ) | u R | u R Im ( Q e k ) Q e k dt + Θ R ( u R , t ) i g ( | u R | ) u R dW ( t ) . Here, Θ R ( u, t ) := θ R ( k u k C ([0 ,t ]; H ) ) , R > , with a cut-off function θ R , that is, a positive C ∞ function on R + which has a compact support, and θ R ( x ) = ( , for x ≥ R, , for x ∈ [0 , . Lemma 1.
Let Assumption 1 hold, d ≤ , and f ǫ ( | x | ) = log( | x | + ǫ ) . Assume in additionthat g ∈ C b ( R ) when W ( t ) is L ( O ; C ) -valued and that g ∈ C b ( R ) when W ( t ) is L ( O ; R ) -valued.Suppose that the Q -Wiener process W ( t ) satisfies P i ∈ N + k Q e i k H < ∞ , and u ∈ H is F -measurable and has any finite p th moment. Then there exists a unique global solution to (5) withcontinuous H -valued path. Proof
Let S ( t ) = exp( i ∆ t ) be the C -group generated by i ∆ . For fixed
R >
0, we use thefollowing notations, for t ∈ [0 , T ] , Γ Rdet u ( t ) : = i Z t S ( t − s ) (cid:16) Θ R ( u, s ) λf ǫ ( | u ( s ) | ) u ( s ) (cid:17) ds, Γ Rmod u ( t ) : = − Z t S ( t − s ) (cid:16) Θ R ( u, s ) X k ∈ N + | Q e k | (cid:16) | g ( | u ( s ) | ) | u ( s ) (cid:17)(cid:17) ds − i Z t S ( t − s ) (cid:16) Θ R ( u, s ) X k ∈ N + g ( | u ( s ) | ) g ′ ( | u ( s ) | ) | u ( s ) | u ( s ) Im ( Q e k ) Q e k (cid:17) ds, Γ RSto u ( t ) : = i Z t S ( t − s ) (cid:16) Θ R ( u, s ) g ( | u ( s ) | ) u ( s ) (cid:17) dW ( s ) . W e look for a fixed point of the following operator given byΓ R u ( t ) := S ( t ) u + Γ Rdet u ( t ) + Γ Rmod u ( t ) + Γ RSto u ( t ) , u ∈ M p F (Ω; C ([0 , r ]; H )) , where r will be chosen later. The unitary property of S ( · ) yields that k S ( · ) u k M p F (Ω; C ([0 ,r ]; H )) ≤ k u k H . Now, we define a stopping time τ = inf { t ∈ [0 , T ] : k u k C ([0 ,t ]; H ) ≥ R } ∧ r. By using theproperties of log( · ) and the Sobolev embedding H ֒ → L ∞ ( O ), we have k Γ Rdet u k C ([0 ,r ]; H ) ≤ Cr | λ | max( | log( ǫ ) | , log( ǫ + 4 R ))( sup t ∈ [0 ,r ] k u ( t ) k H )+ Cr | λ | (1 + ǫ − ) (cid:16) sup t ∈ [0 ,r ] k u ( t ) k H + sup t ∈ [0 ,r ] k u ( t ) k H (cid:17) ≤ C ( ǫ, λ ) r (1 + 2 R + R ) , and k Γ Rmod u k C ([0 ,r ]; H ) ≤ C ( λ, g ) r X k ∈ N + k Q e k k H (cid:16) R + R + R (cid:17) . k Γ Rdet u k M p F (Ω; C ([0 ,r ]; H )) + k Γ Rmod u k M p F (Ω; C ([0 ,r ]; H )) ≤ C ( λ, g ) r X k ∈ N + k Q e k k H (cid:16) R + R + R (cid:17) . The Burkerholder inequality yields that for p ≥ k Γ Rsto u k M p F (Ω; C ([0 ,r ]; H )) ≤ C ( g ) (cid:16) E h(cid:16) Z r X k ∈ N + k g ( | u ( s ) | ) u ( s ) Q e k k H ds (cid:17) p i(cid:17) p ≤ C ( g ) r X k ∈ N + k Q e k k H ! (cid:16) R + R (cid:17) . Therefore, Γ R is well-defined on M p F (Ω; C ([0 , r ]; H )) . Now we turn to show the contractivity of Γ R . Let u , u ∈ M p F (Ω; C ([0 , r ]; H )) , and definethe stopping times τ j = inf { t ∈ [0 , T ] : k u j k C ([0 ,r ]; H ) ≥ R } ∧ r, j = 1 , . For a fixed ω, let usassume that τ ≤ τ without the loss of generality. Then direct calculation leads to k Γ Rdet u − Γ Rdet u k C ([0 ,r ]; H ) ≤ C ( λ ) r (cid:13)(cid:13)(cid:13)(cid:16) Θ R ( u , · ) − Θ R ( u , · ) (cid:17) f ǫ ( | u | ) u (cid:13)(cid:13)(cid:13) C ([0 ,r ]; H ) + C ( λ ) r (cid:13)(cid:13)(cid:13) Θ R ( u , · ) (cid:16) f ǫ ( | u | ) u − f ǫ ( | u | ) u (cid:17)(cid:13)(cid:13)(cid:13) C ([0 ,r ]; H ) ≤ C ( λ ) r k u − u k C ([0 ,r ]; H ) k f ǫ ( | u | ) u k C ([0 ,τ ]; H ) + C ( λ ) r k f ǫ ( | u | ) u − f ǫ ( | u | ) u k C ([0 ,τ ]; H ) ≤ C ( λ, ǫ ) r k u − u k C ([0 ,r ]; H ) (1 + R ) , and k Γ Rmod u − Γ Rmod u k C ([0 ,r ]; H ) ≤ C ( λ, ǫ ) X k ∈ N + k Q e k k H r k u − u k C ([0 ,r ]; H ) (1 + R ) . By applying the Burkerholder inequality, we obtain k Γ RSto u − Γ RSto u k M p F (Ω; C ([0 ,r ]; H )) ≤ Cr (cid:13)(cid:13)(cid:13) Θ R ( u , · )( g ( | u | ) u − g ( | u | ) u ) (cid:13)(cid:13)(cid:13) L p (Ω; L ([0 ,r ]; L )) + Cr (cid:13)(cid:13)(cid:13) (Θ R ( u , · ) − Θ R ( u , · )) g ( | u | ) u (cid:13)(cid:13)(cid:13) L p (Ω; L ([0 ,r ]; L )) ≤ C ( λ, ǫ )( X k ∈ N + k Q e k k H ) r k u − u k M p F (Ω; C ([0 ,r ]; H )) (1 + R ) . where L is the space of Hilbert–Schmidt operators form U = Q ( L ( O )) to H . Combiningall the above estimates, we have k Γ R u − Γ R u k M p F (Ω; C ([0 ,r ]; H )) ≤ C ( λ, ǫ ) X k ∈ N + k Q e k k H ! ( r + r ) k u − u k M p F (Ω; C ([0 ,r ]; H )) (1 + R ) , r > Q, R, λ, ǫ such that Γ R is a strictcontraction in M p F (Ω; C ([0 , r ]; H )) and has a fixed point u R, satisfying Γ R ( u R, ) = u R, . Assume that we have found the fixed point on each interval [( l − r, lr ] , l ≤ k for some k ≥ u R,k = S ( · ) u + Γ Rdet u R,k + Γ
Rmod u R,k + Γ stodet u R,k , on [0 , kr ] . In order to extend u R,k to [ kr, ( k + 1) r ], we repeat the previous arguments to showthat on the interval [ kr, ( k + 1) r ] , there exists a fixed point of the map Γ R defined byΓ R u ( t ) := S ( t ) u R,k ( kr ) + Γ R,kdet u ( t ) + Γ R,kmod u ( t ) + Γ R,kSto u ( t ) , u ∈ M p F (Ω; C ([0 , r ]; H )) . Here we use the following notations, Γ R,kdet u ( t ) : = i Z t S ( t − s ) (cid:16)e Θ R ( u, k, s ) λf ǫ ( | u ( s ) | ) u ( s ) (cid:17) ds, Γ R,kmod u ( t ) : = − Z t S ( t − s ) (cid:16)e Θ R ( u, k, s ) X j ∈ N + | Q e j | (cid:16) | g ( | u ( s ) | ) | u ( s ) (cid:17)(cid:17) ds − i Z t S ( t − s ) (cid:16) e Θ R ( u, k, s ) X k ∈ N + g ( | u ( s ) | ) | g ′ ( | u ( s ) | ) | u ( s ) | u ( s ) Im ( Q e k ) Q e k (cid:17) ds, Γ R,kSto u ( t ) : = i Z t S ( t − s ) (cid:16)e Θ R ( u, k, s ) g ( | u ( s ) | ) u ( s ) (cid:17) dW k ( s ) , w here t ∈ [0 , r ], W k ( s ) = W ( s + kr ) − W ( kr ) , u ∈ M p F kr (Ω; C ([0 , r ]; H )) and e Θ R ( u, k, s ) = θ R ( k u R,k k C ([0 ,kr ]; H ) + k u k C ([0 ,r ]; H ) ) . For any different v , v ∈ M p F kr (Ω; C ([0 , r ]; H )), we define the stopping times τ i = inf { t ∈ [0 , T − kr ] : k u R,k k C ([0 ,kr ]; H ) + k v i k C ([0 ,t ]; H ) ≥ R } ∧ r, i = 1 , τ ≤ τ forthe convenience. Then the same procedures yield that this map is a strict contraction and hasa fixed point u R,k +1 for a small r > . Now, we define a process u R as u R ( t ) := u R,k ( t ) for t ∈ [0 , kr ] and u R ( t ) := u R,k +1 ( t ) for t ∈ [ kr, kr + 1] . It can be checked that u R satisfies (5)by the induction assumption and the definition of Γ R . Meanwhile, the uniqueness of the mildsolution can be obtained by repeating the previous arguments. (cid:3) Proposition 1.
Let the condition of Lemma 1 hold. There exists a unique local mild solu-tion to (4) with continuous H -valued path. And the solution is defined on a random interval [0 , τ ∗ ǫ ( u , ǫ )) , where τ ∗ ǫ ( u , ǫ, ω ) is a stopping time such that τ ∗ ǫ ( u , ǫ, ω ) = + ∞ or lim t → τ ∗ ǫ k u ǫ ( t ) k H =+ ∞ . Proof
When e g = 1, one can follow the same steps in the proof of [13, Theorem 3.1] to complete theproof. It suffices to consider the multiplicative noise case. Let { u R } R ∈ N + ∈ M F (Ω; C ([0 , T ]; H ))be a sequence of solution constructed in Lemma 1. And define a stopping time sequence τ R :=inf { t ∈ [0 , T ] : k u R k C ([0 ,t ]; H ) ≥ R } ∧ T. Then τ R > k u R k C ([0 ,t ]; H ) isan increasing, continuous and F t -adapted process. We claim that if R ≤ R , then τ R ≤ τ R and u R = u R , a.s. on [0 , τ R ] . Let τ R ,R := inf { t ∈ [0 , T ] : k u R k C ([0 ,t ]; H ) ≥ R } ∧ T. Thenit holds that τ R ,R ≤ τ R and Θ R ( u R , t ) = Θ R ( u R , t ) on t ∈ [0 , τ R ,R ] . This implies that { ( u R , τ R ,R ) } is a solution of (5) and that u R = u R , a.s. on { t ≤ τ R ,R } . Thus we concludethat τ R = τ R ,R , a.s. and that u R = u R for { t ≤ τ R } .7ow consider the triple { u, ( τ R ) R ∈ N + , τ ∞ } defined by u ( t ) := u R ( t ) for t ∈ [0 , τ R ] and τ ∞ =sup R ∈ N + τ R . From Lemma 1, we know that u ∈ M p F (Ω , C ([0 , τ ]; H )) satisfies (4) for t ≤ τ R . Theuniqueness of the local solution also holds. If we assume that ( u, τ ) and ( v, σ ) are local mildsolutions of (4), then u ( t ) = v ( t ) , a.s. on { t < σ ∧ τ } . Let R , R ∈ N + . Set τ R ,R := inf { t ∈ [0 , T ] : max( k u k C ([0 ,t ]; H ) , k v k C ([0 ,t ]; H ) ) ≥ R } ∧ σ R ∧ σ R . Then we have on { t ≤ τ R ,R } , ( u, τ R ,R ) , ( v, τ R ,R ) are local mild solutions of (4). The uniqueness in Lemma 1 leads to u = v on { t ≤ τ R ,R } . Letting R , R → ∞ , we complete the proof. (cid:3) If we assume that g ∈ C s p ( R ) , u ∈ H s , s ≥
2, following the same procedures, we can alsoobtain the local existence of the solution u ǫ in C ([0 , τ ∗ ǫ ]; H s ) when d ≤
3. When d ≥
3, one needsto use another regularization function log( | x | + ǫ ǫ | x | ) and additional assumption on g . In this case,we can get the local well-posedness in M p F (Ω; C ([0 , r ]; H )) based on Lemma 8 in Appendix andprevious arguments. Since its proof is similar to that in Proposition 1, we omit these details hereand leave them to readers. Proposition 2.
Let Assumption 1 hold, and f ǫ ( | x | ) = log( | x | + ǫ ǫ | x | ) . Suppose that d ∈ N + , u ∈ H is F -measurable and has any finite p th moment for p ≥ . Assume in addition that Q -Wienerprocess W satisfies P k ∈ N + k Q e k k L α + k Q e k k H < ∞ when e g = 1 , and P k ∈ N + k Q e k k W , ∞ + k Q e k k H < ∞ when e g ( x ) = i g ( | x | ) x . Then there exists a unique local mild solution to (4) withcontinuous H -valued path. And the solution is defined on a random interval [0 , τ ∗ ǫ ( u , ǫ, ω )) ,where τ ∗ ǫ ( u , ǫ, ω ) is a stopping time such that τ ∗ ( u , ǫ, ω ) = + ∞ or lim t → τ ∗ ǫ k u ǫ ( t ) k H = + ∞ . Due to the blow-up alternative results in section 2, it suffices to prove that sup t ∈ [0 ,τ ∗ ǫ ) k u ǫ ( t ) k H < ∞ , or sup t ∈ [0 ,τ ∗ ǫ ) k u ǫ ( t ) k H < ∞ , a.s. under corresponding assumptions. In the following, wepresent several a priori estimates in strong sense to achieve our goal. To simplify the presenta-tion, we omit some procedures like mollifying the unbounded operator ∆ and taking the limiton the regularization parameter. More precisely, the mollifier e Θ m , m ∈ N + may be defined bythe Fourier transformation (see e.g. [13]) F ( e Θ m v )( ξ ) = e θ ( | ξ | m ) b v ( ξ ) , ξ ∈ R d , where e θ is a positive C ∞ function on R + , has a compact support satisfying θ ( x ) = 0 , for x ≥ θ ( x ) = 1 , for 0 ≤ x ≤ . Another choice of mollifier is via Yosida approximation Θ m := m ( m − ∆) − for m ∈ N + (see e.g. [15]). This kind of procedure is introduced to make thatthe Itˆo formula can be applied rigorously to deducing several a priori estimates. If O becomesa bounded domain equipped with periodic or homogenous Dirichlet boundary condition, themollifier can be chosen as the Galerkin projection, and the approximated equation becomes theGalerkin approximation (see e.g. [9, 10, 11, 12]).In this section, we assume that u has finite p -moment for all p ≥ d ≤ f ǫ ( x ) = log( x + ǫ ) , we assume that P k k Q e k k H < ∞ . When d ∈ N + and f ǫ ( x ) = log( x + ǫ ǫx ) , we assume that Q ∈ L for the additive noise and P k k Q e k k W , ∞ < ∞ for the multiplicative noise. L and H Lemma 2.
Let
T > . Under the condition of Proposition 1 or Proposition 2, assume that ( u ǫ , τ ∗ ǫ ) be a local mild solution in H . Then for any p ≥ , there exists a positive constant ( Q, T, λ, p, u ) > such that E h sup t ∈ [0 ,τ ∗ ǫ ∧ T ) k u ǫ ( t ) k p i ≤ C ( Q, T, λ, p, u ) . Proof
Take any stopping time τ < τ ∗ ǫ ∧ T, a.s.
Using the Itˆo formula to M k ( u ǫ ( t )) , where M ( u ǫ ( t )) := k u ǫ ( t ) k and k ∈ N + or k ≥
2, we obtain that for t ∈ [0 , τ ] and the case e g = 1 ,M k ( u ǫ ( t ))= M k ( u ǫ ) + 2 k ( k − Z t M k − ( u ǫ ( s )) X i ∈ N + h u ǫ ( s ) , Q e i i ds + k Z t M k − ( u ǫ ( s )) X i ∈ N + k Q e i k ds + 2 k Z t M k − ( u ǫ ( s )) h u ǫ ( s ) , dW ( s ) i , and for t ∈ [0 , τ ] and the case e g ( x ) = i g ( | x | ) x , M k ( u ǫ ( t ))= M k ( u ǫ ) + 2 k ( k − Z t M k − ( u ǫ ( s )) X i ∈ N + h u ǫ ( s ) , i g ( | u ǫ ( s ) | ) u ǫ ( s ) Q e i i ds + 2 k Z t M k − ( u ǫ ( s )) h u ǫ ( s ) , i g ( | u ǫ ( s ) | ) u ǫ ( s ) dW ( s ) i + 2 k Z t M k − ( u ǫ ( s )) × D u ǫ ( s ) , − i X k ∈ N + Im ( Q e k ) Q e k g ′ ( | u ǫ ( s ) | ) g ( | u ǫ ( s ) | ) | u ǫ ( s ) | u ǫ ( s ) E ds. In particular, if W ( t, x ) is real-valued and e g ( x ) = i g ( | x | ) x , we have M k ( u ǫ ( t )) = M k ( u ǫ ) , for t ∈ [0 , T ] a.s. By Assumption 1 and conditions in Propositions 1 and 2, using the martingaleinequality, the H¨older inequality and the Young inequality and Gronwall’s inequality, we achievethat for all k ≥ , sup t ∈ [0 ,τ ] E h M k ( u ǫ ( t )) i ≤ C ( T, k, u , Q ) . Next, taking the supreme over t and repeating the above procedures, we have that E h sup t ∈ [0 ,τ ] M k ( u ǫ ( t )) i ≤ E h M k ( u ǫ ) i + C ( k ) E h Z τ M k − ( u ǫ ( s )) X i ∈ N + (1 + k u ǫ ( s ) k ) k Q e i k U ds i + C ( k ) E h(cid:16) Z τ M k − ( u ǫ ( s ))(1 + k u ǫ ( s ) k ) X i ∈ N + k Q e i k U ds (cid:17) i , where U = L for additive noise case and U = L ∞ for multiplicative noise case. Applying theestimate of sup t ∈ [0 ,τ ] E h M k ( u ǫ ( t )) i , we complete the proof by taking p = 2 k . (cid:3) Lemma 3.
Let
T > . Under the condition of Proposition 1 or Proposition 2, assume that ( u ǫ , τ ∗ ǫ ) is a local mild solution in H . Then for any p ≥ , there exists C ( Q, T, λ, p, u ) > suchthat E h sup t ∈ [0 ,τ ∗ ǫ ∧ T ) k u ǫ k p H i ≤ C ( Q, T, λ, p, u ) . roof Take any stopping time τ < τ ∗ ǫ ∧ T, a.s.
Applying the Itˆo formula to the kinetic energy K ( u ǫ ( t )) := k∇ u ǫ ( t ) k , for k ∈ N + and using integration by parts, we obtain that for t ∈ [0 , τ ] ,K k ( u ǫ ( t ))= K k ( u ǫ ) + k Z t K k − ( u ǫ ( s )) h∇ u ǫ ( s ) , i λf ′ ǫ ( | u ǫ ( s ) | ) Re (¯ u ǫ ( s ) ∇ u ǫ ( s )) u ǫ ( s ) i ds + 12 k ( k − Z t K k − ( u ǫ ( s )) X i ∈ N + h∇ u ǫ ( s ) , ∇ Q e i i ds + k Z t K k − ( u ǫ ( s )) X i ∈ N + h∇ Q e i , ∇ Q e i i ds + k Z t K k − ( u ǫ ( s )) h∇ u ǫ ( s ) , ∇ dW ( s ) i for the additive noise case, and K k ( u ǫ ( t ))= K k ( u ǫ ) + k Z t K k − ( u ǫ ( s )) h∇ u ǫ ( s ) , i λf ′ ǫ ( | u ǫ ( s ) | ) Re (¯ u ǫ ( s ) ∇ u ǫ ( s )) u ǫ ( s ) i ds + 12 k ( k − Z t K k − ( u ǫ ( s )) X i ∈ N + h∇ u ǫ ( s ) , i ∇ ( g ( | u ǫ ( s ) | ) u ǫ ( s ) Q e i ) i ds + k Z t K k − ( u ǫ ( s )) X i ∈ N + h∇ ( g ( | u ǫ ( s ) | ) u ǫ ( s ) Q e i ) , ∇ ( g ( | u ǫ ( s ) | ) u ǫ ( s ) Q e i ) i ds + k Z t K k − ( u ǫ ( s )) h∇ u ǫ ( s ) , i ∇ g ( | u ǫ ( s ) | ) u ǫ ( s ) dW ( s ) i + k Z t K k − ( u ǫ ( s )) h ∆ u ǫ ( s ) , X i ∈ N + | Q e i | ( g ( | u ǫ ( s ) | )) u ǫ ( s ) i ds − k Z t K k − ( u ǫ ( s )) h ∆ u ǫ ( s ) , i X i ∈ N + g ( | u ǫ ( s ) | ) g ′ ( | u ǫ ( s ) | ) | u ǫ ( s ) | u ǫ ( s ) Im ( Q e i ) Q e i i ds f or the multiplicative noise case. Applying integration by parts, in the multiplicative noise case,we further obtain K k ( u ǫ ( t )) ≤ K k ( u ǫ ) + k Z t K k − ( u ǫ ( s )) h∇ u ǫ , i λf ′ ǫ ( | u ǫ ( s ) | ) Re (¯ u ǫ ∇ u ǫ ) u ǫ ( s ) i ds + C k Z t K k − ( u ǫ ( s )) X i ∈ N + (cid:16) h∇ u ǫ , g ( | u ǫ | ) u ǫ ∇ Q e i ) i + h∇ u ǫ , g ′ ( | u ǫ | ) Re (¯ u ǫ ∇ u ǫ ) u ǫ Q e i ) i (cid:17) ds + C k Z t K k − ( u ǫ ( s )) X i ∈ N + (cid:16) k g ( | u ǫ | ) u ǫ ∇ Q e i k ++ k g ( | u ǫ | ) ∇ u ǫ Q e i k + k g ′ ( | u ǫ | ) Re (¯ u ǫ ∇ u ǫ ) u ǫ Q e i k (cid:17) ds + k Z t K k − ( u ǫ ( s )) h∇ u ǫ , i ∇ g ( | u ǫ | ) u ǫ dW ( s ) i C k Z t K k − ( u ǫ ( s )) X i ∈ N + (cid:16) |h∇ u ǫ , ( ∇ Im ( Q e i ) Q e i + ∇ Q e i Im ( Q e i )) g ′ ( | u ǫ | ) g ( | u ǫ | ) | u ǫ | u ǫ i| + |h∇ u ǫ , Im ( Q e i ) Q e i g ′′ ( | u ǫ | ) g ( | u ǫ | ) | u ǫ | u ǫ Re (¯ u ǫ ∇ u ǫ ) i| + |h∇ u ǫ , Im ( Q e i ) Q e i | g ′ ( | u ǫ | ) | | u ǫ | u ǫ Re (¯ u ǫ ∇ u ǫ ) i| + |h∇ u ǫ , Im ( Q e i ) Q e i g ′ ( | u ǫ | ) g ( | u ǫ | )(2 Re (¯ u ǫ ∇ u ǫ ) u ǫ + | u ǫ | ∇ u ǫ ) i| (cid:17) ds. By using the property of g in Assumption 1 and conditions on Q , and applying H¨older’s, Young’sand Burkholder’s inequalities, we achieve that for small ǫ > , E [ sup r ∈ [0 ,t ] K k ( u ǫ ( r ))] ≤ E [ K k ( u ǫ )] + C k | λ | E h Z t sup r ∈ [0 ,s ] K k − ( u ǫ ( r )) k∇ u ǫ ( r ) k ds i + C k T E h sup r ∈ [0 ,t ] K k − ( u ǫ ( r )) X i ∈ N + k∇ u ǫ ( r ) k (1 + k∇ u ǫ ( r ) k + k u ǫ ( r ) k ) k Q e i k U i + C k E h(cid:16) Z t K k − ( u ǫ ( s )) X i ∈ N + k∇ u ǫ ( r ) k (1 + k∇ u ǫ ( r ) k + k u ǫ ( r ) k k∇ Q e i k U ds (cid:17) i ≤ E [ K k ( u ǫ )] + C ( T, λ, k, ǫ , Q ) + ǫ E [ sup r ∈ [0 ,t ] K k ( u ǫ ( r ))]+ C k E h Z t sup r ∈ [0 ,s ] K k ( u ǫ ( r )) dr i , where k Q e i k U = k Q e i k H for the additive noise case and k Q e i k U = k Q e i k H + k Q e i k W , ∞ for the multiplicative noise case. Applying Gronwall’s inequality, we complete the proof by taking p = 2 k . ✷ From the above proofs of Lemmas 2 and 3, it is not hard to see that to obtain ǫ -independentestimates, the boundedness restriction sup x ≥ | g ( x ) | < ∞ may be not necessary in the case that W ( t, x ) is real-valued. We present such result in the following which is the key of the globalwell-posedness of an SlogS equation with super-linear growth diffusion in next section. Lemma 4.
Let
T > and ( u ǫ , τ ∗ ǫ ) be a local mild solution in H s , s ≥ for any p ≥ . Assumethat u ∈ H ∩ L α , for some α ∈ (0 , , is F -measurable and has any finite p th moment,and W ( t, x ) is real-valued with P i k Q e i k H + k Q e i k W , ∞ < ∞ . Let e g ( x ) = i g ( | x | ) x, g ∈C b ( R ) ∩ C ( R ) satisfy the growth condition and the embedding condition, sup x ∈ [0 , ∞ ) | g ′ ( x ) x | ≤ C g , k vg ( | v | ) k L q ≤ C d (1 + k v k H + k v k L α ) , for some q ≥ , where C g > depends on g , and C d > depends on O , d , k v k . Then it holdsthat M ( u ǫt ) = M ( u ) for t ∈ [0 , τ ǫ ) . Furthermore, there exists a positive constant C ( Q, T, λ, p, u ) such that E h sup t ∈ [0 ,τ ∗ ǫ ∧ T ) k u ǫ ( t ) k p H i ≤ C ( Q, T, λ, p, u ) . Proof
The proof is similar to those of Lemmas 2 and 3. We only need to modify the estimationinvolved with g. The mass conservation is not hard to be obtained since the calculations in11emmas 2 only use the assumptions that g ∈ C ( R ) and W ( t, x ) is real-valued. Therefore, wefocus on the estimate in H . We only show estimation about E h K k ( u ǫ ( t )) i since the proof on E h sup t ∈ [0 ,τ ∗ ǫ ∧ T ) K k ( u ǫ ( t )) i is similar. Then following the same steps in Lemma 3, we get thatfor q + q ′ = , E h K k ( u ǫ ( t )) i ≤ E h K k ( u ǫ ) i + k E h Z t K k − ( u ǫ ( s )) h∇ u ǫ , i λf ′ ǫ ( | u ǫ ( s ) | ) Re (¯ u ǫ ∇ u ǫ ) u ǫ ( s ) i ds i + E h C k Z t K k − ( u ǫ ( s )) X i ∈ N + (cid:16) h∇ u ǫ , g ( | u ǫ | ) u ǫ ∇ Q e i ) i + h∇ u ǫ , g ′ ( | u ǫ | ) Re (¯ u ǫ ∇ u ǫ ) u ǫ Q e i ) i (cid:17) ds i + C k E h Z t K k − ( u ǫ ( s )) X i ∈ N + (cid:16) k g ( | u ǫ | ) u ǫ ∇ Q e i k + k g ′ ( | u ǫ | ) Re (¯ u ǫ ∇ u ǫ ) u ǫ Q e i k (cid:17) ds i ≤ E h K k ( u ǫ ) i + C k E h Z t K k ( u ǫ ( s )) ds i + E h C k Z t K k − ( u ǫ ( s )) X i ∈ N + (cid:16) k g ( | u ǫ | ) u ǫ k L q k∇ Q e i k L q ′ + k∇ u ǫ k k Q e i k (cid:17) ds i . Applying the embedding condition on g , mass conservation law and the procedures in the proofof Proposition 3, we complete the proof by taking p = 2 k and Gronwall’s inequality. ✷ The embedding condition is depending on the assumption on Q and d . One example whichsatisfies the embedding condition on g and is not bounded is g ( x ) = log( c + x ) , for x ≥ c > . Let us verify this example on O = R d . If the domain O is bounded, one can obtain thesimilar estimate. We apply the Gagliardo–Nirenberg interpolation inequality and get that for q > η > , k g ( | v | ) v k qL q ≤ Z c + | v | ≥ | log( c + | v | ) | q | v | q dx + Z c + | v | ≤ | log( c + | v | ) | q | v | q dx ≤ C (cid:16) k v k qL q + k v k q + qηL q + qη + k v k q − qηL q − qη (cid:17) ≤ C (cid:16) k v k − α k∇ v k α + ( k v k − α k∇ v k α ) η + ( k v k − α k∇ v k α ) − η (cid:17) q ≤ C (cid:16) k∇ v k qα + k∇ v k ( q + qη ) α + k∇ v k ( q − qη ) α (cid:17) , where α = d ( q − q , α = d ( q (1+ η ) − q (1+ η ) and α = d ( q (1 − η ) − q (1 − η ) satisfies α i ∈ (0 , , i = 0 , , . When q = 2 , similar calculations, together with the interpolation inequality in Lemma 5, yield that k g ( | v | ) v k ≤ C (cid:16) k v k + k v k ηL η + k v k − ηL − η (cid:17) ≤ C (cid:16) k∇ v k + k∇ v k (2+2 η ) α + k v k dηα L α (cid:17) , where α = d (2(1+ η ) − η ) ∈ (0 ,
1) and α ∈ ( dη − η , . L α -estimate and modified energy Beyond the L and H estimates, we also need the uniform boundedness in L α , α > { u ǫ } ǫ> when O = R d . We would like to mention that when O
12s a bounded domain, such estimate in L α , α > Lemma 5.
Let d ∈ N + and η ∈ (0 , . Then for α > dη − η , it holds that for some C = C ( d ) > , k v k L − η ≤ C k v k − dη α (1 − η ) k v k dη α (1 − η ) L α , v ∈ L ∩ L α . Proof
Using the Cauchy-Schwarz inequality and α > dη − η , we have that for any r > , k v k − ηL − η ≤ Z | x |≤ r | v ( x ) | − η dx + Z | x |≥ r | x | α (2 − η ) | v ( x ) | − η | x | α (2 − η ) dx ≤ Cr dη k v k − η + C k v k − ηL α ( Z | x |≥ r | x | α (2 − η ) η dx ) η ≤ Cr dη k v k − η + Cr − α (2 − η )+ dη k v k − ηL α . Let r = ( k v k L α k v k ) α , we complete the proof. ✷ Proposition 3.
Let
T > , O = R d , d ∈ N + . Under the condition of Proposition 1 or Proposi-tion 2, let ( u ǫ , τ ∗ ǫ ) be a local mild solution in H s , s ≥ for any p ≥ . Let u ∈ L α ∩ H , for some α ∈ (0 , , have any finite p th moment. Then the solution u ǫ of regularized problem satisfies for α ∈ (0 , , E h sup t ∈ [0 ,τ ∗ ǫ ∧ T ) k u ǫ ( t ) k pL α i ≤ C ( Q, T, λ, p, u ) . Proof
We first introduce the stopping time τ R = inf n t ∈ [0 , T ] : sup s ∈ [0 ,t ] k u ǫ ( s ) k L α ≥ R o ∧ τ ∗ ǫ , then show that E h sup t ∈ [0 ,τ R ] k u ǫ ( t ) k pL α i ≤ C ( T, u , Q, p ) independent of R . After taking R → ∞ ,we get τ R = τ ∗ ǫ , a.s. For simplicity, we only prove uniform upper bound when p = 1.Taking 0 < t ≤ t ≤ τ R , and applying the Itˆo formula to k u ǫ k L α = R R d (1 + | x | ) α | u ǫ | dx , weget k u ǫ ( t ) k L α = k u ǫ k L α + Z t h (1 + | x | ) α u ǫ ( s ) , i ∆ u ǫ ( s ) i ds + Z t X i ∈ N + h (1 + | x | ) α Q e i , Q e i i ds + Z t h (1 + | x | ) α u ǫ ( s ) , i f ǫ ( | u ǫ ( s ) | ) u ǫ ( s ) i ds + Z t h (1 + | x | ) α u ǫ ( s ) , dW ( s ) i for additive noise case, and k u ǫ ( t ) k L α = k u ǫ k L α + 2 Z t h (1 + | x | ) α u ǫ ( s ) , i ∆ u ǫ ( s ) + i f ǫ ( | u ǫ ( s ) | ) u ǫ ( s ) i ds − Z t h (1 + | x | ) α u ǫ ( s )( s ) , ( g ( | u ǫ ( s ) | )) u ǫ ( s ) X i | Q e i | i ds Z t X i h (1 + | x | ) α g ( | u ǫ ( s ) | ) u ǫ ( s ) Q e i , g ( | u ǫ ( s ) | ) u ǫ ( s ) Q e i i ds − Z t h (1 + | x | ) α u ǫ ( s ) , i X k ∈ N + g ( | u ǫ ( s ) | ) g ′ ( | u ǫ ( s ) | ) | u ǫ ( s ) | u ǫ ( s ) Im ( Q e k ) Q e k i ds + 2 Z t h (1 + | x | ) α u ǫ ( s ) , i g ( | u ǫ ( s ) | ) u ǫ ( s ) dW ( s ) i for multiplicative noise case. Using integration by parts, then taking supreme over t ∈ [0 , t ] andapplying the Burkerholder inequality, we deduce E h sup t ∈ [0 ,t ] k u ǫ ( t ) k L α i ≤ E h k u ǫ k L α i + C α E h Z T (cid:12)(cid:12)(cid:12) h (1 + | x | ) α − xu ǫ ( s ) , ∇ u ǫ ( s ) i (cid:12)(cid:12)(cid:12) ds i + C (cid:16) E h Z t X i ∈ N + k (1 + | x | ) α u ǫ ( s ) k k (1 + | x | ) α Q e i k ds i(cid:17) for additive noise case, and E h sup t ∈ [0 ,t ] k u ǫ ( t ) k L α i ≤ E h k u ǫ k L α i + C α E h Z T (cid:12)(cid:12)(cid:12) h (1 + | x | ) α − xu ǫ ( s ) , ∇ u ǫ ( s ) i (cid:12)(cid:12)(cid:12) ds i + C (cid:16) E h Z t X i ∈ N + k (1 + | x | ) α u ǫ ( s ) k k Q e i k L ∞ ds i(cid:17) for multiplicative noise case. Then Young’s and Gronwall’s inequalities, together with a prioriestimate of u ǫ in H , lead to the desired result. ✷ Corollary 1.
Under the condition of Lemma 4, the solution u ǫ of regularized problem satisfiesfor α ∈ (0 , , E h sup t ∈ [0 ,τ ∗ ǫ ∧ T ) k u ǫ ( t ) k pL α i ≤ C ( Q, T, λ, p, u ) . It is not possible to obtain the uniform bound of the exact solution in L α for α ∈ (1 ,
2] like thedeterministic case. The main reason is that the rough driving noise leads to low H¨older regularityin time and loss of uniform estimate in H for the mild solution. We can not expect that the mildsolution of (4) enjoys ǫ -independent estimate in H . More precisely, we prove that applying theregularization f ǫ ( | x | ) = log( | x | + ǫ ) in Proposition 1, one can only expect ǫ -dependent estimatein H . We omit the tedious calculation and procedures, and present a sketch of the proof forLemma 6 and Propostion 4 in Appendix. Lemma 6.
Let
T > and d = 1 . Under the condition of Proposition 1, assume that ( u ǫ , τ ∗ ǫ ) is the local mild solution in H . In addition assume that P i ∈ N + k Q e i k W , ∞ < ∞ for themultiplicative noise. Then for any p ≥ , there exists a positive C ( Q, T, λ, p, u ) such that E h sup t ∈ [0 ,τ ∗ ǫ ∧ T ) k u ǫ ( t ) k p H i ≤ C ( Q, T, λ, p, u )(1 + ǫ − p ) . roposition 4. Assume that O = R d . Let T > and d = 1 . Under the condition of Proposition1, assume that ( u ǫ , τ ∗ ǫ ) is the local mild solution in H . Let u ∈ L α , for some α ∈ (1 , . In addi-tion assume that P i k Q e i k L α < ∞ in the additive noise case and that P i ∈ N + k Q e i k W , ∞ < ∞ for the multiplicative noise. Then the solution u ǫ ( t ) of regularized problem satisfies for α ∈ (1 , , E h sup t ∈ [0 ,τ ∗ ǫ ∧ T ) k u ǫ ( t ) k pL α i ≤ C ( Q, T, λ, p, u )(1 + ǫ − p ) . The above results indicate that both spatial and temporal regularity for SLogS equation arerougher than deterministic LogS equation.In the following, we present the behavior of the regularized energy for the RSlogS equation.When applying f ǫ ( | x | ) = log( | x | + ǫ ) , the modified energy of (4) becomes H ǫ ( u ǫ ( t )) := K ( u ǫ ) − λ F ǫ ( | u ǫ | )with F ǫ ( | u ǫ | ) = R O (cid:16) ( ǫ + | u ǫ | ) log( ǫ + | u ǫ | ) −| u ǫ | − ǫ log ǫ (cid:17) dx. When using another regularizationfunction f ǫ ( | u | ) = log( ǫ + | u | ǫ | u | ) , its regularized entropy in the modified energy H ǫ becomes F ǫ ( | u ǫ | ) = Z O (cid:16) | u ǫ | log( | u ǫ | + ǫ | u ǫ | ǫ ) + ǫ log( | u ǫ | + ǫ ) − ǫ log( ǫ | u ǫ | + 1) − ǫ log( ǫ ) (cid:17) dx. In general, the modified energy is defined by the regularized entropy e F ( ρ ) = R O R ρ f ǫ ( s ) dsdx, where f ǫ ( · ) is a suitable approximation of log( · ) . We remark that the regularized energy is well-defined when O is a bounded domain. The additional constant term ǫ log( ǫ ) ensures that theregularized energy is still well-defined when O = R d . We leave the proof of Proposition 5 inAppendix.
Proposition 5.
Let
T > . Under the condition of Proposition 1 or Proposition 2, assume that ( u ǫ , τ ∗ ǫ ) is the local mild solution in H . Then for any p ≥ , there exists a positive constant C ( Q, T, λ, p, u ) such that E h sup t ∈ [0 ,τ ǫ ∧ T ) | H ǫ ( u ǫ ( t )) | p i ≤ C ( Q, T, λ, p, u ) . Below, we present the global existence of the unique mild solution for Eq. (4) in H based onProposition 1, Lemma 6, Proposition 2 and Lemma 3, as well as a standard argument in [13]. Proposition 6.
Let Assumption 1 hold and ( u ǫ , τ ∗ ǫ ) be the local mild solution in Proposition 1or Proposition 2. Then the mild solution u ǫ in H is global, i.e., τ ∗ ǫ = + ∞ , a.s. In additionassume that d = 1 in Proposition 1, the mild solution u ǫ in H is global.
3. Well-posedness for SLogS equation
Based on the a priori estimates of the regularized problem, we are going to prove the strongconvergence of any sequence of the solutions of the regularized problem. This immediately impliesthat the existence and uniqueness of the mild solution in L for SLogS equation.15 .1. Well-posedness for SLogS equation via strong convergence approximation In this part, we not only show the strong convergence of a sequence of solutidregularizedproblems, but also give the explicit strong convergence rate. The strong convergence rate of theregularized SLogS equation will make a great contribution to the numerical analysis of numericalschemes for the SLogS equation. And this topic will be studied in a companion paper. For thestrong convergence result, we only present the mean square convergence rate result since theproof of the strong convergence rate in L q (Ω) , q ≥ f ǫ in Lemmas 7 and 8 will be frequently used.In the multiplicative noise case, Assumption 2 is needed to obtain the strong convergence rateof the solution of Eq. (4). We remark that the assumption can be weaken if one only wants toobtain the strong convergence instead of deriving a convergence rate. Some sufficient conditionfor (3) in Assumption 2 is (cid:12)(cid:12)(cid:12) ( g ′ ( | x | ) g ( | x | ) | x | − g ′ ( | y | ) g ( | y | ) | y | )( | x | − | y | ) (cid:12)(cid:12)(cid:12) ≤ C g | x − y | , x, y ∈ C or (cid:12)(cid:12)(cid:12) ( g ′ ( | x | ) g ( | x | ) | x | − g ′ ( | y | ) g ( | y | ) | y | )( | x | + | y | ) (cid:12)(cid:12)(cid:12) ≤ C g | x − y | , x, y ∈ C . Functions like 1 , c + x , xc + x , xc + x , log( c + | x | c | x | ) with c >
0, etc., will satisfy Assumption 2.The main idea of the proof lies on showing that for a decreasing sequence { ǫ n } n ∈ N + satisfyinglim n →∞ ǫ n = 0, { u ǫ n } n ∈ N + must be a Cauchy sequence in C ([0 , T ]; L p (Ω; H )) , p ≥ . As a conse-quence, we obtain that there exists a limit process u in C ([0 , T ]; L p (Ω; H )) which is shown to beindependent of the sequence { u ǫ n } n ∈ N + and is the unique mild solution of the mild form of (1). [Proof of Theorem 1] Based on Proposition 6, we can construct a sequence of mild solutions { u ǫ n } n ∈ N + of Eq. (4) with f ǫ n ( | x | ) = log( ǫ n + | x | ǫ n | x | ). Here the decreasing sequence { ǫ n } n ∈ N + satisfies lim n →∞ ǫ n = 0 . We use the following steps to complete the proof. For simplicity, we onlypresent the details for p = 2 since the procedures for p > { u ǫ n } n ∈ N + is a Cauchy sequence in L (Ω; C ([0 , T ]; H )) . Fix different n, m ∈ N + such that n < m. Subtracting the equation of u ǫ n from the equationof u ǫ m , we have that d ( u ǫ m − u ǫ n ) = i ∆( u ǫ m − u ǫ n ) dt + i λ ( f ǫ m ( | u ǫ m | ) u ǫ m − f ǫ n ( | u ǫ n | ) u ǫ n ) dt for additive noise case, and d ( u ǫ m − u ǫ n )= i ∆( u ǫ m − u ǫ n ) dt + i λ ( f ǫ m ( | u ǫ m | ) u ǫ m − f ǫ n ( | u ǫ n | ) u ǫ n ) dt − X k ∈ N + | Q e k | (cid:16) | g ( | u ǫ m | ) | u ǫ m − | g ( | u ǫ n | ) | u ǫ n (cid:17) dt − i X k ∈ N + Im ( Q e k ) Q e k (cid:16) g ′ ( | u ǫ m | ) g ( | u ǫ m | ) | u ǫ m | u ǫ m − g ′ ( | u ǫ n | ) g ( | u ǫ n | ) | u ǫ n | u ǫ n (cid:17) dt + i (cid:16) g ( | u ǫ m | ) u ǫ m − g ( | u ǫ n | ) u ǫ n (cid:17) dW ( t )for multiplicative noise case. Then using the Itˆo formula to k u ǫ m ( t ) − u ǫ n ( t ) k , the properties of f ǫ in Lemma 8, the mean value theorem and the Gagliardo–Nirenberg interpolation inequality,we obtain that for η ′ (2 − d ) ≤ , k u ǫ m ( t ) − u ǫ n ( t ) k (6)16 Z t h u ǫ m − u ǫ n , i λf ǫ m ( | u ǫ m | ) u ǫ m − f ǫ n ( | u ǫ n | ) u ǫ n i ds ≤ Z t | λ |k u ǫ m ( s ) − u ǫ n ( s ) k ds + 4 | λ | Z t | Im h u ǫ m ( s ) − u ǫ n ( s ) , ( f ǫ m ( | u ǫ n | ) − f ǫ n ( | u ǫ n | ) u ǫ n i| ds ≤ Z t | λ |k u ǫ m ( s ) − u ǫ n ( s ) k ds + 4 | λ | Z t k u ǫ m ( s ) − u ǫ n ( s ) k L (cid:13)(cid:13)(cid:13) ( ǫ m − ǫ n ) | u ǫ n | ǫ m + | u ǫ n | (cid:13)(cid:13)(cid:13) L ∞ ds + 2 | λ | Z t k log(1 + ( ǫ n − ǫ m ) | u ǫ n | ǫ m | u ǫ n | ) | u ǫ n |k ds ≤ Z t | λ |k u ǫ m ( s ) − u ǫ n ( s ) k ds + 4 | λ | ǫ n Z t k u ǫ m ( s ) − u ǫ n ( s ) k L d + 2 | λ | Cǫ η ′ n Z t k u ǫ n k η ′ L η ′ ds for additive noise case, and k u ǫ m ( t ) − u ǫ n ( t ) k = Z t h u ǫ m − u ǫ n , i λf ǫ m ( | u ǫ m | ) u ǫ m − f ǫ n ( | u ǫ n | ) u ǫ n i ds − Z t h u ǫ m − u ǫ n , X k ∈ N + | Q e k | (cid:16) | g ( | u ǫ m | ) | u ǫ m − | g ( | u ǫ n | ) | u ǫ n (cid:17) i ds − Z t h u ǫ m − u ǫ n , i X k ∈ N + Im ( Q e k ) Q e k (cid:16) g ′ ( | u ǫ m | ) g ( | u ǫ m | ) | u ǫ m | u ǫ m − g ′ ( | u ǫ n | ) g ( | u ǫ n | ) | u ǫ n | u ǫ n (cid:17) i ds + 2 Z t h u ǫ m − u ǫ n , i (cid:16) g ( | u ǫ m | ) u ǫ m − g ( | u ǫ n | ) u ǫ n (cid:17) dW ( s ) i + Z t h g ( | u ǫ m | ) u ǫ m − g ( | u ǫ n | ) u ǫ n , X k ∈ N + | Q e k | (cid:16) g ( | u ǫ m | ) u ǫ m − g ( | u ǫ n | ) u ǫ n (cid:17) i ds ≤ Z t | λ |k u ǫ m ( s ) − u ǫ n ( s ) k ds + 4 | λ | ǫ n Z t k u ǫ m ( s ) − u ǫ n ( s ) k L ds + 2 | λ | Cǫ η ′ n Z t k u ǫ n k η ′ L η ′ ds + 2 Z t h u ǫ m − u ǫ n , i (cid:16) g ( | u ǫ m | ) u ǫ m − g ( | u ǫ n | ) u ǫ n (cid:17) dW ( s ) i + Z t h ( g ( | u ǫ m | ) − g ( | u ǫ n | )) u ǫ n , X k ∈ N + | Q e k | ( g ( | u ǫ m | ) − g ( | u ǫ n | )) u ǫ m i ds − Z t h u ǫ m − u ǫ n , i X k ∈ N + Im ( Q e k ) Q e k (cid:16) g ′ ( | u ǫ m | ) g ( | u ǫ m | ) | u ǫ m | u ǫ m − g ′ ( | u ǫ n | ) g ( | u ǫ n | ) | u ǫ n | u ǫ n (cid:17) i ds for multiplicative noise case. By using (2) and (3) in Assumption 2 and the assumptions on Q ,we have that k u ǫ m ( t ) − u ǫ n ( t ) k (7) ≤ Z t (4 | λ | + C ( g, Q )) k u ǫ m ( s ) − u ǫ n ( s ) k ds + 4 | λ | ǫ n Z t k u ǫ m ( s ) − u ǫ n ( s ) k L ds + 2 | λ | Cǫ η ′ n Z t k u ǫ n k η ′ L η ′ ds + Z t h u ǫ m − u ǫ n , i (cid:16) g ( | u ǫ m | ) u ǫ m − g ( | u ǫ n | ) u ǫ n (cid:17) dW ( s ) i . Next we show the strong convergence of the sequence { u ǫ n } n ∈ N + in the following differentcases. 17ase 1: O is a bounded domain. By using the H¨older inequality k u ǫ m ( s ) − u ǫ n ( s ) k L ≤|O| k u ǫ m ( s ) − u ǫ n ( s ) k on (6) and (7), and using the Gronwall’s inequality, we getsup t ∈ [0 ,T ] k u ǫ m ( t ) − u ǫ n ( t ) k ≤ C ( λ, T, |O| )( ǫ n + ǫ η ′ n )(1 + sup t ∈ [0 ,T ] k u ǫ n k η ′ L η ′ )for additive noise case. In the multiplicative noise case, taking supreme over t and then takingexpectation on (7), together with the Burkholder and Young inequalities, we get that for a small κ > E h sup t ∈ [0 ,T ] k u ǫ m ( t ) − u ǫ n ( t ) k i ≤ C ( λ, T, |O| , Q )( ǫ n + ǫ η ′ n ) + C E h sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12) Z t h u ǫ m − u ǫ n , i (cid:16) g ( | u ǫ m | ) u ǫ m − g ( | u ǫ n | ) u ǫ n (cid:17) dW ( s ) i (cid:12)(cid:12)(cid:12)i ≤ C ( λ, T, |O| , Q )( ǫ n + ǫ η ′ n ) + C E h(cid:16) Z T X i k Q e i k L ∞ k u ǫ m − u ǫ n k ds (cid:17) i ≤ C ( λ, T, |O| , Q )( ǫ n + ǫ η ′ n ) + C E h sup s ∈ [0 ,T ] k u ǫ m − u ǫ n k (cid:16) Z T X i k Q e i k L ∞ k u ǫ m − u ǫ n k ds (cid:17) i ≤ C ( λ, T, |O| , Q )( ǫ n + ǫ η ′ n ) + κ E h sup t ∈ [0 ,T ] k u ǫ m ( t ) − u ǫ n ( t ) k i + C ( κ ) E h Z T X i k Q e i k L ∞ k u ǫ m − u ǫ n k ds i . Taking κ < , we have that E h sup t ∈ [0 ,T ] k u ǫ m ( t ) − u ǫ n ( t ) k i ≤ C ( Q, T, λ, p, u , |O| )( ǫ n + ǫ η ′ n ) . Case 2: O = R d . Since u ∈ L α , α ∈ (0 , , using the interpolation inequality in Lemma 5implies that for any η ∈ [0 ,
1) and α > ηd − η ) (i.e., η ∈ (0 , α α + d )), k u ǫ m ( t ) − u ǫ n ( t ) k ≤ Z t | λ |k u ǫ m ( s ) − u ǫ n ( s ) k ds + 4 | λ | Z t ǫ η n k u ǫ m ( s ) − u ǫ n ( s ) kk u ǫ n k − ηL − η ds + 2 | λ | Cǫ η ′ n Z t k u ǫ n k η ′ L η ′ ds ≤ Z t | λ |k u ǫ m ( s ) − u ǫ n ( s ) k ds + 2 | λ | Z t k u ǫ m ( s ) − u ǫ n ( s ) k + 2 | λ | ǫ ηm Z t k u ǫ n k − ηL − η ds + 2 | λ | Cǫ η ′ n Z t k u ǫ n k η ′ L η ′ ds ≤ Z t | λ |k u ǫ m ( s ) − u ǫ n ( s ) k ds + 2 | λ | Cǫ ηn Z t k u ǫ n k dηα L α k u ǫ n k − η − dηα ds + 2 | λ | Cǫ η ′ n Z t k u ǫ n k η ′ L η ′ ds for additive noise case, and k u ǫ m ( t ) − u ǫ n ( t ) k Z t C k u ǫ m ( s ) − u ǫ n ( s ) k ds + Cǫ ηm Z t k u ǫ n k dηα L α k u ǫ n k − η − dηα ds + 2 | λ | Cǫ η ′ n Z t k u ǫ n k η ′ L η ′ ds + (cid:12)(cid:12)(cid:12) Z t h u ǫ m − u ǫ n , i (cid:16) g ( | u ǫ m | ) u ǫ m − g ( | u ǫ n | ) u ǫ n (cid:17) dW ( s ) i (cid:12)(cid:12)(cid:12) for multiplicative noise case. Then taking supreme over t , taking expectation, using (2), Lemma2 and Proposition 3, and applying Gronwall’s inequality, we have that for α ∈ (0 , , η ∈ (0 , α α + d )and η ′ ( d − ≤ E h sup t ∈ [0 ,T ] k u ǫ m ( t ) − u ǫ n ( t ) k i (8) ≤ C ( T, Q, u , g ) E h sup [0 ,T ] (cid:16) k u ǫ n k dηα L α k u ǫ n k − η − dηα + k u ǫ n k η ′ L η ′ (cid:17)i ( ǫ ηn + ǫ η ′ n ) ≤ C ( T, Q, u , g, α, η ) ǫ min( η,η ′ ) n . Step 2: The limit process u of { u ǫ n } n ∈ N + in M F ( C ([0 , T ]; H )) satisfies (1) in mild form. Weuse the multiplicative noise case to present all the detailed procedures. It suffices to prove thateach term in the mild form of RSlogS equation (4) convergenes to the corresponding part in S ( t ) u + i λ Z t S ( t − s ) log( | u | ) uds − Z t S ( t − s )( g ( | u | )) u X k | Q e k | ds − i Z t S ( t − s ) g ′ ( | u | ) g ( | u | ) | u | u X k Im ( Q e k ) Q e k ds + i Z t S ( t − s ) g ( | u | ) udW ( s ):= S ( t ) u + V + V + V + V . We first claim that all the terms V - V make sense. By Lemma 3 and Proposition 3, we havethat for p ≥ , sup n E " sup t ∈ [0 ,T ] k u ǫ n ( t ) k p H + sup n E " sup t ∈ [0 ,T ] k u ǫ n ( t ) k pL α ≤ C ( u , T, Q ) . By applying the Fourier transform and Parseval’s theorem, using the Fatou theorem and strongconvergence of ( u ǫ n ) n ∈ N + in M F ( C ([0 , T ]; H )), we obtain E h sup t ∈ [0 ,T ] k u ( t ) k H i + E h sup t ∈ [0 ,T ] k u ( t ) k L α i ≤ sup n E h sup t ∈ [0 ,T ] k u ǫ n ( t ) k H i + sup n E [ sup t ∈ [0 ,T ] k u ǫ n ( t ) k L α ] ≤ C ( u , T, Q ) . Then the Gagliardo–Nirenberg interpolation inequality yields that for small η ′ , η > k log( | u | ) u k = Z | u | ≥ (log( | u | )) | u | dx + Z | u | ≤ (log( | u | )) | u | dx ≤ Z | u | ≥ | u | η ′ dx + Z | u | ≤ | u | − η dx ≤ C ( k u k η ′ L η + k u k − ηL − η ) ≤ C ( k u k − ηL − η + k∇ u k dη ′ k u k η ′ +2 − dη ′ ) . O = R d , we use the weighted version of the interpolation inequality in Lemma 5 to dealwith the term k u k − ηL − η , and have that for small η < α α + d , k log( | u | ) u k ≤ C ( k∇ u k dη ′ k u k η ′ +2 − dη ′ + k u k dηα L α k u k − η − dηα ) . This implies that V makes sense in M F ( C ([0 , T ]; H )) by Proposition 3, Lemmas 2 and 3. Mean-while, we can show that V - V ∈ M F ( C ([0 , T ]; H )) by using the Minkowski and Burkerholderinequalities due to our assumption on g and Q .Next, we show that the mild form of u ǫ n converges to S ( t ) u + V + V + V + V . To provethat lim n →∞ E h sup t ∈ [0 ,T ] (cid:13)(cid:13)(cid:13) Z t S ( t − s ) f ǫ n ( | u ǫ n | ) u ǫ n ds − V (cid:13)(cid:13)(cid:13) i = 0 , we use the following decomposition of f ǫ n ( | u ǫ n | ) u ǫ n − log( | u | ) u. When | u | > | u ǫ n | , f ǫ n ( | u ǫ n | ) u ǫ n − log( | u | ) u =( f ǫ n ( | u ǫ n | ) − f ǫ n ( | u | )) u ǫ n + f ǫ n ( | u | )( u ǫ n − u ) + ( f ǫ n ( | u | ) − log( | u | )) u, and when | u | < | u ǫ n | ,f ǫ n ( | u ǫ n | ) u ǫ n − log( | u | ) u =(log( | u ǫ n | ) − log( | u | )) u + log( | u ǫ n | )( u ǫ n − u ) + ( f ǫ n ( | u ǫ n | ) − log( | u ǫ n | )) u ǫ n . For convenience, let us show the estimate for | u | > | u ǫ n | , the other case will be estimated ina similar way. By using the H¨older inequality and the mean-valued theorem, we have that forsmall γ > , (cid:12)(cid:12)(cid:12) ( f ǫ n ( | u ǫ n | ) − f ǫ n ( | u | )) u ǫ n (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) ( | u | − | u ǫ n | ǫ n + | u ǫ n | ) − γ ( f ǫ n ( | u ǫ n | ) − f ǫ n ( | u | )) + γ u ǫ n (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) ( | u | − | u ǫn | ) − γ ( | u | + | u ǫn | ) − γ | u ǫ n | ( ǫ n + | u ǫ n | ) − γ ( f ǫ n ( | u ǫ n | ) + f ǫ n ( | u | )) + γ (cid:12)(cid:12)(cid:12) . Then it implies that for small enough η > , Z | u | > | u ǫn | (cid:12)(cid:12)(cid:12) ( f ǫ n ( | u ǫ n | ) − f ǫ n ( | u | )) u ǫ n (cid:12)(cid:12)(cid:12) dx ≤ Z | u | > | u ǫn | | u ǫ n | ( ǫ n + | u ǫ n | ) − γ | u − u ǫ n | − γ ( | u | + | u ǫ | ) − γ ( f ǫ n ( | u ǫ n | ) + f ǫ n ( | u | )) γ dx ≤ Z | u | > | u ǫn | ,ǫ n + | u ǫn |≤ | u ǫ n | ( ǫ n + | u ǫ n | ) − γ | u − u ǫ n | − γ ( | u | + | u ǫ | ) − γ ( f ǫ n ( | u ǫ n | ) + f ǫ n ( | u | )) γ dx + Z | u | > | u | ǫn ,ǫ n + | u ǫn |≥ | u ǫ n | ( ǫ n + | u ǫ n | ) − γ | u − u ǫ n | − γ ( | u | + | u ǫ | ) − γ ( f ǫ n ( | u ǫ n | ) + f ǫ n ( | u | )) γ dx ≤ C Z | u | > | u ǫn | ,ǫ n + | u ǫn |≤ | u ǫ n | ( ǫ n + | u ǫ n | ) − γ | u − u ǫ n | − γ ( | u | + | u ǫ | ) − γ (( ǫ n + | u ǫ n | ) − η + ( ǫ n + | u | ) η ) dx + C Z | u | > | u | ǫn ,ǫ n + | u ǫn |≥ | u ǫ n | ( ǫ n + | u ǫ n | ) − γ ( | u | + | u ǫ | ) − γ | u + u ǫ | − γ ( ǫ n + | u | ) η dx. − γ + 2 η ≤
2, using the H¨older inequality and the weighted interpolationinequality in Lemma 5, we have that for α > ηd γ − η , Z | u | > | u ǫn | (cid:12)(cid:12)(cid:12) ( f ǫ n ( | u ǫ n | ) − f ǫ n ( | u | )) u ǫ n (cid:12)(cid:12)(cid:12) dx ≤ C Z R d | u ǫ n | γ − η | u − u ǫ n | − γ ( | u | + | u ǫ | ) − γ dx + C Z R d | u ǫ n | γ − η | u − u ǫ n | − γ ( | u | + | u ǫ | ) − γ ( ǫ n + | u | ) η dx ≤ C k u − u ǫ n k − γ (cid:13)(cid:13)(cid:13) ( | u | γ − η + | u | γ )( | u | + | u ǫ | ) − γ (cid:13)(cid:13)(cid:13) L γ ≤ C k u − u ǫ n k − γ (cid:16) k u k γ + k u k γ − ηL − η γ (cid:17) ≤ C k u − u ǫ n k − γ (cid:16) k u k γ + k u k ηdα (1+ r ) L α k u k γ − η − ηdα (1+ γ ) (cid:17) . For the term f ǫ n ( | u | )( u ǫ n − u ) , we similarly have that for η ′ < d − , and η < α α + d , Z | u | > | u ǫ | | f ǫ n ( | u | )( u ǫ n − u ) | dx ≤ Z | u | > | u ǫ | ,ǫ n + | u | ≤ ( ǫ n + | u | ) − η | u ǫ n − u | dx + Z | u | > | u ǫ | ,ǫ n + | u | ≥ ( ǫ n + | u | ) η | u ǫ n − u | dx ≤ C k u ǫ n − u k (cid:16) k u ǫ n − u k + k u k η ′ L η ′ + k u ǫ n k − ηL − η + k u k − ηL − η (cid:17) ≤ C k u ǫ n − u k (cid:16) k u ǫ n − u k + k∇ u k η ′ d k u k η ′ − η ′ d + k u k dη α L α k u k − η − dη α + k u ǫ n k dη α L α k u ǫ n k − η − dη α (cid:17) . For the term ( f ǫ n ( | u | ) − log( | u | )) u, the mean-valued theorem, the property that log(1+ | x | ) ≤ | x | and the Gagliardo–Nirenberg interpolation inequality k u k L q ( R d ) ≤ C k u k − α k∇ u k α , q = 2 dd − α , for α ∈ (0 , q = 2 η ′ + 2, yield that for η ′ ( d − ≤ η ≤ α α − d , Z O | ( f ǫ n ( | u | ) − log( | u | )) u | dx ≤ Cǫ η ′ n k u k η ′ +2 L η ′ +2 + Cǫ ηn k u k − ηL − η ≤ Cǫ η ′ n k∇ u k η ′ d k u k η ′ − η ′ d + Cǫ ηn k u ǫ n k dηα L α k u ǫ n k − η − dηα . Combining the above estimates, using the a priori estimate of u ǫ and u in Lemmas 2 and 3 andProposition 3, and applying the strong convergence of u ǫ n (8),we obtain thatlim n →∞ E h sup t ∈ [0 ,T ] (cid:13)(cid:13)(cid:13) Z t S ( t − s ) f ǫ n ( | u ǫ n | ) u ǫ n − log( | u | ) uds (cid:13)(cid:13)(cid:13) i = 0 . The Minkowski inequality, (2) and (3) yield that (cid:13)(cid:13)(cid:13) − Z t S ( t − s ) 12 ( g ( | u ǫ n | )) u ǫ n X k | Q e k | ds − V (cid:13)(cid:13)(cid:13) X k k Q e k k L ∞ Z T k g ( | u ǫ n | )) u ǫ n − g ( | u | )) u k ds ≤ C g T X k k Q e k k L ∞ sup t ∈ [0 ,T ] k u ǫ n ( t ) − u ( t ) k , and (cid:13)(cid:13)(cid:13) − i Z t S ( t − s )( g ′ ( | u ǫ n | ) g ( | u ǫ n | ) | u ǫ n | u ǫ n X k Im ( Q e k ) Q e k ds − V (cid:13)(cid:13)(cid:13) ≤ C g T X k k Q e k k L ∞ sup t ∈ [0 ,T ] k u ǫ n ( t ) − u ( t ) k . The Burkerholder inequality and the unitary property of S ( · ) yield that E h sup t ∈ [0 ,T ] (cid:13)(cid:13)(cid:13) i Z t S ( t − s ) g ( | u ǫ n | ) u ǫ n dW ( s ) − V (cid:13)(cid:13)(cid:13) i ≤ C E h Z T X k k Q e k k L ∞ k g ( | u ǫ n | ) u ǫ n − g ( | u | ) u k ds i ≤ C E h sup t ∈ [0 ,T ] k u ǫ n ( t ) − u ( t ) k i . Combining the above estimates and the strong convergence of u ǫ n , we complete the proof of step2. Step 3: u is independent of the choice of the sequence of { u ǫ n } n ∈ N + . Assume that e u and u aretwo different limit processes of two different sequences of { u ǫ n } n ∈ N + and { u ǫ m } m ∈ N + , respectively.Then by step 2, they both satisfies Eq. (1). By repeating the procedures in step 1, it is not hardto obtain that e u = u. ✷ The procedures in the above proof immediately yield the following convergence rate resultfor u ǫ in the regularized problem (4) and the H¨older regularity estimate of u ǫ and u . Corollary 2.
Let the condition of Theorem 1 hold. Assume that u ǫ is the mild solution inProposition 6, ǫ ∈ (0 , . For p ≥ , there exists C ( Q, T, λ, p, u ) > such that for any η ′ ( d − ≤ , E h sup t ∈ [0 ,T ] k u ( t ) − u ǫ ( t ) k p i ≤ C ( Q, T, λ, p, u )( ǫ p + ǫ η ′ p ) when O is bounded domain, and E h sup t ∈ [0 ,T ] k u ( t ) − u ǫ ( t ) k p i ≤ C ( Q, T, λ, p, u , α )( ǫ αp α + d + ǫ η ′ p ) when O = R d . Corollary 3.
Let the condition of Theorem 1 hold. Assume that u ǫ is the mild solution inProposition 6, ǫ ∈ (0 , and u is the mild solution of Eq. (1) . For p ≥ , there exists C ( Q, T, λ, p, u ) > such that for ǫ ∈ [0 , , E h k u ǫ ( t ) − u ǫ ( s ) k p i ≤ C ( Q, T, λ, p, u ) | t − s | p . Proof
By means of the mild form of u ǫ , ǫ ∈ [0 , u ǫ in H ∩ L α inLemmas 3 and 3, and in Step 2 of the proof of Theorem 1, and the Burkholder inequalty, weobtain the desirable result. ✷ .2. Well-posedness of SlogS equation with super-linearly growing diffusion coefficients In this part, we extends the scope of e g , which allows the diffusion with super-linear growth,for the well-posedness of SlogS equation driven by conservative multiplicative noise. For instance,it includes the example e g ( x ) = i g ( | x | ) x = i x log( c + | x | ) , for c > Theorem 2.
Let W ( t ) be L ( O ; R ) -valued and g ∈ C b ( R ) ∩ C ( R ) satisfy the growth conditionand the embedding condition, sup x ∈ [0 , ∞ ) | g ′ ( x ) x | ≤ C g , k vg ( | v | ) k ≤ C d (1 + k v k H + k v k L α ) for some q ≥ , α ∈ [0 , , where C g > depends on g , C d > depends on O , d , k v k and v ∈ H ∩ L α . Assume that d = 1 , u ∈ H ∩ L α , α ∈ (0 , , and P i ∈ N + k Q e i k H + k Q e i k W , ∞ < ∞ .Then there exists a unique mild solution u in C ([0 , T ]; H ) for Eq. (1) satisfying E h sup t ∈ [0 ,T ] k u ( t ) k p H i + E h sup t ∈ [0 ,T ] k u ( t ) k pL α i ≤ C ( Q, T, λ, p, u ) . Proof
By Proposition 6 and Lemma 3, we can introduce the truncated sample spaceΩ R ( t ) := ( ω : sup s ∈ [0 ,t ] k u ǫ m ( s ) k L ∞ ≤ R, sup s ∈ [0 ,t ] k u ǫ n ( s ) k L ∞ ≤ R ) , where n ≤ m. The Gagliardo–Nirenberg interpolation inequality in d = 1, the priori estimate in H and the continuity in L of u ǫ n imply that u ǫ n are continuous in L ∞ a.s. Define a stoppingtime τ R := inf { t ≥ s ∈ [0 ,t ] k u ǫ m ( s ) k L ∞ , sup s ∈ [0 ,t ] k u ǫ n ( s ) k L ∞ ) ≥ R } ∧ T. Then on Ω R ( T ) , we have τ R = T. Let us take f ǫ ( x ) = log( x + ǫ ) , x > R ( t ) → Ω as R → ∞ and that for any p ≥ P (cid:16) sup t ∈ [0 ,T ] min( k u ǫ m ( t ) k L ∞ , k u ǫ n ( t ) k L ∞ ) ≥ R (cid:17) ≤ C R p (cid:16) E h sup t ∈ [0 ,T ] k u ǫ m ( t ) k pL ∞ i + sup t ∈ [0 ,T ] E h k u ǫ m ( t ) k pL ∞ i(cid:17) . Step 1: { u ǫ n } n ∈ N + forms a Cauchy sequence in M F (Ω; C ([0 , T ]; H )). Following the samesteps like the proof of Theorem 1, applying the Itˆo formula on Ω R ( t ) for t ∈ (0 , τ R ) yields that k u ǫ m ( t ) − u ǫ n ( t ) k ≤ Z t | λ |k u ǫ m ( s ) − u ǫ n ( s ) k ds + 4 | λ | ǫ n Z t k u ǫ m ( s ) − u ǫ n ( s ) k L ds + Z t X k ∈ N + h| Q e k | ( g ( | u ǫ n | ) − g ( | u ǫ m | )) u ǫ m , ( g ( | u ǫ n | ) − g ( | u ǫ m | )) u ǫ n i ds + Z t h u ǫ m − u ǫ n , i (cid:16) g ( | u ǫ m | ) u ǫ m − g ( | u ǫ n | ) u ǫ n (cid:17) dW ( s ) i . R ( t ) yields that E h I Ω R ( t ) k u ǫ m ( t ) − u ǫ n ( t ) k i ≤ Z t | λ | E h I Ω R ( t ) k u ǫ m ( s ) − u ǫ n ( s ) k i ds + 4 | λ | ǫ n Z t E h I Ω R ( t ) k u ǫ m ( s ) − u ǫ n ( s ) k L i ds + Z t X k ∈ N + | Q e k | L ∞ E h I Ω R ( t ) h ( g ( | u ǫ n | ) − g ( | u ǫ m | )) u ǫ m , ( g ( | u ǫ n | ) − g ( | u ǫ m | )) u ǫ n i i ds. Making use of the assumptions on g , we get E h I Ω R ( t ) k u ǫ m ( t ) − u ǫ n ( t ) k i ≤ Z t | λ | E h I Ω R ( t ) k u ǫ m ( s ) − u ǫ n ( s ) k i ds + 4 | λ | ǫ n Z t E h I Ω R ( t ) k u ǫ m ( s ) − u ǫ n ( s ) k L i ds + C ( Q ) Z t E h I Ω R ( t ) Z O ( | u ǫ m | + | u ǫ n | ) | u ǫ n || u ǫ m || u ǫ m ( t ) − u ǫ n ( t ) | dx i ds ≤ Z t | λ | E h I Ω R ( t ) k u ǫ m ( s ) − u ǫ n ( s ) k i ds + 4 | λ | ǫ n Z t E h I Ω R ( t ) k u ǫ m ( s ) − u ǫ n ( s ) k L i ds + C ( Q ) Z t E h (1 + R ) I Ω R ( t ) k u ǫ m ( t ) − u ǫ n ( t ) k i ds. If O is bounded, then H¨older inequality and Gronwall’s inequality yield that E h I Ω R ( t ) k u ǫ m ( t ) − u ǫ n ( t ) k i ≤ C ( u , Q, T, λ ) e (1+ R ) T ǫ n . On the other hand, the Chebyshev inequality and the a priori estimate lead to E h I Ω cR ( t ) k u ǫ m ( t ) − u ǫ n ( t ) k i ≤ ( P (Ω cR ( t ))) p (cid:16) E h k u ǫ m ( t ) − u ǫ n ( t ) k q i(cid:17) q , where p + q = 1 . From the above estimate, choosing p ≫ p and denote by κ = pp , we concludethat E h k u ǫ m ( t ) − u ǫ n ( t ) k i ≤ C ( u , Q, T, λ, p, p ) (cid:16) e (1+ R ) T ǫ n + R − κ (cid:17) Then one may take R = ( cT | log( ǫ ) | ) for c ∈ (0 ,
1) and get E h k u ǫ m ( t ) − u ǫ n ( t ) k i ≤ C ( u , Q, T, λ, p, p )( ǫ − cn + ( cT | log( ǫ n ) | ) − κ ) . By further applying the Burkerholder inequality to the stochastic integral, we achieve that forany κ > , E h sup t ∈ [0 ,T ] k u ǫ m ( t ) − u ǫ n ( t ) k i ≤ C ( u , Q, T, λ, p, p ) | log( ǫ n ) | − κ . When O = R d , we just repeat the procedures in the proof of the case that g is bounded andobtain that for η ∈ (0 , α α + d ] and α ∈ (0 , , E h I Ω R ( t ) k u ǫ m ( t ) − u ǫ n ( t ) k i Z t | λ | E h I Ω R ( t ) k u ǫ m ( s ) − u ǫ n ( s ) k i ds + Cǫ ηm Z t E h I Ω R ( t ) k u ǫ n k dηα L α k u ǫ n k − η − dηα i ds + C ( Q ) Z t E h (1 + R ) I Ω R ( t ) k u ǫ m ( t ) − u ǫ n ( t ) k i ds. By using Gronwall’s inequality and the estimate of P ( I c Ω R ( t ) ), we immediately have that for η ∈ (0 , α α + d ) , α ∈ (0 ,
1] and any κ > , E h k u ǫ m ( t ) − u ǫ n ( t ) k i ≤ C ( u , Q, T, λ, p, p ) (cid:16) e (1+ R ) T ǫ ηn + R − κ (cid:17) Taking R = ( ηcT | log( ǫ n ) | ) for cη ∈ (0 ,
1) and using the Burkerholder inequality, we have for η ∈ (0 , α α + d ) , α ∈ (0 ,
1] and any κ > , E h sup t ∈ [0 ,T ] k u ǫ m ( t ) − u ǫ n ( t ) k i ≤ C ( u , Q, T, λ, p, p , η ) | log( ǫ n ) | − κ . Step 2. u is the mild solution.Let us use the same notations and procedures as in step 2 of the proof in Theorem 1. Toshow that the mild form u ǫ n converges to S ( t ) u + V + V + V + V . We only need to estimate V and V since V = 0. DefineΩ R ( t ) := { ω : sup s ∈ [0 ,t ] k u ǫ n ( s ) k L ∞ ≤ R , sup s ∈ [0 ,t ] k u ( s ) k L ∞ ≤ R } , and a stopping time τ R := inf { t ≥ s ∈ [0 ,t ] k u ( s ) k L ∞ , sup s ∈ [0 ,t ] k u ǫ n ( s ) k L ∞ ) ≥ R } ∧ T. Then on Ω R ( T ) , we have τ R = T . The Minkowski inequality and the properties of g yield thaton Ω R ( t ) , for a small enough η > (cid:13)(cid:13)(cid:13) − Z t S ( t − s )( g ( | u ǫ n | )) u ǫ n X k | Q e k | ds − V (cid:13)(cid:13)(cid:13) ≤ X k | Q e k | L ∞ Z T k g ( | u ǫ n | )) u ǫ n − g ( | u | )) u k ds ≤ C g T X k | Q e k | L ∞ (cid:16) R (cid:17) sup t ∈ [0 ,T ] k u ǫ n − u k . On the other hand, for any p > E h I Ω cR ( t ) (cid:13)(cid:13)(cid:13) − Z t S ( t − s )( g ( | u ǫ n | )) u ǫ n X k | Q e k | ds − V (cid:13)(cid:13)(cid:13) i ≤ C ( u , Q, T, p ) R − p . Taking R = O ( | log( ǫ n ) | κ p ) , κ < κ, we have thatlim n →∞ sup t ∈ [0 ,T ] E h(cid:13)(cid:13)(cid:13) Z t − S ( t − s )( g ( | u ǫ n | )) u ǫ n X k | Q e k | ds − V (cid:13)(cid:13)(cid:13) i = 0 , n →∞ E h sup t ∈ [0 ,T ] (cid:13)(cid:13)(cid:13) − Z t S ( t − s )( g ( | u ǫ n | )) u ǫ n X k | Q e k | ds − V (cid:13)(cid:13)(cid:13) i = 0 . The Burkerholder inequality and the unitary property of S ( · ) yield that E h sup t ∈ [0 ,τ R ] (cid:13)(cid:13)(cid:13) Z t i S ( t − s ) g ( | u ǫ n | ) u ǫ n dW ( s ) − V (cid:13)(cid:13)(cid:13) i ≤ C E h Z T X k k Q e k k L ∞ k g ( | u ǫ n | ) u ǫ n − g ( | u | ) u k ds i ≤ C (1 + R ) E h sup t ∈ [0 ,T ] k u ǫ n ( t ) − u ( t ) k i . On the other hand, the Chebyshev inequality, together with the a priori estimate of u ǫ n , impliesthat E h sup t ≥ τ R (cid:13)(cid:13)(cid:13) Z t i S ( t − s ) g ( | u ǫ n | ) u ǫ n dW ( s ) − V (cid:13)(cid:13)(cid:13) i = E h sup t ∈ [0 ,T ] I Ω cR (cid:13)(cid:13)(cid:13) Z t S ( t − s ) g ( | u ǫ n | ) u ǫ n dW ( s ) − V (cid:13)(cid:13)(cid:13) i ≤ C ( u , Q, T, p ) R − p . Taking R = O ( | log( ǫ n ) | κ p ) , κ < κ, we have thatlim n →∞ E h sup t ∈ [0 ,T ] (cid:13)(cid:13)(cid:13) Z t i S ( t − s ) g ( | u ǫ n | ) u ǫ n dW ( s ) − V (cid:13)(cid:13)(cid:13) i = 0 . Combining the above estimates and the strong convergence of u ǫ n , we complete the proof. ✷ Remark 1.
One may extend the scope of e g to an abstract framework by similar arguments. Herethe assumption d = 1 lies on the fact that in H is an algebra by Sololev embedding theorem.When considering the case d ≥ , one may use H s , s > d as the underlying space for the localwell-posedness. However, as stated in Lemma 6, it seems impossible to get the uniform bound of u ǫ in H s for s ≥ .
4. Appendix
The original problem and regularized problem can be rewritten into the equivalent evolutionforms du = Audt + F ( u ) dt + G ( u ) dW ( t ) , (9) u (0) = u , where A = i ∆ , F is the Nemystkii operator of drift coefficient function, and G are the Nemystkiioperator of diffusion coefficient function. Then the mild solution of the above evolution is definedas follows. 26 efinition 1. A continuous H -valued F t adapted process u is a solution to (9) if it satisfies P -a.s for all t ∈ [0 , T ] ,u ( t ) = S ( t ) u + Z t S ( t − s ) F ( u ( s )) ds + Z t G ( u ( s )) dW ( s ) , where S ( t ) is the C -group generated by A. Definition 2.
A local mild solution of (9) is ( u, τ ) := ( u, τ n , τ ) satisfying τ n ր τ, a.s., as n → ∞ , u ∈ M p F (Ω; C ([0 , τ ); H s ) , s > , p ≥ and that u ( t ) = S ( t ) u + Z t S ( t − s ) F ( u ( s )) ds + Z t S ( t − s ) G ( u ( s )) dW ( s ) , a.s., for t ≤ τ n in H for n ∈ N + . Solutions of (9) are called unique, if P (cid:16) u ( t ) = u ( t ) , ∀ t ∈ [0 , σ ∧ σ ) (cid:17) = 0 . for all local mild solution ( u , σ ) and ( u , σ ) . The local solution ( u, τ ) is called a global mildsolution if τ = T, a.s. and u ∈ M p F (Ω; C ([0 , T ]; H s ) . Lemma 7.
Let ǫ ∈ (0 , . Then f ǫ ( x ) = log( | x | + ǫ ) , x ∈ C , satisfies | Im [( f ǫ ( x ) − f ǫ ( x ))(¯ x − ¯ x )] | ≤ | x − x | , Proof
Without loss of generality, we assume that 0 < | x | ≤ | x | . Notice that Im [( f ǫ ( x ) − f ǫ ( x ))(¯ x − ¯ x )] = 12 (log( ǫ + | x | ) − log( ǫ + | x | )) Im (¯ x x − ¯ x x ) . Direct calculation yields that | Im (¯ x x − ¯ x x ) | ≤ | x || x − x | . Using the fact that | log( ǫ + | x | ) − log( ǫ + | x | ) | = 2 | log(( ǫ + | x | ) ) − log(( ǫ + | x | ) ) | , we obtain | Im [( f ǫ ( x ) − f ǫ ( x ))(¯ x − ¯ x )] |≤ | log(( ǫ + | x | ) ) − log(( ǫ + | x | ) ) || x || x − x | . The mean value theorem leads to the desired result. ✷ Lemma 8.
Let ǫ ∈ (0 , . Then f ǫ ( | x | ) = log( | x | + ǫ ǫ | x | ) satisfies the following properties, | f ǫ ( | x | ) | ≤ | log( ǫ ) | , | d | x | f ǫ ( | x | ) | ≤ − ǫ ) | x | ( ǫ + | x | )(1 + ǫ | x | ) , | Im [( f ǫ ( | x | ) x − f ǫ ( | x | ) x )(¯ x − ¯ x )] | ≤ − ǫ ) | x − x | . roof The proof of first and second estimates are derived by the property of log( · ) . The lastestimate is proven by similar arguments in the proof of Lemma 7. ✷ [Proof of Proposition 5] Due to Lemma 3, it suffices to prove E h sup t ∈ [0 ,T ] ( F ǫ ( | u ǫ ( t ) | )) p i ≤ C ( u , T, Q, p ) . Let us take f ǫ ( | x | ) = log( | x | + ǫ ) as an example to illustrate the procedures. The desirableestimate in case that f ǫ ( | x | ) = log( | x | + ǫ ǫ | x | ) can be obtained similarly. Using the property oflogarithmic function, we have that for small η > , | F ǫ ( | u ǫ ( t ) | ) | = (cid:12)(cid:12)(cid:12) Z O (cid:16) ( ǫ + | u ǫ ( t ) | ) log( ǫ + | u ǫ ( t ) | ) − | u ǫ ( t ) | − ǫ log( ǫ ) (cid:17) dx (cid:12)(cid:12)(cid:12) ≤ k u ǫ ( t ) k + Z O | u ǫ ( t ) | log( ǫ + | u ǫ ( t ) | ) dx + (cid:12)(cid:12)(cid:12) Z O ǫ (log( ǫ + | u ǫ ( t ) | ) − log( ǫ )) dx (cid:12)(cid:12)(cid:12) ≤ k u ǫ ( t ) k + k u ǫ ( t ) k − ηL − η + k u ǫ ( t ) k ηL η , where we use the following estimation, for any small enough η > Z O | u ǫ ( t ) | log( ǫ + | u ǫ ( t ) | ) dx = Z ǫ + | u ǫ | ≥ f ǫ ( | u ǫ | ) | u ǫ | dx + Z ǫ + | u ǫ | ≤ f ǫ ( | u ǫ | ) | u ǫ | dx ≤ Z ǫ + | u ǫ | ≥ ( ǫ + | u ǫ | ) η | u ǫ | dx + Z ǫ + | u ǫ | ≤ ( ǫ + | u ǫ | ) − η | u ǫ | dx ≤ k u k − ηL − η + k u k ηL η . Then by the Gagliardo–Nirenberg interpolation inequality in a bounded domain O , i.e., k u k L q ( O ) ≤ C k u k − α k∇ u k α + C k u k , q = 2 dd − α , for α ∈ (0 , , we have that Z O F ǫ ( | u ǫ ( s ) | ) dx ≤ | Z O (cid:16) ( ǫ + | u ǫ | ) log( ǫ + | u ǫ | ) − | u ǫ | − ǫ log ǫ (cid:17) dx |≤ C (1 + k u ǫ k ηL η ) ≤ C ( k u ǫ k + k∇ u ǫ k dη η +2 k u ǫ k − dη η +2 ) . Taking p th moment and applying Lemma 3, we complete the proof for the case that O is abounded domain.When O = R d , we need to control k u ǫ k L − η . By using the weighted interpolation inequalityin Lemma 5 with α > dη − η , α ∈ (0 , k u k L q ( R d ) ≤ C k u k − α k∇ u k α , q = 2 dd − α , for α ∈ (0 , , where q = 2 η + 2 . Based on Lemma 3 and Lemma 3, we complete the proof by using the Younginequality and taking p th moment. ✷ [Sketch Proof of Lemma 6] Due to the loss of the regularity of the solution in time, wecan not establish the bound in H through ∂u ǫ ∂t like in the deterministic case. According to28emma 3, it suffices to bound k ∆ u ǫ k . We present the procedures of the estimation of E [ k ∆ u k ]for the conservative multiplicative noise case. One can easily follow the procedures to obtain theestimate of E [sup t ∈ [0 ,τ ] k ∆ u ( t ) k ] for both additive and multiplicative noises.By using the Itˆo formula to k ∆ u ǫ k we obtain that k ∆ u ǫ ( t ) k = k ∆ u ǫ k + 2 Z t h ∆ u ǫ ( s ) , II det i ds + 2 Z t h ∆ u ǫ ( s ) , II mod i ds + 2 II Sto , where II Sto := Z t h ∆ u ǫ , i g ( | u ǫ | ) ∇ u ǫ ∇ dW ( s ) i + Z t h ∆ u ǫ , i g ( | u ǫ | ) u ǫ ∆ dW ( s ) i + Z t h ∆ u ǫ , i g ′ ( | u ǫ | ) Re (¯ u ǫ ∇ u ǫ ) ∇ u ǫ dW ( s ) + Z t h ∆ u ǫ , i g ′ ( | u ǫ | ) Re (¯ u ǫ ∇ u ǫ ) u ǫ ∇ dW ( s ) i + Z t h ∆ u ǫ , i g ′′ ( | u ǫ | )( Re (¯ u ǫ ∇ u ǫ )) u ǫ dW ( s ) + Z t h ∆ u ǫ , i g ′ ( | u ǫ | )( Re (¯ u ǫ ∆ u ǫ )) u ǫ dW ( s ) i + Z t h ∆ u ǫ , g ′ ( | u ǫ | ) |∇ u | u ǫ dW ( s ) i ,II det := i ∆ u ǫ + i λf ( | u ǫ | )∆ u ǫ dt + i λf ′ ( | u ǫ | ) Re (¯ u ǫ ∇ u ǫ ) ∇ u ǫ + i λf ′′ ( | u ǫ | )( Re (¯ u ǫ ∇ u ǫ )) u ǫ + i λf ′ ( | u ǫ | ) Re (¯ u ǫ ∆ u ǫ ) u ǫ , and II mod is the summation of all terms involving the second derivative of the Itˆo modifiedterm produced by the Stratonovich integral. Here for simplicity, we omit the presentation of theexplicit form for II Stra . Taking expectation and using the Gagliardo–Nirenberg interpolation inequality k∇ v k L ≤ C k ∆ v k k∇ v k in d = 1, we obtain that E h k ∆ u ǫ ( t ) k i ≤ E h k ∆ u ǫ (0) k i + C ( λ, p ) ǫ − E h Z t k ∆ u ǫ ( r ) k (1 + k∇ u ǫ k L ) dr i + C ( λ, p ) E h Z t k ∆ u ǫ ( r ) k (cid:16) k X i ∆ Q e i Q e i ( g ( | u ǫ | )) u ǫ k + k X i |∇ Q e i | ( g ( | u ǫ | )) u ǫ k + k X i ∇ Q e i Q e i g ( | u ǫ | ) g ′ ( | u ǫ | ) | u ǫ | ∇ u ǫ k + k X i | Q e i | g ( | u ǫ | ) g ′ ( | u ǫ | ) |∇ u ǫ | u ǫ k + k X i | Q e i | ( g ( | u ǫ | ) g ′′ ( | u ǫ | ) + ( g ′ ( | u ǫ | )) ) |∇ u ǫ | | u ǫ | k + k X i ∇ Q e i Q e i ( g ( | u ǫ | )) ∇ u ǫ k (cid:17) dr i + C ( λ, p ) E h Z t X i ∈ N (cid:16) k g ( | u ǫ | ) ∇ u ǫ ∇ Q e i k + k g ( | u ǫ | ) u ǫ ∆ Q e i k + k g ′ ( | u ǫ | ) |∇ u ǫ | u ǫ Q e i k + k g ′ ( | u ǫ | ) ∇ u ǫ | u ǫ | Q e i k + k g ′′ ( | u ǫ | ) |∇ u ǫ | | u ǫ | Q e i k + k g ′ ( | u ǫ | ) |∇ u ǫ | u ǫ Q e i k (cid:17) dr i E h k ∆ u ǫ (0) k i + C ( λ, p ) ǫ − E h Z t k ∆ u ǫ ( r ) k (1 + k∇ u ǫ k L ) dr i + C ( λ, p ) E h Z t k ∆ u ǫ ( r ) k A ( r ) dr i + C ( λ, p ) E h Z t B ( r ) dr i . Now applying the H¨older inequality, using the properties of g , using the Gagliardo–Nirenberginterpolation inequality, we obtain that for a small η > ,A ( r ) ≤ X i k ∆ Q e i kk Q e i k L ∞ k ( g ( | u ǫ | )) u ǫ k L ∞ + X i k∇ Q e i k L k ( g ( | u ǫ | )) u ǫ k L ∞ + X i k∇ Q e i k L k Q e i k L ∞ k g ( | u ǫ | ) k L ∞ k∇ u ǫ k L + X i k Q e i k L ∞ k∇ u ǫ k L k g ( | u ǫ | ) g ′ ( | u ǫ | ) u ǫ k L ∞ + X i k Q e i k L ∞ k∇ u ǫ k L k ( g ( | u ǫ | ) g ′′ ( | u ǫ | ) + ( g ′ ( | u ǫ | )) ) | u ǫ | k L ∞ + X i k∇ Q e i k L k Q e i k ∞ k g ( | u ǫ | ) k L ∞ k∇ u ǫ k L ≤ X i (cid:16) k∇ Q e i k L + k∇ Q e i k L ∞ + k ∆ Q e i k + k Q e i k L ∞ (cid:17)(cid:16) k∇ u k L + k∇ u k L (cid:17) × (cid:16) k ( g ( | u ǫ | )) u ǫ k L ∞ + k g ( | u ǫ | ) k L ∞ + k g ( | u ǫ | ) g ′ ( | u ǫ | ) u ǫ k L ∞ + k ( g ( | u ǫ | ) g ′′ ( | u ǫ | ) + ( g ′ ( | u ǫ | )) ) | u ǫ | k L ∞ + k g ( | u ǫ | ) k L ∞ (cid:17) ≤ C X i (cid:16) k∇ Q e i k L + k∇ Q e i k L ∞ + k ∆ Q e i k + k Q e i k L ∞ (cid:17)(cid:16) k∇ u ǫ k L + k∇ u ǫ k L (cid:17) . Similarly, we have that for a small η > B ( r ) ≤ X i (cid:16) k g ( | u ǫ | ) k L ∞ k∇ u ǫ k L k∇ Q e i k L + k g ( | u ǫ | ) u ǫ k L ∞ k ∆ Q e i k + k g ′ ( | u ǫ | ) u ǫ k L ∞ k∇ u ǫ k L k Q e i k L ∞ + k g ′ ( | u ǫ | ) | u ǫ | k L ∞ k∇ u ǫ k L k Q e i k L + k g ′′ ( | u ǫ | ) | u ǫ | k L ∞ k∇ u ǫ k L k Q e i k L ∞ + k g ′ ( | u ǫ | ) u ǫ k L ∞ k∇ u ǫ k L k Q e i k L ∞ (cid:17) ≤ C X i (cid:16) k∇ Q e i k L + k∇ Q e i k L ∞ + k ∆ Q e i k + k Q e i k L ∞ (cid:17)(cid:16) k∇ u ǫ k L (cid:17) . Combining the above estimates, and using the Young inequality and Gronwall inequality implythat E h k ∆ u ǫ ( t ) k i ≤ C ( u , T, Q, p, η )(1 + ǫ − ) . Now, taking supreme over t , then taking expectation, and applying the Burkerholder inequalityto the III sto dW ( t ) , we achieve that for sufficient small η > , E h sup t ∈ [0 ,τ ] k ∆ u ǫ ( t ) k i ≤ C ( u , T, Q, η )(1 + ǫ − ) . ✷ [Proof of Proposition 4] We follow the steps in the proof of Proposition 3 to present theproof in the case of p = 2. For convenience, we present the proof for the multiplicative noise30ase. Applying the Itˆo formula to k u ǫ k L α = R R d (1 + | x | ) α | u ǫ | dx , using integration by parts,then taking supreme over t , and applying Burkerholder inequality, we deduce that E h sup t ∈ [0 ,T ] k u ǫ ( t ) k L α i ≤ E h k u k L α i + 2 α E h Z T (cid:12)(cid:12)(cid:12) h (1 + | x | ) α − xu ǫ ( s ) , ∇ u ǫ i (cid:12)(cid:12)(cid:12) ds i + E h sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12) Z t h (1 + | x | ) α u ǫ ( s ) , i g ( | u ǫ ( s ) | ) u ǫ ( s ) dW ( s ) i (cid:12)(cid:12)(cid:12)i ≤ E h k u k L α i + C α E h Z T (cid:12)(cid:12)(cid:12) h (1 + | x | ) α − xu ǫ ( s ) , ∇ u ǫ i (cid:12)(cid:12)(cid:12) ds i + C E h Z T X i ∈ N + (cid:12)(cid:12)(cid:12) h (1 + | x | ) α u ǫ ( s ) , i g ( | u ǫ ( s ) | ) u ǫ ( s ) Q e i i (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ds i . By H¨older’s inequality, for α ∈ (1 , (cid:12)(cid:12)(cid:12) h (1 + | x | ) α − xu ǫ , ∇ u ǫ i (cid:12)(cid:12)(cid:12) ≤ C k u ǫ k L α k (1 + | x | ) α − ∇ u ǫ k . Integration by parts and H¨older’s inequality yield that for some small η > , k (1 + | x | ) α − ∇ u ǫ k = h (1 + | x | ) α − ∇ u ǫ , ∇ u ǫ i = −h (1 + | x | ) α − u ǫ , ∆ u ǫ i − α − h (1 + | x | ) α − x ∇ u ǫ , u ǫ i≤ k u ǫ k L α − , k ∆ u ǫ k + C ( η ) | α − |k u ǫ k L α + η | α − |k∇ u ǫ k L α − , . Combining the above estimates in Proposition 3 and using Young’s inequality, we achieve that E h sup t ∈ [0 ,T ] k u ǫ ( t ) k L α i ≤ e CT (1 + ǫ − ) , if α ∈ (1 ,
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