Ergodicity of Stochastic Burgers’ System with Dissipative Term

Abstract

A 2-dimensional stochastic Burgers equation with dissipative term perturbed by Wiener noise is considered. The aim is to prove the well-posedness, existence, and uniqueness of invariant measure as well as strong law of large numbers and convergence to equilibrium.

1. Introduction

The paper is concerned with the 2-dimensional Burgers equation in a bounded domain with Wiener noise as the body forces like this (()𝑢)[]𝑢()[]𝑢()𝑑𝑢=𝜈Δ𝑢+𝑢⋅∇𝑑𝑡+𝑑𝑊,on0,𝑇×𝐷,𝑡,𝑥=0,𝑡∈0,𝑇,𝑥∈𝜕𝐷,0,𝑥=𝑢₀(𝑥),𝑥∈𝐷, (1) where 𝑢(𝑡,𝑥)=(𝑢¹(𝑡,𝑥),𝑢²(𝑡,𝑥)) is the velocity field, 𝜈>0 is viscid coefficient, Δ denotes the Laplace operator, ∇ represents the gradient operator, 𝑊 stands for the 𝑄-Wiener process, and 𝐷 is a regular bounded open domain of ℝ². Burgers equation has received an extensive amount of attention since the studies by Burgers in the 1940s (and it has been considered even earlier by Beteman [1] and Forsyth [2]). But it is well known that the Burgers’ equation is not a good model for turbulence since it does not perform any chaos. Even if a force is added to equation, all solutions will converge to a unique stationary solution as time goes to infinity. However, if the force is a random one, the result is completely different. So, several authors have indeed suggested to use the stochastic Burgers’ equation to model turbulence, see [3–6]. The stochastic equation has also been proposed in [7] to study the dynamics of interfaces.

So far, most of the monographs concerning the equation focus on one-dimensional case, for example, Bertini et al. [8] solved the equation with additive space-time white noise by an adaptation of the Hopf-cole transformation. Da Prato et al. [9] studied the equation via a different approach based on semigroup property for the heat equation on a bounded interval. The more general equation with multiplicative noise was considered by Da Prato and Debussche [10]. With a similar method, Gyöngy and Nualart [11] extended the Burgers equation from bounded interval to real line. A large deviation principle for the solution was obtained by Gourcy [12]. Concerning the ergodicity, an important paper by Weinan et al. [13] proved that there exists a unique stationary distribution for the solutions of the random inviscid Burgers equation, and typical solutions are piecewise smooth with a finite number of jump discontinuities corresponding to shocks. For model with jumps, Dong and Xu [14] proved that the global existence and uniqueness of the strong, weak, and mild solutions for a one-dimensional Burgers equation perturbed by Lévy noise. When the noise is fractal, Wang et al. [15] get the well-posedness.

The main aim in our paper is to study the large time behavior of stochastic system. There are lots of the literature about the topic (see [16–20]).

Burgers system is a well-known model for mechanics problems. But as far as we know, there are no results about the long-term behavior of stochastic Burgers’ system. We think that the difficulty lies in the fact that the dissipative term Δ𝑢 cannot dominate the nonlinear term (𝑢⋅∇)𝑢. However, in many practical cases, we cannot ignore the energy dissipation and external forces, especially considering the long-term behavior. Therefore, we introduce dissipative term 𝑓(𝑢) and study the ergodicity of the following equation: ()(𝑢[]𝑢()[]𝑢()𝑑𝑢=Δ𝑢+𝑢⋅∇𝑢−𝑓)𝑑𝑡+𝑑𝑊,on0,𝑇×𝐷,𝑡,𝑥=0,𝑡∈0,𝑇,𝑥∈𝜕𝐷,0,𝑥=𝑢₀(𝑥),𝑥∈𝐷, (2) where 𝑓(𝑢)=𝜗|𝑢(𝑡,𝑥)|²𝑢(𝑡,𝑥),  𝜗>0,|⋅| denote the absolute value or norm for the real number or two-dimensional vector, respectively.

We believe that our work is new and is worth researching. The methods and results in this paper can be applied to stochastic reaction diffusion equations and stochastic real valued Ginzburg Landau equation in high dimensions. But we cannot extend our result to dynamical systems with state-delays. Since in order to show the existence of an invariant measure, we should consider the segments of a solution. In contrast to the scalar solution process, the process of segments is a Markov process. We show that the process of segments is also Feller and that there exists a solution of which the segments are tight. Then, we apply the Krylov-Bogoliubov method. Since the segment process has values in the infinite-dimensional space 𝐶([−𝑟,0],𝐻), boundedness in probability does not automatically imply tightness. For solution processes of infinite-dimensional equations, one often uses compactness of the orbits of the underlying deterministic equation to obtain tightness. For an infinite-dimensional formulation of the functional differential equation, however, such a compactness property does not hold. For ergodicity of stochastic delay equations, we can see [21]. We believe that stochastic Burgers’ system with state-delays is a very interesting problem.

In order to study ergodicity of problem (2), we use a remarkable dissipativity property of the stochastic dynamic to obtain the existence of the invariant measure. For uniqueness, we try to use the method from [22] to prove that the distributions 𝑃(𝑡,𝑥,⋅) induced by the solution are equivalent. It is well known that the equivalence of the distributions implies uniqueness, a strong law of large numbers, and the convergence to equilibrium.

The remaining of this paper is organized as follows. Some preliminaries are presented in Section 2, the local existence and global existence are presented, respectively, in Sections 3 and 4. In Section 5, we obtain the existence and uniqueness of the invariant measure as well as strong law of large numbers, and convergence to equilibrium. As usual, constants 𝐶 may change from one line to the next; we denote by 𝐶_𝑎 a constant which depends on some parameter 𝑎.

2. Preliminaries on the Burgers Equation

Let 𝑢(𝑡,𝑥)=(𝑢¹(𝑡,𝑥),𝑢²(𝑡,𝑥)) be a row vector valued function on [0,∞)×ℝ². And it denotes the following: |𝑢|²∶=²_𝑖=1𝑢^𝑖2,𝜕_𝑖𝑢^𝑗∶=𝜕𝑢^𝑗𝜕𝑥_𝑖,𝑖,𝑗=1,2. (3) Let [𝐶^∞(𝐷)]² be infinitely differentiable 2-dimensional vector field on 𝐷, and let [𝐶^∞₀(𝐷)]² be infinitely differentiable 2-dimensional vector field with compact support strictly contained in 𝐷. We denote by 𝐻^𝛼 the closure of [𝐶^∞(𝐷)]² in [𝐻^𝛼(𝐷)]², whose norms are denoted by ‖⋅‖_𝐻^𝛼, when 𝛼≠0. Let 𝐻¹₀,𝐻 be the closure of [𝐶^∞₀(𝐷)]² in [𝐻¹(𝐷)]² and [𝐿²(𝐷)]² whose norms are denoted by ‖⋅‖_𝐻¹ and ‖⋅‖_𝐻, respectively. Without confusion, set ⟨⋅,⋅⟩ as the inner product in 𝐻 or 𝐿²(𝐷). For 𝑝>0, let ‖⋅‖_𝐿^𝑝 be the norm of vector filed in Lebesgue spaces [𝐿^𝑝(𝐷)]². |⋅|_𝐻^𝛼 represents the norm in the usual sobolev spaces 𝐻^𝛼(𝐷) for real valued functions on 𝐷 and 𝛼∈ℝ; |⋅|_𝐿^𝑝 stands for the norm in the usual Lebesgue spaces 𝐿^𝑝(𝐷) for real valued functions on 𝐷. Denote 𝐴∶=−Δ; then 𝐴∶𝐷(𝐴)⊂𝐻→𝐻 and 𝐷(𝐴)=[𝐻²(𝐷)]²∩𝐻¹₀. Since 𝐻¹₀ coincides with 𝐷(𝐴^1/2), we can endow 𝐻¹₀ with the norm ‖𝑢‖_𝐻¹=‖𝐴^1/2𝑢‖_𝐻. The operator 𝐴 is positive self-adjoint with compact resolvent; we denote by 0<𝛼₁≤𝛼₂≤⋯ the eigenvalues of 𝐴, and by 𝑒₁,𝑒₂,… the eigenvectors which is a corresponding complete orthonormal system in 𝐻 satisfying (i)𝑒_𝑖∈𝐶^∞₀(𝐷)²,(ii)𝑒_𝑖(𝑥)≤𝐶,∇𝑒_𝑖(𝑥)≤𝐶√𝛼_𝑖,𝑥∈𝐷,𝑖=1,2,…., (4) for some positive constant C. We remark that ‖𝑢‖²_𝐻1≥𝛼₁‖𝑢‖²_𝐻. We define the bilinear operator 𝐵(𝑢,𝑣)∶𝐻¹×𝐻¹→𝐻⁻¹ as ⟨𝐵(𝑢,𝑣),𝑧⟩=_𝐷𝑧(𝑥)⋅(𝑢(𝑥)⋅∇)𝑣(𝑥)𝑑𝑥, (5) for all 𝑧∈𝐻¹. Then, (2) is equivalent to the following abstract equation: 𝑑𝑢(𝑡)+𝐴𝑢(𝑡)+𝐵(𝑢(𝑡),𝑢(𝑡))+𝑓(𝑢(𝑡))𝑑𝑡=𝑑𝑊(𝑡). (6)𝑊 is the 𝑄 Wiener process having the following representative: 𝑊(𝑡)=^∞_𝑛=1𝜆_𝑛𝑒_𝑛𝛽_𝑛(𝑡),𝑡∈[0,𝑇], (7) in which ∑^∞_𝑛=1𝜆_𝑛<∞ and 𝛽_𝑘 are a sequence of mutually independent 1-dimensional Brownian motions in a fixed probability space (Ω,ℱ,𝑃) adapted to a filtration {ℱ_𝑡}_𝑡≥0.

It can be derived from [23] that the solution to the linear problem corresponding to (2) with the following initial condition: 𝑢()[]𝑢()𝑑𝑢=Δ𝑢𝑑𝑡+𝑑𝑊,𝑡,𝑥=0,𝑡∈0,𝑇,𝑥∈𝜕𝐷,0,𝑥=𝑢₀(𝑥),𝑥∈𝐷, (8) is unique, and when 𝑢₀=0, it has the form of 𝑊_𝐴(𝑡)=^𝑡₀𝑒^{(𝑡−𝑠)𝐴}𝑑𝑊(𝑠). (9) Let 𝑣(𝑡)=𝑢(𝑡)−𝑊_𝐴(𝑡),𝑡≥0, (10) then 𝑢 is a solution to (2) if and only if it solves the following evolution equation: 𝜕𝑣𝑑𝑡+𝐴𝑣+𝐵𝑣+𝑊_𝐴,𝑣+𝑊_𝐴+𝑓𝑣+𝑊_𝐴𝑣()[]𝑣(0)=0,𝑡,𝑥=0,𝑡∈0,𝑇,𝑥∈𝜕𝐷,=𝑢₀. (11) So, we see that when 𝑤∈Ω is fixed, this equation is in fact a deterministic equation. From now on, we will study the equation of the form (11) to get the existence and uniqueness of the solution a.s. 𝑤∈Ω.

3. Local Existence in Time

Definition 1 (see Definition 5.1.1 in [24]). We say a (ℱ(𝑡))_𝑡≥0 adapted process 𝑣(𝑡) is a mild solution to (11), if 𝑣(𝑡)∈𝐶([0,𝑇];𝐻¹₀) and it satisfies 𝑣(𝑡)=𝑒^𝑡𝐴𝑣₀+^𝑡₀𝑒^{(𝑡−𝑠)𝐴}𝐵𝑣+𝑊_𝐴,𝑣+𝑊_𝐴−𝑑𝑠_𝑡0𝑒^{(𝑡−𝑠)𝐴}𝑓𝑣+𝑊_𝐴𝑑𝑠,𝑡∈[0,𝑇]. (12)

Lemma 2. For any 𝜃∈(0,1), if ∑^∞_𝑖=1𝜆_𝑖(𝛼_𝑖)^𝜃<∞, then 𝐴^1/2𝑊_𝐴 has a version which is 𝛼-Hölder continuous with respect to 𝑡∈[0,𝑇],  𝑥∈𝐷 with any 𝛼∈]0,𝜃/2[.

Proof. Let 𝑇>0 and 𝑠,𝑡∈[0,𝑇]; then 𝐸𝐴^1/2𝑊_𝐴(𝑡,𝑥)−𝐴^1/2𝑊_𝐴(𝑠,𝑥)²=^∞_𝑖=1𝜆_𝑖^𝑡_𝑠𝐴^1/2𝑆(𝑡−𝜏)𝑒_𝑖(𝑥)²+𝑑𝑠_∞𝑖=1𝜆_𝑖^𝑠₀𝐴^1/2[𝑆(𝑡−𝜏)−𝑆(𝑠−𝜏)]𝑒_𝑖(𝑥)²𝑑𝜏=∶𝐼₁(𝑡,𝑠,𝑥)+𝐼₂(𝑡,𝑠,𝑥). (13) Then, we have 𝐼₁(𝑡,𝑠,𝑥)≤𝐶^∞_𝑖=1𝜆_𝑖𝛼_𝑖^𝑡_𝑠𝑒^{−2(𝑡−𝜏)𝛼𝑖}𝑑𝜏=𝐶^∞_𝑖=1𝜆_𝑖𝛼_𝑖1−𝑒^{−2(𝑡−𝑠)𝛼𝑖}2𝛼_𝑖≤𝐶^∞_𝑖=1𝜆_𝑖𝛼_𝑖^𝜃|𝑡−𝑠|^𝜃,𝐼₂()≤1𝑡,𝑠,𝑥2𝐶^∞_𝑖=1𝜆_𝑖𝛼_𝑖^𝑠₀𝑒^{−(𝑡−𝜏)𝛼𝑖}−𝑒^{−(𝑠−𝜏)𝛼𝑖2}𝑑𝜏=𝐶^∞_𝑖=1𝜆_𝑖𝛼_𝑖12𝛼_𝑖𝑒^{−(𝑡−𝑠)𝛼𝑖}−1²−𝑒^{−𝑡𝛼𝑖}−𝑒^{−𝑠𝛼𝑖2}≤𝐶^∞_𝑖=1𝜆_𝑖𝛼_𝑖^𝜃|𝑡−𝑠|^𝜃. (14) So, by the estimate of 𝐼₁ and 𝐼₂, we arrive at 𝐸𝐴^1/2𝑊_𝐴(𝑡,𝑥)−𝐴^1/2𝑊_𝐴(𝑠,𝑥)²≤𝐶^∞_𝑖=1𝜆_𝑖𝛼_𝑖^𝜃|𝑡−𝑠|^𝜃. (15) For 𝑡∈[0,𝑇],  𝑥,𝑦∈𝐷, we get 𝐸𝐴^1/2𝑊_𝐴(𝑡,𝑥)−𝐴^1/2𝑊_𝐴𝑡,𝑦²=^∞_𝑖=1𝜆_𝑖𝛼_𝑖^𝑡₀𝑒^{−2𝛼𝑖(𝑡−𝑠)}𝑒_𝑖(𝑥)−𝑒_𝑖𝑦²≤𝑑𝑠_∞𝑖=1𝜆_𝑖𝑒_𝑖(𝑥)−𝑒_𝑖𝑦²≤^∞_𝑖=1𝜆_𝑖𝛼_𝑖^𝜃𝑥−𝑦^𝜃. (16) Therefore, 𝐸𝐴^1/2𝑊_𝐴(𝑡,𝑥)−𝐴^1/2𝑊_𝐴𝑠,𝑦²≤𝐶|𝑡−𝑠|^𝜃+𝑥−𝑦^𝜃. (17) As 𝐴^1/2𝑊_𝐴(𝑡,𝑥)−𝐴^1/2𝑊_𝐴(𝑠,𝑦) is a Gaussian random variable, we obtain 𝐸𝐴^1/2𝑊_𝐴(𝑡,𝑥)−𝐴^1/2𝑊_𝐴𝑠,𝑦^2𝑚≤𝐶|𝑡−𝑠|^𝑚𝜃+𝑥−𝑦^𝑚𝜃, (18) for 𝑚=1,2,… By Kolmogorov’ test theorem, we get the conclusion.

Remark 3. An example of the noise satisfying condition of Lemma 2 is 𝑑𝑊(𝑡)=^∞_𝑛=1𝜆_𝑛𝑒_𝑛𝑑𝛽_𝑛(𝑡), (19) where {𝛽_𝑛} is a sequence of independent 1-dimensional Brownian motion, and {𝜆_𝑛} satisfies 𝜆_𝑛=𝑛^−(1+2𝜃),𝛼_𝑛=𝑛∀𝑛∈ℕ. (20) It is so because the eigenvalues 𝛼_𝑛 of the operator 𝐴, in 2-dimensional space, behave like 𝑛.

Remark 4. Another example of stochastic noise satisfying Lemma 2 is 𝐴^−𝛾𝐿𝑑𝑊(𝑡), (21) where 𝑊(𝑡)=∑^∞_𝑛=1𝑒_𝑛𝛽_𝑛(𝑡),  𝐿 is an isomorphism in 𝐻, and 1𝛾≥2+𝜃. (22)

To prove the local existence of the solution of (1) in sense of Definition 1, we introduce the space ℬ_𝑚 defined by ℬ_𝑚=𝑣∶𝑣∈𝐶0,𝑇^∗;𝐻¹₀,‖𝑣‖_𝐻¹≤𝑚,∀𝑡∈0,𝑇^∗, (23) where 𝑇^∗≥0 which in fact is a stopping time and 𝑚>0, 𝑝>0.

Lemma 5. For 𝑢₀=(𝑢¹(0),𝑢²(0)),‖𝑢₀‖_𝐻1<𝑚, and 𝑢^𝑖(0) is adapted to ℱ₀, 𝑖=1,2; then there exists a unique mild solution 𝑣 in sense of Definition 1 to (11) in ℬ_𝑚.

Proof. Choose a 𝑣 in ℬ_𝑚, and set ℒ(𝑣)∶=𝑒^−𝑡𝐴𝑢₀+^𝑡₀𝑒^{−(𝑡−𝑠)𝐴}𝑣+𝑊_𝐴⋅∇𝑣+𝑊_𝐴𝑑𝑠−^𝑡₀𝑒^{−(𝑡−𝑠)𝐴}𝑓𝑣+𝑊_𝐴𝑑𝑠. (24) Then, ‖ℒ(𝑣)‖_𝐻¹≤𝑒^−𝑡𝐴𝑢_0𝐻1+^𝑡₀𝑒^{−(𝑡−𝑠)𝐴}𝑣+𝑊_𝐴⋅∇𝑣+𝑊_𝐴𝑑𝑠_𝐻1+^𝑡₀𝑒^{−(𝑡−𝑠)𝐴}𝑓𝑣+𝑊_𝐴𝑑𝑠_𝐻1. (25) For the second term on the right hand side of (25), 𝑒^{−(𝑡−𝑠)𝐴}𝑣+𝑊_𝐴⋅∇𝑣+𝑊_𝐴𝐻1=𝑒^{−(𝑡−𝑠)𝐴}[𝑢⋅∇]𝑢_𝐻1≤12𝑒^{−(𝑡−𝑠)𝐴}𝜕₁𝑢¹²_𝐻1+12𝑒^{−(𝑡−𝑠)𝐴}𝜕₂𝑢²²_𝐻1+𝑒^{−(𝑡−𝑠)𝐴}𝑢²𝜕₂𝑢¹_𝐻1+𝑒^{−(𝑡−𝑠)𝐴}𝑢¹𝜕₁𝑢²_𝐻1∶=𝐼₁+𝐼₂+𝐼₃+𝐼₄. (26) In the following, we will estimate 𝐼_𝑖, respectively, 𝑖=1,2,3,4. Since {𝑒^−𝑡𝐴}_𝑡≥0 is contraction on 𝐿^𝑝(𝐷),𝑝≥1, it is known that 𝑒^−𝑡𝐴𝑧_{𝑊𝑠2,𝑟}≤𝐶₁𝑡^{(𝑠1−𝑠2)/2}|𝑧|_𝑊^𝑠₁^,𝑟, (27) for all 𝑧∈𝑊^𝑠1,𝑟(𝐷),  𝑠₁,𝑠₂∈ℝ,  𝑠₁≤𝑠₂,  𝑟≥1, and 𝐶₁ only depends on 𝑠₁,  𝑠₂, and 𝑟. Before calculating each 𝐼_𝑖, we outline the Sobolev embedding principle in fractional Sobolev spaces as follows: 𝑊^𝜂1,𝑝1(𝐷)⊂𝑊^𝜂2,𝑞1(𝐷), (28) when 1𝑝₁−1𝑛𝜂₁−𝜂₂≤1𝑞₁≤1𝑝₁, (29) where 𝑛 is the dimension of the spatial. Let 𝜂₁=3/4,  𝑝₁=2,  𝜂₂=1/4, 𝑞₁=4 satisfying (29) such that 𝑊^3/4,2(𝐷)⊂𝑊^1/4,4(𝐷). (30)
For 𝐼₁, by (27) and Theorem A.8 in [25], we get 𝐼₁≤𝐶₁|𝑡−𝑠|^−7/8𝜕₁𝑢¹²_𝐻−3/4=𝐶₁|𝑡−𝑠|^−7/8𝐴^1/8𝑢¹²_𝐻=𝐶₁|𝑡−𝑠|^−7/82𝑢¹𝐴^1/8𝑢¹+𝑅_𝐻, (31) where 𝑅=𝐴^1/8𝑢¹²−2𝐴^1/8𝑢¹, (32) satisfying |𝑅|_𝐻≤𝐴^1/16𝑢¹²_𝐿4≤𝑢¹²_𝐻1. (33) The last inequality follows by (30). For the other term added to 𝑅, we have 2𝑢¹𝐴^1/8𝑢¹_𝐻≤𝑢₁²_𝐿4+𝐴^1/8𝑢¹²_𝐿4≤2𝑢¹²_𝐻1. (34) So, by (31)–(34), we have 𝐼₁≤3𝐶₁|𝑡−𝑠|^−7/8𝑢¹²_𝐻1. (35) Similarly, we get for 𝐼₂ that 𝐼₂≤3𝐶₁|𝑡−𝑠|^−7/8𝑢²²_𝐻1. (36) For 𝐼₃, by Theorem A.8 in [25], we get 𝐼₃≤𝑒^{−(𝑡−𝑠)𝐴}𝑢²𝐴^1/2𝑢¹_𝐻1=𝑒^{−(𝑡−𝑠)𝐴}𝐴^1/4𝑢²𝐴^1/4𝑢¹−𝐴^1/4𝑢¹𝐴^1/4𝑢²−𝑅_1𝐻1, (37) where 𝑅₁=𝐴^1/4𝑢²𝐴^1/4𝑢¹−𝐴^1/4𝑢²𝐴^1/4𝑢¹−𝑢²𝐴^1/2𝑢¹. (38)
For 𝑅₁, we have 𝑒^{−(𝑡−𝑠)𝐴}𝑅_1𝐻1≤𝐶₁|𝑡−𝑠|^−1/2𝑅_1𝐻≤𝐶₁|𝑡−𝑠|^−1/2𝐴^1/4𝑢¹_𝐿4⋅𝐴^1/4𝑢²_𝐿4≤𝐶₁|𝑡−𝑠|^−1/2𝑢¹²_𝐻1+𝑢²²_𝐻1. (39) For the first term on the right hand side of (37), by (27), we have 𝑒^{−(𝑡−𝑠)𝐴}𝐴^1/4𝑢²𝐴^1/4𝑢¹_𝐻1=𝑒^{−(𝑡−𝑠)𝐴}𝐴^3/4(𝑢²𝐴^1/4𝑢¹)_𝐻≤𝐶₁|𝑡−𝑠|^−3/4𝑢²𝐴^1/4𝑢¹_𝐻≤𝐶₁|𝑡−𝑠|^−3/4𝑢²²_𝐿4+𝐴^1/4𝑢¹²_𝐿4≤𝐶₁|𝑡−𝑠|^−3/4𝑢²²_𝐻1+𝑢¹²_𝐻1. (40) For the second term on the right hand side of (37), by (27), we obtain 𝑒^{−(𝑡−𝑠)𝐴}[𝐴^1/4𝑢²⋅𝐴^1/4𝑢¹]_𝐻1≤𝐶₁|𝑡−𝑠|^−1/2𝐴^1/4𝑢²⋅𝐴^1/4𝑢¹_𝐻≤𝐶₁|𝑡−𝑠|^−1/2𝑢¹²_𝐻1+𝑢²²_𝐻1. (41) From (37) to (41), we get for 𝐼₃ that 𝐼₃≤𝐶|𝑡−𝑠|^−1/2+|𝑡−𝑠|^−3/4𝑢¹²_𝐻1+𝑢²²_𝐻1. (42) Analogously, for 𝐼₄, we get 𝐼₄≤𝐶|𝑡−𝑠|^−1/2+|𝑡−𝑠|^−3/4𝑢¹²_𝐻1+𝑢²²_𝐻1. (43) By (26), (35), (36), (42), and (43), we have 𝑒^{−(𝑡−𝑠)𝐴}𝑣+𝑊_𝐴⋅∇𝑣+𝑊_𝐴𝐻1≤𝐶|𝑡−𝑠|^−1/2+|𝑡−𝑠|^−3/4+|𝑡−𝑠|^−7/8×𝑢¹²_𝐻1+𝑢²²_𝐻1. (44) As 𝑢=𝑣+𝑊_𝐴, by (44), for 𝑡≤𝑇^∗, we have 𝑑𝑠𝑒^𝑡₀^{−(𝑡−𝑠)𝐴}𝑣+𝑊_𝐴⋅∇𝑣+𝑊_𝐴𝐻1≤𝐶𝑡^1/8+𝑡^1/4+𝑡^1/2sup_{𝑡∈0,𝑇∗}‖𝑣‖²_𝐻1+sup_{[]𝑡∈0,𝑇}𝑊_𝐴²_𝐻1. (45) Since by Lemma 2, sup_{𝑡∈[0,𝑇]}𝑊_𝐴²_𝐻1<∞. (46) For the last term on the right hand side of (25), we have 𝑒^{−(𝑡−𝑠)𝐴}𝑓𝑣+𝑊_𝐴𝐻1≤𝐶|𝑡−𝑠|^−1/2𝑣+𝑊_𝐴³_𝐿6≤𝐶|𝑡−𝑠|^−1/2𝑊_𝐴³_𝐻1+‖𝑣‖³_𝐻1. (47) Therefore, ^𝑡₀𝑒^{−(𝑡−𝑠)𝐴}𝑓𝑣+𝑊_𝐴𝑑𝑠_𝐻1≤𝐶1+𝑚^3𝑡₀|𝑡−𝑠|^−1/2𝑑𝑠≤𝐶1+𝑚³𝑇^∗1/2. (48) So by (25), (45), and (48), when 𝑇^∗ is small enough, ‖ℒ(𝑣)‖_𝐻¹≤𝑚. (49) For each 𝑣₁,𝑣₂∈ℬ_𝑚, set 𝑢_𝑖=𝑣_𝑖+𝑊_𝐴, 𝑖=1,2. To simplify the notation in the following calculation, we denote 𝑢_𝑖=(𝑢¹_𝑖,𝑢²_𝑖), 𝑖=1,2. Then, ℒ𝑣₁−ℒ𝑣₂=^𝑡₀𝑒^{−(𝑡−𝑠)𝐴}𝑢₁⋅∇𝑢₁−𝑢₂⋅∇𝑢₂+𝑑𝑠_𝑡0𝑒^{−(𝑡−𝑠)𝐴}𝑓𝑢₁−𝑓𝑢₂𝑑𝑠. (50) So, ℒ𝑣₁−ℒ𝑣_2𝐻1≤^𝑡₀𝑒^{−(𝑡−𝑠)𝐴}𝑢₁⋅∇𝑢₁−𝑢₂⋅∇𝑢_2𝐻1+𝑑𝑠_𝑡0𝑒^{−(𝑡−𝑠)𝐴}𝑓𝑢₁−𝑓𝑢_2𝐻1𝑑𝑠. (51) In order to simplify the notation, we set 𝑢₁⋅∇𝑢₁−𝑢₂⋅∇𝑢₂=𝑓₁+𝑓₂,𝑓₃+𝑓₄, (52) where 𝑓₁=12𝜕₁𝑢¹₁²−𝑢¹₂²,𝑓₂=𝑢²₁𝜕₂𝑢¹₁−𝑢²₂𝜕₂𝑢¹₂,𝑓₃=12𝜕₂𝑢²₁²−𝑢²₂²,𝑓₄=𝑢¹₁𝜕₁𝑢²₁−𝑢¹₂𝜕₁𝑢²₂. (53) Then, we estimate 𝑓_𝑖, 𝑖=1,2,3,4, respectively. For 𝑓₁, we have 𝑒^{−(𝑡−𝑠)𝐴}𝑓_1𝐻1=𝑒^{−(𝑡−𝑠)𝐴}𝐴^1/2𝑢¹₁−𝑢¹₂𝑢¹₁+𝑢¹_2𝐻1≤𝐶|𝑡−𝑠|^−7/8𝐴^1/8𝑢¹₁²−𝑢¹₂²_𝐻=𝐶|𝑡−𝑠|^−7/8𝐴^1/8𝑢¹₁−𝑢²₁𝑢¹₁+𝑢²_1𝐻=𝐶|𝑡−𝑠|^−7/8𝐴^1/8𝑢¹₁−𝑢²₁𝑢¹₁+𝑢²_1𝐻+𝐶|𝑡−𝑠|^−7/8𝐴^1/8𝑢¹₁+𝑢²₁𝑢¹₁−𝑢²₁+𝑅_2𝐻. (54) We first consider 𝑅_2𝐻≤𝐶𝐴^1/16𝑢¹₁−𝑢¹_2𝐿4⋅𝐴^1/16𝑢¹₁+𝑢¹_2𝐿4≤𝐶𝑢¹₁−𝑢¹_2𝐻1⋅𝑢¹₁+𝑢¹_2𝐻1. (55) For the other term added to 𝑅₂, 𝐴^1/8𝑢¹₁+𝑢¹₂𝑢¹₁−𝑢¹_2𝐻≤𝑢¹₁+𝑢¹_2𝐻1⋅𝑢¹₁−𝑢¹_2𝐻1. (56) By (54)–(56), 𝑒^{−(𝑡−𝑠)𝐴}𝑓_1𝐻1≤𝐶|𝑡−𝑠|^−7/8𝑢¹₁−𝑢¹_2𝐻1⋅𝑢¹₁+𝑢¹_2𝐻1. (57) Analogously, for 𝑓₃, 𝑒^{−(𝑡−𝑠)𝐴}𝑓_3𝐻1≤𝐶|𝑡−𝑠|^−7/8𝑢¹₁−𝑢¹_2𝐻1⋅𝑢¹₁+𝑢¹_2𝐻1. (58) For 𝑓₂, by (53), we have 𝑒^{−(𝑡−𝑠)𝐴}𝑓_2𝐻1=𝑒^{−(𝑡−𝑠)𝐴}𝑢²₁𝜕₂𝑢¹₁−𝑢²₂𝜕₂𝑢¹_2𝐻1≤𝑒^{−(𝑡−𝑠)𝐴}𝑢²₁𝜕₂𝑢¹₁−𝜕₂𝑢¹_2𝐻1+𝑒^{−(𝑡−𝑠)𝐴}𝑢²₁−𝑢²₂𝜕₂𝑢¹_2𝐻1. (59) For the first term on the right hand side of (59), we have 𝑒^{−(𝑡−𝑠)𝐴}𝑢²₁𝜕₂𝑢¹₁−𝜕₂𝑢¹_2𝐻1≤𝑒^{−(𝑡−𝑠)𝐴}𝑢²₁𝐴^1/2𝑢¹₁−𝑢¹_2𝐻1=𝑒^{−(𝑡−𝑠)𝐴}𝐴^1/4𝑢²₁𝐴^1/4𝑢¹₁−𝑢¹₂−𝐴^1/4𝑢²₁,𝐴^1/4𝑢¹₁−𝑢¹₂−𝑅_3𝐻1. (60) For 𝑅₃, 𝑒^{−(𝑡−𝑠)𝐴}𝑅_3𝐻1≤|𝑡−𝑠|^−1/2𝑅_3𝐻≤|𝑡−𝑠|^−1/2𝐴^1/4𝑢²_1𝐿4⋅𝐴^1/4𝑢¹₁−𝑢¹_2𝐿4=|𝑡−𝑠|^−1/2𝑢²_1𝐻1⋅𝑢¹₁−𝑢¹_2𝐻1. (61) For the first term on the right hand side of (60), we arrive at 𝑒^{−(𝑡−𝑠)𝐴}𝐴^1/4𝑢²₁𝐴^1/4𝑢¹₁−𝑢¹_2𝐻1=𝑒^{−(𝑡−𝑠)𝐴}𝐴^3/4𝑢²₁𝐴^1/4𝑢¹₁−𝑢¹_2𝐻≤|𝑡−𝑠|^−3/4𝑢²₁𝐴^1/4𝑢¹₁−𝑢¹_2𝐻≤|𝑡−𝑠|^−3/4𝑢²_1𝐿4⋅𝐴^1/4𝑢¹₁−𝑢¹_2𝐿4≤|𝑡−𝑠|^−3/4𝑢²_1𝐻1⋅𝑢¹₁−𝑢¹_2𝐻1. (62) For the second term on the right hand side of (60), we obtain 𝑒^{−(𝑡−𝑠)𝐴}𝐴^1/4𝑢²₁𝐴^1/4𝑢¹₁−𝑢¹_2𝐻1≤|𝑡−𝑠|^−1/2[𝐴^1/4𝑢²₁]⋅𝐴^1/4𝑢¹₁−𝑢¹_2𝐻≤|𝑡−𝑠|^−1/2𝐴^1/4𝑢²_1𝐿4⋅𝐴^1/4𝑢¹₁−𝑢¹_2𝐿4≤|𝑡−𝑠|^−1/2𝑢²_1𝐻1⋅𝑢¹₁−𝑢¹_2𝐻1. (63) By (59)–(63), we get for 𝑓₂ that 𝑒^{(𝑡−𝑠)𝐴}𝑓_2𝐻1≤𝐶|𝑡−𝑠|^−1/2+|𝑡−𝑠|^−3/4×𝑢_1𝐻1+𝑢_2𝐻1𝑢₁−𝑢_2𝐻1. (64) Similarly, we get for 𝑓₄ that 𝑒^{(𝑡−𝑠)𝐴}𝑓_4𝐻1≤𝐶|𝑡−𝑠|^−1/2+|𝑡−𝑠|^−3/4×𝑢_1𝐻1+𝑢_2𝐻1𝑢₁−𝑢_2𝐻1. (65) By (52), (53), (57), (58), (64), and (65), we have 𝑒^{−(𝑡−𝑠)𝐴}𝑢₁⋅∇𝑢₁−𝑢₂⋅∇𝑢_2𝐻1≤⁴_𝑖=1𝑒^{−(𝑡−𝑠)𝐴}𝑓_𝑖𝐻1≤𝐶|𝑡−𝑠|^−1/2+|𝑡−𝑠|^−3/4+|𝑡−𝑠|^−7/8×𝑢_1𝐻1+𝑢_2𝐻1𝑢₁−𝑢_2𝐻1≤𝐶(2𝑚+1)|𝑡−𝑠|^−1/2+|𝑡−𝑠|^−3/4+|𝑡−𝑠|^−7/8×𝑣₁−𝑣_2𝐻1. (66) For the second term on the right hand side of (51), we have 𝑓𝑢₁−𝑓𝑢₂=ℎ₁,ℎ₂, (67) where ℎ₁=𝑢₁²𝑢¹₁−𝑢₂²𝑢¹₂,ℎ₂=𝑢₁²𝑢²₁−𝑢₂²𝑢²₂. (68) Then, 𝑒^{(𝑡−𝑠)𝐴}ℎ_1𝐻1≤𝐶|𝑡−𝑠|^−1/2𝑢₁²𝑢¹₁−𝑢₂²𝑢¹_2𝐻≤𝐶|𝑡−𝑠|^−1/2𝑢₁²𝑢¹₁−𝑢₂²𝑢¹_1𝐻+𝐶|𝑡−𝑠|^−1/2𝑢₂²𝑢¹₁−𝑢₂²𝑢¹_2𝐻≤𝐶|𝑡−𝑠|^−1/2𝑣₁−𝑣₂⋅𝑣₁+𝑣₂+2𝑊_𝐴⋅𝑢¹_1𝐻+𝐶|𝑡−𝑠|^−1/2𝑢₂²⋅𝑣¹₁−𝑣¹_2𝐻≤𝐶|𝑡−𝑠|^−1/2𝑣₁−𝑣_2𝐿4⋅𝑣₁+𝑣₂+2𝑊_𝐴⋅𝑢¹_1𝐿4+𝐶|𝑡−𝑠|^−1/2𝑣₁−𝑣_2𝐿4⋅𝑢₂²_𝐿4≤𝐶|𝑡−𝑠|^−1/2𝑣₁−𝑣_2𝐿4𝑣₁²_𝐿8+𝑣₂²_𝐿8||+1≤𝐶𝑡−𝑠^−1/2𝑣₁−𝑣_2𝐻1𝑣₁²_𝐻1+𝑣₂²_𝐻1+1. (69) Similarly, we can get the same estimate for ℎ₂. So, we have ^𝑡₀𝑒^{(𝑡−𝑠)𝐴}𝑓𝑢₁−𝑓𝑢_2𝐻1𝑑𝑠≤𝐶1+𝑚²𝑇^∗1/2sup_{𝑡∈[0,𝑇∗]}𝑣₁(𝑡)−𝑣₂(𝑡)_𝐻1. (70) By (51), (66), and (70), we have ℒ𝑣₁−ℒ𝑣_2𝐻1≤𝐶𝑇^∗1/2+𝑇^∗1/4+𝑇^∗1/8⋅sup_{𝑡∈0,𝑇∗}𝑣₁−𝑣_2𝐻1. (71) By (49), (71), and fixed point principle, we get the conclusion.

Remark 6. By making some minor modifications in the proof of Lemma 5, we can see that the conclusion in Lemma 5 is also true for (1). Our original aim is to get the global well-posedness of (1), but we find that the dissipative term Δ𝑢 cannot dominate the nonlinear term (𝑢⋅∇)𝑢. So, we introduce the dissipative term |𝑢|²𝑢 which will also play an important role in obtaining the ergodicity.

4. Global Existence

Theorem 7. With conditions in Lemma 2, for 𝑣∈𝐶([0,𝑇];𝐻¹₀) satisfying (12), when 𝜗>1/16, one has ‖𝑣‖_𝐻¹≤𝐶_𝑇+𝑣₀²_𝐻1𝑒^𝐶𝑇. (72) Subsequently, one gets the existence of the global solution belonging to 𝐶([0,𝑇];𝐻¹₀).

Proof. Let {𝑢⁰_𝑛}_𝑛≥1 be a sequence of vectors which satisfies 𝑢⁰_𝑛=(𝑢_𝑛^0,1,𝑢_𝑛^0,2) and 𝑢_𝑛^0,𝑖∈𝐶^∞₀(𝐷),  𝑖=1,2,𝑛≥1, such that 𝑢⁰_𝑛⟶𝑢₀,as𝑛⟶∞, (73) in sense of ‖⋅‖_𝐻¹. Let {𝑊_𝑛}_𝑛≥1 be a sequence of regular process, such that 𝐴^𝑎/2𝑊^𝑛_𝐴∶=𝐴^𝑎/2𝑡₀𝑒^{(𝑡−𝑠)𝐴}𝑑𝑊_𝑛(𝑠)⟶𝐴^𝑎/2𝑊_𝐴(𝑡),as𝑛→∞, (74) in 𝐶(𝑇×𝐷) when 𝑎=0 or 𝑎=1. For ℎ=(ℎ₁,ℎ₂), ℎ_𝑖∈𝐶([0,𝑇]×𝐷;ℝ),  ‖ℎ‖_{𝐶(𝑇×𝐷)}∶=∑²_𝑖=1|ℎ_𝑖|_{𝐶(𝑇×𝐷)}, where |ℎ_𝑖|_{𝐶(𝑇×𝐷)}=sup_{(𝑡,𝑥)∈[0,𝑇]×𝐷}|ℎ_𝑖|. Then, by (74), we have sup_{𝑛≥1}𝑊^𝑛_{𝐴𝐶(𝑇×𝐷)}<∞,sup_{𝑛≥1}sup_{𝑡∈[0,𝑇]}𝐴^1/2𝑊^𝑛_𝐴<∞. (75)(76) If 𝑣_𝑛 satisfies 𝑣_𝑛=𝑒^𝑡𝐴𝑢⁰_𝑛+^𝑡₀𝑒^{(𝑡−𝑠)𝐴}𝑣_𝑛+𝑊_𝐴⋅∇𝑣_𝑛+𝑊_𝐴−𝑑𝑠_𝑡0𝑒^{(𝑡−𝑠)𝐴}𝑓𝑣_𝑛+𝑊_𝐴, (77) then, 𝑣_𝑛 is regular, such that 𝜕𝑣_𝑛𝜕𝑡+𝐴𝑣_𝑛+𝐵𝑣_𝑛+𝑊^𝑛_𝐴,𝑣_𝑛+𝑊^𝑛_𝐴+𝑓𝑣_𝑛+𝑊^𝑛_𝐴=0. (78) Taking inner product with respect to 𝑣_𝑛 in (78), we have 𝜕𝑣_𝑛𝜕𝑡,𝑣_𝑛+𝐴𝑣_𝑛,𝑣_𝑛+𝐵𝑣_𝑛+𝑊^𝑛_𝐴,𝑣_𝑛+𝑊^𝑛_𝐴,𝑣_𝑛+𝑓𝑣_𝑛+𝑊^𝑛_𝐴,𝑣_𝑛=0. (79) For simplicity, we calculate the third term on the left hand side of (79) first as follows: 𝐵𝑣_𝑛+𝑊^𝑛_𝐴,𝑣_𝑛+𝑊^𝑛_𝐴,𝑣_𝑛=𝑣¹_𝑛+𝑊^𝑛_𝐴,1𝜕₁𝑣¹_𝑛+𝑊^𝑛_𝐴,1,𝑣¹_𝑛+𝑣²_𝑛+𝑊^𝑛_𝐴,2𝜕₂𝑣¹_𝑛+𝑊^𝑛_𝐴,1,𝑣¹_𝑛+𝑣¹_𝑛+𝑊^𝑛_𝐴,1𝜕₁𝑣²_𝑛+𝑊^𝑛_𝐴,2,𝑣^𝑛₂+𝑣²_𝑛+𝑊^𝑛_𝐴,2𝜕₂𝑣²_𝑛+𝑊^𝑛_𝐴,2,𝑣²_𝑛=𝐼₁+𝐼₂+𝐼₃+𝐼₄, (80) where 𝑊^𝑛_𝐴=(𝑊^𝑛_𝐴,1,𝑊^𝑛_𝐴,2). For 𝐼₁, we have 𝐼₁=𝑣¹_𝑛+𝑊^𝑛_𝐴,1𝜕₁𝑣¹_𝑛+𝑊^𝑛_𝐴,1,𝑣¹_𝑛=𝑣¹_𝑛𝜕₁𝑣¹_𝑛,𝑣¹_𝑛+𝑊^𝑛_𝐴,1𝜕₁𝑣¹_𝑛,𝑣¹_𝑛+𝑣¹_𝑛𝜕₁𝑊^𝑛_𝐴,1,𝑣¹_𝑛+𝑊^𝑛_𝐴,1𝜕₁𝑊^𝑛_𝐴,1,𝑣¹_𝑛. (81) In the following, we estimate the four terms for 𝐼₁, respectively. For the first term, 𝑣¹_𝑛𝜕₁𝑣¹_𝑛,𝑣¹_𝑛=_𝐷𝑣¹_𝑛²𝜕₁𝑣¹_𝑛=𝑑𝑥_𝐷𝜕₁𝑣¹_𝑛³3𝑑𝑥=0. (82) For the second term, by (75), we have ⟨𝑊^𝑛_𝐴,1𝜕₁𝑣¹_𝑛,𝑣¹_𝑛⟩≤𝐶𝑣¹_𝑛²_𝐻+𝜀_𝐷𝜕₁𝑣¹_𝑛²𝑣𝑑𝑥≤𝐶¹_𝑛²_𝐻+𝜀𝑣¹_𝑛²_𝐻1. (83) similarly, for the third term, 𝑣¹_𝑛𝜕₁𝑊^𝑛_𝐴,1,𝑣¹_𝑛=_𝐷𝑣¹_𝑛²𝜕₁𝑊^𝑛_𝐴,1=𝑑𝑥_𝐷𝑊^𝑛_𝐴,1𝜕₁𝑣¹_𝑛²𝑑𝑥≤𝐶_𝐷𝑣¹_𝑛𝜕₁𝑣¹_𝑛𝑣𝑑𝑥≤𝐶¹_𝑛²_𝐻+𝜀𝑣¹_𝑛²_𝐻1. (84) For the last term, by (75) and (76), 𝑊^𝑛_𝐴,1𝜕₁𝑊^𝑛_𝐴,1,𝑣¹_𝑛≤𝐶_𝐷𝜕₁𝑣¹_𝑛𝑣𝑑𝑥≤𝐶+𝜀¹_𝑛²_𝐻1. (85) By (81)–(85), it follows that 𝐼₁≤𝐶1+𝑣_𝑛²_𝐻+4𝜀𝑣_𝑛²_𝐻1. (86) Similarly, 𝐼₄≤𝐶1+𝑣_𝑛²_𝐻+4𝜀𝑣_𝑛²_𝐻1. (87) For 𝐼₃, 𝐼₃=⟨𝑣¹_𝑛𝜕₁𝑣²_𝑛,𝑣²_𝑛⟩+⟨𝑣¹_𝑛𝜕₁𝑊^𝑛_𝐴,2,𝑣²_𝑛⟩+⟨𝑊^𝑛_𝐴,1𝜕₁𝑣²_𝑛,𝑣²_𝑛⟩+⟨𝑊^𝑛_𝐴,1𝜕₁𝑊^𝑛_𝐴,2,𝑣²_𝑛⟩. (88) For the first term on the right hand side of (88), we deduce that 𝑣¹_𝑛𝜕₁𝑣²_𝑛,𝑣²_𝑛=12_𝐷𝑣¹_𝑛𝜕₁𝑣²_𝑛²=1𝑑𝑥2_𝐷𝜕₁𝑣¹_𝑛⋅𝑣²_𝑛²≤1𝑑𝑥2𝑣²_𝑛²_𝐿4⋅𝑣¹_𝑛𝐻1≤14𝜖𝑣²_𝑛⁴_𝐿4+1𝑣4𝜖^1𝑛2𝐻1, (89) where 𝜖>0. For the second term on the right hand side of (88), we have 𝑣¹_𝑛𝜕₁𝑊^𝑛_𝐴,2,𝑣²_𝑛≤𝜀𝑣^𝑛2_𝐻1+𝐶𝑣^𝑛2_𝐻. (90) Analogously, for the third term on the right hand side of (88), we see that 𝑊^𝑛_𝐴,1𝜕₁𝑣²_𝑛,𝑣²_𝑛≤𝐶𝑣^𝑛2_𝐻+𝜀𝑣^𝑛2_𝐻1. (91) For the last term, by (75) and (76), we have 𝑊^𝑛_𝐴,1𝜕_𝑥𝑊^𝑛_𝐴,2,𝑣²_𝑛≤𝐶+𝜀𝑣^𝑛2_𝐻1. (92) By (88)–(92), we get 𝐼₃≤1𝑣4𝜖^2𝑛4𝐿4+𝜖4𝑣¹_𝑛²_𝐻1+3𝜀𝑣_𝑛²_𝐻1+𝐶𝑣_𝑛²_𝐻+𝐶. (93) Analogously, for 𝐼₂, it follows that 𝐼₂≤1𝑣4𝜖^1𝑛4𝐿4+𝜖4𝑣²_𝑛²_𝐻1+3𝜀𝑣_𝑛²_𝐻1+𝐶𝑣_𝑛²_𝐻+𝐶. (94) By (80) and the estimates of 𝐼₁,𝐼₂,𝐼₃, and 𝐼₄, see (86), (87), (93), and (94), we have 𝐵𝑣_𝑛+𝑊^𝑛_𝐴,𝑣_𝑛+𝑊^𝑛_𝐴,𝑣_𝑛≤𝐶1+𝑣_𝑛²_𝐻+𝜖4+14𝜀𝑣_𝑛²_𝐻1+1𝑣4𝜖^𝑛4𝐿4. (95) For the last term on the left hand side of (79), we have ⟨𝑓𝑣_𝑣+𝑊^𝑛_𝐴,𝑣_𝑛⟩=𝜗𝑣_𝑛⁴_𝐿4+3𝜗_𝐷𝑣_𝑛²𝑣¹_𝑛𝑊^𝑛_𝐴,1+𝑣²_𝑛𝑊^𝑛_𝐴,2𝑑𝑥+𝜗_𝐷𝑣_𝑛²𝑊^𝑛_𝐴²𝑑𝑥+𝜗_𝐷𝑣¹_𝑛𝑊^𝑛_𝐴,1+𝑣²_𝑛𝑊^𝑛_𝐴,2𝑊^𝑛_𝐴²𝑑𝑥+2𝜗_𝐷𝑊^𝑛_𝐴,1²𝑣¹_𝑛²+𝑊^𝑛_𝐴,2²𝑣²_𝑛²𝑑𝑥+4𝜗_𝐷𝑊^𝑛_𝐴,1𝑊^𝑛_𝐴,2𝑣¹_𝑛𝑣²_𝑛≤(𝑣𝑑𝑥𝜗+𝜀)_𝑛⁴_𝐿4+𝐶1+𝑣_𝑛²_𝐻. (96) By (79), (95), and (96), we get 12𝜕𝑣𝜕𝑡^𝑛2𝐻+𝑣_𝑛²_𝐻1+𝜗𝑣_𝑛⁴_𝐿4≤𝐶1+𝑣_𝑛²_𝐻+𝜖4+14𝜀𝑣_𝑛²_𝐻1+1𝑣4𝜖+𝜀_𝑛⁴_𝐿4. (97) Rearranging the above inequality, we deduce that 12𝜕𝑣𝜕𝑡^𝑛2𝐻+𝜖1−4−14𝜀𝑣_𝑛²_𝐻1+1𝜗−𝑣4𝜖−𝜀_𝑛⁴_𝐿4≤𝐶1+𝑣_𝑛²_𝐻. (98) Let 𝜖∈(1/4𝜗,4), and 𝜀 be small enough, such that 𝜖1−41−14𝜀>0,𝜗−4𝜖−𝜀>0. (99) So, we integrate with respect to 𝑡 on both sides of (98) to obtain 𝑣_𝑛(𝑡)²_𝐻+𝐶_𝜖^𝑡₀𝑣_𝑛(𝑠)²_𝐻1≤𝑣𝑑𝑠_𝑛(0)²_𝐻+𝐶𝑡+𝐶^𝑡₀𝑣_𝑛(𝑠)²_𝐻𝑑𝑠, (100) where 𝐶_𝜖=2(1−𝜖/4−14𝜀), by Gronwall’s inequality, we arrive at 𝑣_𝑛(𝑡)²_𝐻≤𝑣_𝑛(0)²_𝐻+𝐶𝑡𝑒^𝐶𝑡≤𝐶_𝑇. (101) By (100) and (101), we have ^𝑡₀𝑣_𝑛(𝑠)²_𝐻1𝑑𝑠≤𝐶_𝑇. (102) Multiplying 𝐴𝑣_𝑛 on both sides of (78), and integrating with respect to 𝑥∈𝐷, we have 𝜕𝑣_𝑛𝜕𝑡,𝐴𝑣_𝑛+𝐴𝑣_𝑛,𝐴𝑣_𝑛+𝑓𝑣_𝑛+𝑊^𝑛_𝐴,𝐴𝑣_𝑛=𝐵𝑣_𝑛+𝑊^𝑛_𝐴,𝑣_𝑛+𝑊^𝑛_𝐴,𝐴𝑣_𝑛, (103) which is equivalent to 12𝜕𝑣𝜕𝑡^𝑛2𝐻1+𝑣_𝑛²_𝐻2=−𝑓𝑣_𝑛+𝑊^𝑛_𝐴,𝐴𝑣_𝑛+⟨𝐵𝑣_𝑛+𝑊^𝑛_𝐴,𝑣_𝑛+𝑊^𝑛_𝐴,𝐴𝑣_𝑛⟩. (104) We first estimate the second term on the right hand side of (104) as follows: 𝐵𝑣_𝑛+𝑊^𝑛_𝐴,𝑣_𝑛+𝑊^𝑛_𝐴,𝐴𝑣_𝑛=𝑣¹_𝑛+𝑊^𝑛_𝐴,1𝜕₁𝑣¹_𝑛+𝑊^𝑛_𝐴,1,𝐴𝑣¹_𝑛+𝑣²_𝑛+𝑊^𝑛_𝐴,2𝜕₂𝑣¹_𝑛+𝑊^𝑛_𝐴,1,𝐴𝑣¹_𝑛+𝑣¹_𝑛+𝑊^𝑛_𝐴,1𝜕₁𝑣²_𝑛+𝑊^𝑛_𝐴,2,𝐴𝑣²_𝑛+𝑣²_𝑛+𝑊^𝑛_𝐴,2𝜕₂𝑣²_𝑛+𝑊^𝑛_𝐴,2,𝐴𝑣²_𝑛=𝐽₁+𝐽₂+𝐽₃+𝐽₄. (105) For 𝐽₁, we have 𝐽₁=𝑣¹_𝑛𝜕₁𝑣¹_𝑛,𝐴𝑣¹_𝑛+𝑣¹_𝑛𝜕₁𝑊^𝑛_𝐴,1,𝐴𝑣¹_𝑛+𝑊^𝑛_𝐴,1𝜕₁𝑣¹_𝑛,𝐴𝑣¹_𝑛+𝑊^𝑛_𝐴,1𝜕₁𝑊^𝑛_𝐴,1,𝐴𝑣¹_𝑛=𝑘₁+𝑘₂+𝑘₃+𝑘₄. (106) For 𝑘₁, we have 𝑘₁≤𝜀𝑣¹_𝑛²_𝐻2+𝐶𝑣¹_𝑛²_𝐿4⋅𝑣¹_𝑛²_𝑊1,4. (107) By interpolation inequality, there exists some 𝐶>0, such that 𝑣¹_𝑛𝐿4≤𝐶𝑣¹_𝑛𝐻^1/2𝑣¹_𝑛𝐻^1/2₁,𝑣¹_𝑛𝑊1,4≤𝐶𝑣¹_𝑛𝐻^1/4𝑣¹_𝑛𝐻^3/4₂. (108) Then, 𝑘₁≤𝜀𝑣¹_𝑛²_𝐻2+𝐶𝑣¹_𝑛𝐻^3/2⋅𝑣¹_𝑛𝐻1⋅𝑣¹_𝑛𝐻^3/2₂≤𝜀𝑣¹_𝑛²_𝐻2+𝜀𝑣¹_𝑛²_𝐻2+𝐶𝑣¹_𝑛⁶_𝐻𝑣¹_𝑛⁴_𝐻1≤2𝜀𝑣¹_𝑛²_𝐻2+𝐶_𝑇𝑣¹_𝑛⁴_𝐻1, (109) where the last inequality follows from (101). For 𝑘₂, we deduce that 𝑘₂≤𝜀𝑣¹_𝑛²_𝐻2+𝐶_𝐷𝑣¹_𝑛²𝜕₁𝑊^𝑛_𝐴,1²𝑣𝑑𝑥≤𝜀¹_𝑛²_𝐻2+𝐶𝑣¹_𝑛²_𝐻≤𝜀𝑣¹_𝑛²_𝐻2+𝐶_𝑇. (110) For 𝑘₃, we arrive at 𝑘₃≤𝐶_𝐷𝜕₁𝑣¹_𝑛⋅𝐴𝑣¹_𝑛𝑑𝑥≤𝜀𝑣¹_𝑛²_𝐻2+𝐶𝑣¹_𝑛²_𝐻1. (111) For 𝑘₄, we obtain 𝑘₄≤𝐶+𝜀𝑣¹_𝑛²_𝐻2. (112) By (106) and (109)–(112), 𝐽₁≤5𝜀𝑣¹_𝑛²_𝐻2+𝐶_𝑇𝑣¹_𝑛⁴_𝐻1+𝐶𝑣¹_𝑛²_𝐻1+𝐶_𝑇. (113) Similarly, for 𝐽₄, we infer that 𝐽₄≤5𝜀𝑣²_𝑛²_𝐻2+𝐶_𝑇𝑣²_𝑛⁴_𝐻1+𝐶𝑣²_𝑛²_𝐻1+𝐶_𝑇. (114) For 𝐽₂, we have 𝐽₂=𝑣²_𝑛𝜕₂𝑣¹_𝑛,𝐴𝑣¹_𝑛+𝑊^𝑛_𝐴,2𝜕₂𝑣¹_𝑛,𝐴𝑣¹_𝑛+⟨𝑣²_𝑛𝜕₂𝑊^𝑛_𝐴,1,𝐴𝑣¹_𝑛⟩+⟨𝑊^𝑛_𝐴,2𝜕₂𝑊^𝑛_𝐴,1,𝐴𝑣¹_𝑛⟩=𝑙₁+𝑙₂+𝑙₃+𝑙₄. (115) By interpolation inequality and (101), we deduce that 𝑙₁≤𝜀𝑣¹_𝑛²_𝐻2+𝐶𝑣²_𝑛²_𝐿4⋅𝑣¹_𝑛²_𝑊1,4≤𝜀𝑣¹_𝑛²_𝐻2+𝐶𝑣²_𝑛𝐻⋅𝑣²_𝑛𝐻1⋅𝑣¹_𝑛𝐻^1/2⋅𝑣¹_𝑛𝐻^3/2₂≤2𝜀𝑣¹_𝑛²_𝐻2+𝐶_𝑇𝑣²_𝑛⁴_𝐻1. (116) For 𝑙₂, we have 𝑙₂≤𝐶_𝐷𝜕₂𝑣¹_𝑛⋅𝐴𝑣¹_𝑛𝑑𝑥≤𝜀𝑣¹_𝑛²_𝐻2+𝐶𝑣¹_𝑛²_𝐻1. (117) Similarly, for 𝑙₃, 𝑙₃≤𝐶_𝐷𝑣²_𝑛⋅𝐴𝑣¹_𝑛𝑑𝑥≤𝜀𝑣¹_𝑛²_𝐻2+𝐶_𝑇. (118) As for 𝑙₄, we get 𝑙₄≤𝜀𝑣¹_𝑛²_𝐻2+𝐶_𝑇. (119) By (115)-(119), we arrive at 𝐽₂≤5𝜀𝑣¹_𝑛²_𝐻2+𝐶_𝑇𝑣²_𝑛⁴_𝐻1+𝐶𝑣¹_𝑛²_𝐻1+𝐶_𝑇. (120) Analogously to 𝐽₂, we have 𝐽₃≤5𝜀𝑣²_𝑛²_𝐻2+𝐶_𝑇𝑣¹_𝑛⁴_𝐻1+𝐶𝑣²_𝑛²_𝐻1+𝐶_𝑇. (121) By (105) and the estimates of 𝐽₁−𝐽₄, see (113), (114), (120), and (121), we get that 𝐵𝑣_𝑛+𝑊^𝑛_𝐴,𝑣_𝑛+𝑊^𝑛_𝐴,𝐴𝑣_𝑛≤10𝜀𝑣_𝑛²_𝐻2+𝐶_𝑇𝑣_𝑛⁴_𝐻1+𝐶𝑣_𝑛²_𝐻1+𝐶_𝑇. (122) For the first term on the right hand side of (104), we have 𝑓𝑣_𝑛+𝑊^𝑛_𝐴,𝐴𝑣_𝑛≤𝜀𝑣_𝑛²_𝐻2+𝐶𝑣_𝑛+𝑊^𝑛_𝐴⁶_𝐿6≤𝜀𝑣_𝑛²_𝐻2+𝐶𝑣_𝑛⁶_𝐿6+𝐶_𝑇≤𝜀𝑣_𝑛²_𝐻2+𝐶_𝑇𝑣_𝑛²_𝐻1𝑣_𝑛⁴_𝐻+𝐶_𝑇≤𝜀𝑣_𝑛²_𝐻2+𝐶_𝑇1+𝑣_𝑛²_𝐻1. (123) By (104), (122), and (123), 12𝜕𝑣𝜕𝑡^𝑛2𝐻1+𝑣_𝑛²_𝐻2≤11𝜀𝑣_𝑛²_𝐻2+𝐶_𝑇1+𝑣_𝑛²_𝐻1𝑣_𝑛²_𝐻1+𝐶_𝑇. (124) By the Gronwall inequality, we get 𝑣_𝑛(𝑡)²_𝐻1≤𝑣_𝑛(0)²_𝐻1+𝐶_𝑇𝑒^{𝐶𝑇∫𝑡0(1+‖𝑣𝑛(𝑠)‖2𝐻1𝑑𝑠)}≤𝑣_𝑛(0)²_𝐻1+𝐶_𝑇𝑒^𝐶𝑇. (125) Let 𝑛→∞, by Fatou Lemma, ‖𝑣(𝑡)‖²_𝐻1≤‖𝑣(0)‖²_𝐻1+𝐶_𝑇𝑒^𝐶𝑇. (126)

5. Invariant Measures

5.1. Existence

In this section, we will establish the existence of invariant measure for (2). Analogously to [24], we extend the Wiener process 𝑊(𝑡) to ℝ by setting 𝑊(𝑡)∶=𝑊¹(𝑡),𝑡≤0, (127) where 𝑊¹(𝑡) is another 𝐻-valued Wiener process satisfying conditions in Lemma 2 and being independent of 𝑊(𝑡). For any 𝜏≥0, we consider the following equation: 𝑑𝑢_𝜏+𝐴𝑢_𝜏+𝐵𝑢_𝜏,𝑢_𝜏+𝑓𝑢_𝜏[]𝑑𝑡=𝑑𝑊,on0,𝑇×𝐷,𝑢_𝜏(−𝜏)=0. (128) By Theorem 7, we know that there exists unique solution. In order to obtain the invariant measure, we should show that the family of laws {ℒ(𝑢_𝜏(0))}_𝜏≥0 is tight. Since 𝐻^1+𝛿⊂𝐻¹ is compact, for any 𝛿>0, we only need to show that {ℒ(𝑢_𝜏(0))}_𝜏≥0 is bounded in probability in 𝐻^1+𝛿. As we know, 𝑊_𝐴(𝑡)=^𝑡_−∞𝑒^{−(𝑡−𝑠)𝐴}𝑑𝑊(𝑠),𝑡∈ℝ (129) is the mild solution of (8) with the following initial condition: 𝑊_𝐴(0)=⁰_−∞𝑒^𝑠𝐴𝑑𝑊(𝑠). (130) Making the classical change of variable 𝑣_𝜏(𝑡)=𝑢_𝜏(𝑡)−𝑊_𝐴(𝑡), (128) is equivalent to 𝑑𝑣_𝜏(𝑡)𝑑𝑡=𝐴𝑣_𝜏(𝑡)+𝐵𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡),𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡)+𝑓𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡), (131) with initial condition 𝑣_𝜏(−𝜏)=−𝑊_𝐴(−𝜏). (132) In order to get the invariant measure of (131), it is enough to show that 𝑣_𝜏(0) is bounded in probability in 𝐻^1+𝛿, for some 𝛿>0. That is what we have to do in Theorem 8 below.

Theorem 8. With conditions in Lemma 2, when 𝜗>1/4, there exists an invariant measure for (2).

Proof. Multiplying (131) by 𝑣_𝜏 and integrating on 𝐷, we get 12𝑑𝑣𝑑𝑡^𝜏(𝑡)²_𝐻+𝑣_𝜏(𝑡)²_𝐻1+𝑓𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡),𝑣_𝜏(𝑡=𝐵𝑣_𝜏)(𝑡)+𝑊_𝐴(𝑡),𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡),𝑣_𝜏(𝑡). (133) For the third term on the left hand side of (133), we deduce that 𝑓𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡),𝑣_𝜏(𝑡𝑣)=𝜗_𝜏(𝑡)+𝑊_𝐴(𝑡)²𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡),𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡𝑣)−𝜗_𝜏(𝑡)+𝑊_𝐴(𝑡)²𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡),𝑊_𝐴(𝑡𝑣)=𝜗_𝜏(𝑡)+𝑊_𝐴(𝑡)⁴_𝐿4−𝜗𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡)²𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡),𝑊_𝐴(𝑡𝑣)≥𝜗_𝜏(𝑡)_𝐿4−𝑊_𝐴(𝑡)_𝐿4⁴−𝜗𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡)²𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡),𝑊_𝐴(𝑡𝑣)≥𝜗_𝜏(𝑡)⁴_𝐿4−4𝜗𝑣_𝜏(𝑡)³_𝐿4𝑊_𝐴(𝑡)_𝐿4−4𝜗𝑣_𝜏(𝑡)¹_𝐿4𝑊_𝐴(𝑡)³_𝐿4−𝜗𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡)²𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡),𝑊_𝐴(𝑡). (134) Substituting (134) into (133), we have 12𝑑𝑣𝑑𝑡^𝜏(𝑡)²_𝐻+𝑣_𝜏(𝑡)²_𝐻1+𝜗𝑣_𝜏(𝑡)⁴_𝐿4≤4𝜗𝑣_𝜏(𝑡)³_𝐿4𝑊_𝐴(𝑡)_𝐿4+4𝜗𝑣_𝜏(𝑡)_𝐿4𝑊_𝐴(𝑡)³_𝐿4+𝜗𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡)²𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡),𝑊_𝐴(𝑡+𝐵𝑣_𝜏)(𝑡)+𝑊_𝐴(𝑡),𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡),𝑣_𝜏(𝑡). (135) For the third term on the right hand side of (135), we get by the Young inequality that 𝜗𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡)²𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡),𝑊_𝐴(𝑡𝑣)≤𝜀_𝜏(𝑡)⁴_𝐿4+𝐶𝑊_𝐴(𝑡)⁴_𝐿4. (136) For the last term on the right hand side of (135), 𝐵𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡),𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡),𝑣_𝜏(𝑡𝑣)=⟨_𝜏(𝑡)⋅∇𝑣_𝜏(𝑡),𝑣_𝜏(𝑡)⟩+⟨𝑊_𝐴(𝑡)⋅∇𝑣_𝜏(𝑡),𝑣_𝜏(𝑡)⟩+⟨𝑣_𝜏(𝑡)⋅∇𝑊_𝐴(𝑡),𝑣_𝜏(𝑡)⟩+⟨𝑊_𝐴(𝑡)⋅∇𝑊_𝐴(𝑡),𝑣_𝜏(𝑡)⟩=𝑟₁+𝑟₂+𝑟₃+𝑟₄. (137)
Since 𝑣_𝜏(𝑡) is vector field, we denote it by 𝑣_𝜏(𝑡)=(𝑣¹_𝜏(𝑡),𝑣²_𝜏(𝑡)), where 𝑣^𝑖_𝜏(𝑡) is real valued function, 𝑖=1,2. For 𝑟₁, we have 𝑟₁=𝑣¹_𝜏(𝑡)𝜕₁𝑣¹_𝜏(𝑡)+𝑣²_𝜏(𝑡)𝜕₂𝑣¹_𝜏(𝑡),𝑣¹_𝜏(𝑡+𝑣_1𝜏)(𝑡)𝜕₁𝑣²_𝜏(𝑡)+𝑣²_𝜏(𝑡)𝜕₂𝑣²_𝜏(𝑡),𝑣²_𝜏(𝑡=𝑣_2𝜏)(𝑡)𝜕₂𝑣¹_𝜏(𝑡),𝑣¹_𝜏(𝑡+𝑣_1𝜏)(𝑡)𝜕₁𝑣²_𝜏(𝑡),𝑣²_𝜏(𝑡1)≤−2𝜕₂𝑣²_𝜏(𝑡),𝑣¹_𝜏(𝑡)²−12𝜕₁𝑣¹_𝜏(𝑡),𝑣²_𝜏(𝑡)²≤14𝜕₁𝑣¹_𝜏(𝑡)²_𝐻+14𝑣²_𝜏(𝑡)⁴_𝐿4+14𝜕₂𝑣²_𝜏(𝑡)²_𝐻+14𝑣¹_𝜏(𝑡)⁴_𝐿4≤14𝑣_𝜏(𝑡)²_𝐻1+14𝑣_𝜏(𝑡)⁴_𝐿4. (138) Similarly for 𝑟₂, 𝑟₂=𝑊_𝐴,1(𝑡)𝜕₁𝑣¹_𝜏(𝑡)+𝑊_𝐴,2(𝑡)𝜕₂𝑣¹_𝜏(𝑡),𝑣¹_𝜏(𝑡+𝑊_𝐴,1)(𝑡)𝜕₁𝑣²_𝜏(𝑡)+𝑊_𝐴,2(𝑡)𝜕₂𝑣²_𝜏(𝑡),𝑣²_𝜏(𝑡𝜕)=−₁𝑊_𝐴,1(𝑡),𝑣¹_𝜏(𝑡)²−𝜕₂𝑊_𝐴,2(𝑡),𝑣¹_𝜏(𝑡)²−𝜕₁𝑊_𝐴,1(𝑡),𝑣²_𝜏(𝑡)²−𝜕₂𝑊_𝐴,2(𝑡),𝑣²_𝜏(𝑡)²≤𝜀𝑣_𝜏(𝑡)⁴_𝐿4+𝐶𝑊_𝐴(𝑡)²_𝐻1. (139) Analogously to 𝑟₁, we deduce that 𝑟₃=𝑣¹_𝜏(𝑡)𝜕₁𝑊_𝐴,1(𝑡)+𝑣²_𝜏(𝑡)𝜕₂𝑊_𝐴,1(𝑡),𝑣¹_𝜏(𝑡)+⟨𝑣¹_𝜏(𝑡)𝜕₁𝑊_𝐴,2(𝑡)+𝑣²_𝜏(𝑡)𝜕₂𝑊_𝐴,2(𝑡),𝑣²_𝜏(𝑡)⟩≤𝜀𝑣_𝜏(𝑡)⁴_𝐿4+𝐶𝑊_𝐴(𝑡)²_𝐻1. (140) For 𝑟₄, we have 𝑟₄=𝑊_𝐴,1(𝑡)𝜕₁𝑊_𝐴,1(𝑡)+𝑊_𝐴,2(𝑡)𝜕₂𝑊_𝐴,1(𝑡),𝑣¹_𝜏(𝑡+𝑊_𝐴,1)(𝑡)𝜕₁𝑊_𝐴,2(𝑡)+𝑊_𝐴,2(𝑡)𝜕₂𝑊_𝐴,2(𝑡),𝑣²_𝜏(𝑡𝑣)≤𝜀¹_𝜏(𝑡)²_𝐻+𝐶𝑊_𝐴,1(𝑡)𝜕₁𝑊_𝐴,1(𝑡)²_𝐻+𝜀𝑣¹_𝜏(𝑡)²_𝐻+𝐶𝑊_𝐴,2(𝑡)𝜕₂𝑊_𝐴,1(𝑡)²_𝐻+𝜀𝑣²_𝜏(𝑡)²_𝐻+𝐶𝑊_𝐴,1(𝑡)𝜕₁𝑊_𝐴,2(𝑡)²_𝐻+𝜀𝑣²_𝜏(𝑡)²_𝐻+𝐶𝑊_𝐴,2(𝑡)𝜕₂𝑊_𝐴,2(𝑡)²_𝐻≤𝜀𝑣¹_𝜏(𝑡)²_𝐻+𝑊_𝐴,1(𝑡)²_𝐿4⋅𝑊_𝐴,1(𝑡)²_𝑊1,4+𝐶𝑊_𝐴,2(𝑡)²_𝐿4⋅𝑊_𝐴,1(𝑡)²_𝑊1,4+𝜀𝑣²_𝜏(𝑡)²_𝐻+𝐶𝑊_𝐴,1(𝑡)²_𝐿4⋅𝑊_𝐴,2(𝑡)²_𝑊1,4+𝐶𝑊_𝐴,2(𝑡)²_𝐿4⋅𝑊_𝐴,2(𝑡)²_𝑊1,4≤𝜀𝑣_𝜏(𝑡)²_𝐻+𝐶𝑊_𝐴(𝑡)²_𝐿4⋅𝑊_𝐴(𝑡)²_𝑊1,4. (141) Since {𝐴^1/2𝑊_𝐴(𝑡)}_𝑡∈ℝ is a Gaussian process, we infer that 𝐸𝐴^1/2𝑊_𝐴(𝑡)⁴≤𝐶𝐸𝐴^1/2𝑊_𝐴(𝑡)²². (142) Then, with the proof of Lemma 2, we know that ‖𝑊_𝐴(𝑡)‖²_𝑊1,4 is continuous with respect to 𝑡. By (137)–(141), we have 𝐵𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡),𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡),𝑣_𝜏(𝑡≤1)4+𝜀𝑣_𝜏(𝑡)²_𝐻1+14+2𝜀𝑣_𝜏(𝑡)⁴_𝐿4+𝐶𝑊_𝐴(𝑡)²_𝐻1+𝑊_𝐴(𝑡)²_𝐿4𝑊_𝐴(𝑡)²_𝑊1,4. (143) By (135), (136), and (143), we arrive at 12𝑑𝑣𝑑𝑡^𝜏(𝑡)²_𝐻+𝑣_𝜏(𝑡)²_𝐻1+𝜗𝑣_𝜏(𝑡)⁴_𝐿4≤14+3𝜀𝑣_𝜏(𝑡)⁴_𝐿4+14+𝜀𝑣_𝜏(𝑡)²_𝐻1+𝐶𝑊_𝐴(𝑡)⁴_𝑊1,4+𝐶. (144) It is equivalent to 12𝑑𝑣𝑑𝑡^𝜏(𝑡)²_𝐻+34−𝜀𝑣_𝜏(𝑡)²_𝐻1+1𝜗−4−3𝜀𝑣_𝜏(𝑡)⁴_𝐿4≤𝐶1+𝑊_𝐴(𝑡)⁴_𝑊1,4. (145) Since 𝜗>1/4, let 𝜀 be small enough, such that 341−𝜀>0;𝜗−4−3𝜀>0. (146) Then, the above estimates can be changed into 𝑑𝑣𝑑𝑡^𝜏(𝑡)²_𝐻+𝛼₁𝑣_𝜏(𝑡)²_𝐻1+𝐶_𝜈𝑣_𝜏(𝑡)⁴_𝐿4≤𝐶1+𝑊_𝐴(𝑡)⁴_𝑊1,4. (147) By the Gronwall inequality, we get 𝑣_𝜏(𝑡)²_𝐻≤𝑊_𝐴(−𝜏)²_𝐻𝑒^{−𝛼1(𝜏+𝑡)}+𝐶^𝑡_−𝜏1+𝑊_𝐴(𝑠)⁴_𝑊1,4𝑒^{𝛼1(𝑠−𝑡)}≤𝑊𝑑𝑠_𝐴(−𝜏)²_𝐻𝑒^{−𝛼1(𝜏+𝑡)}+𝐶⁰_−∞1+𝑊_𝐴(𝑠)⁴_𝑊1,4𝑒^{𝛼1(𝑠−𝑡)}𝑑𝑠. (148) Similarly to the argument of [26], we will prove that ‖𝑊_𝐴(𝑡)‖_𝑊1,4 has at most polynomial growth, when 𝑡→−∞ a.s. So, we conclude that sup_{0≤𝜏;𝑡≤𝑇}𝑣_𝜏(𝑡)²_𝐻<∞.a.s. (149) Multiplying 𝑒^𝛿𝑡 on both sides of (147) and integrating with respect to 𝑡, we have ^𝑡_−𝜏𝑒^𝛿𝑠𝑣_𝜏(𝑠)²_𝐻1𝑑𝑠≤𝑒^−𝛿𝜏𝑊_𝐴(−𝜏)²_𝐻+𝛼₁^𝑡_−𝜏𝑒^𝛿𝑠𝑣_𝜏(𝑠)²_𝐻𝑑𝑠+𝐶^𝑡_−𝜏1+𝑊_𝐴(𝑡)⁴_𝑊1,4𝑒^𝛿𝑠𝑑𝑠≤𝑒^−𝛿𝜏𝑊_𝐴(−𝜏)²_𝐻+𝛼₁⁰_−∞𝑒^𝛿𝑠𝑣_𝜏(𝑠)²_𝐻𝑑𝑠+𝐶⁰_−∞1+𝑊_𝐴(𝑡)⁴_𝑊1,4𝑒^𝛿𝑠𝑑𝑠. (150) As 1+𝑊⁰_−∞𝐴(𝑡)⁴_𝑊1,4𝑒^𝛿𝑠𝑑𝑠<∞, (151) by (149), we have sup_{0≤𝜏;𝑡≤𝑇}^𝑡_−𝜏𝑒^𝛿𝑠𝑣_𝜏(𝑠)²_𝐻1𝑑𝑠<∞.a.s. (152) By Theorem 7, we know that for problem (131) there exists unique mild solution, which has the following: 𝑣_𝜏(0)=𝑒^𝜏𝐴𝑊_𝐴()+−𝜏_0−𝜏𝑒^𝑡𝐴𝐵𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡),𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡)+𝑑𝑡_0−𝜏𝑒^𝑡𝐴𝑓𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡)𝑑𝑡. (153) Then, for any 𝜁∈(0,𝜃)∩(0,1/4), where the 𝜃 is the parameter in Lemma 2, 𝐴^(1+𝜁)/2𝑣_𝜏(0)_𝐻≤𝑒^𝜏𝐴𝐴^(1+𝜁)/2𝑊_𝐴(−𝜆)_𝐻+⁰_−𝜏𝐴^(1+𝜁)/2𝑒^𝑡𝐴𝐵(𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡),𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡)_𝐻+𝑑𝑡_0−𝜏𝐴^(1+𝜁)/2𝑒^𝑡𝐴𝑓𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡)_𝐻𝑑𝑡. (154) Since 𝐵𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡),𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡)=𝑣_𝜏(𝑡)⋅∇𝑣_𝜏(𝑡)+𝑣_𝜏(𝑡)⋅∇𝑊_𝐴(𝑡)+𝑊_𝐴(𝑡)⋅∇𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡)⋅∇𝑊_𝐴(𝑡), (155) then, 𝐴^(1+𝜁)/2𝑒^𝑡𝐴𝐵𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡),𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡)_𝐻≤𝐴^(1+𝜁)/2𝑒^𝑡𝐴𝑣_𝜏(𝑡)⋅∇𝑣_𝜏(𝑡)_𝐻+𝐴^(1+𝜁)/2𝑒^𝑡𝐴𝑣_𝜏(𝑡)⋅∇𝑊_𝐴(𝑡)_𝐻+𝐴^(1+𝜁)/2𝑒^𝑡𝐴𝑊_𝐴(𝑡)⋅∇𝑣_𝜏(𝑡)_𝐻+𝐴^(1+𝜁)/2𝑒^𝑡𝐴𝑊_𝐴(𝑡)⋅∇𝑊_𝐴(𝑡)_𝐻=𝑧₁+𝑧₂+𝑧₃+𝑧₄. (156) For 𝑧₁, we have 𝑧₁≤𝐴^(1+𝜁)/2𝑒^𝑡𝐴𝑣¹_𝜏(𝑡)𝜕₁𝑣¹_𝜏(𝑡)_𝐻+𝐴^(1+𝜁)/2𝑒^𝑡𝐴𝑣²_𝜏(𝑡)𝜕₂𝑣¹_𝜏(𝑡)_𝐻+𝐴^(1+𝜁)/2𝑒^𝑡𝐴𝑣¹_𝜏(𝑡)𝜕₁𝑣²_𝜏(𝑡)_𝐻+𝐴^(1+𝜁)/2𝑒^𝑡𝐴𝑣²_𝜏(𝑡)𝜕₂𝑣²_𝜏(𝑡)_𝐻=𝑧_1,1+𝑧_1,2+𝑧_1,3+𝑧_1,4. (157) In the following, we use Theorem 6.13 in chapter two of [27] to estimate them respectively as follows: 𝑧_1,1=12𝑒^𝑡𝐴𝐴^(1+𝜁)/2𝜕₁𝑣¹_𝜏²_𝐻≤12𝑒^𝑡𝐴𝐴^(3+2𝜁)/4𝑣¹_𝜏²_𝐻1/2≤𝐶|𝑡|^{−(3+2𝜁)/4}𝑒^𝛿𝑡𝑣¹_𝜏²_𝐻1/2≤𝐶|𝑡|^{−(3+2𝜁)/4}𝑒^𝛿𝑡2𝑣¹_𝜏(𝑡)𝐴^1/4𝑣¹_𝜏(𝑡)+𝑅_4𝐻, (158) the last inequality follows by Theorem A.8 in [25], where 𝛿>0,  𝑅₄=𝐴^1/4(𝑣¹_𝜏)²−2𝑣¹_𝜏𝐴^1/4𝑣¹_𝜏, and |𝑅₄|_𝐻≤𝐶|𝐴^1/8𝑣¹_𝜏(𝑡)|²_𝐿4≤𝐶|𝑣¹_𝜏(𝑡)|²_𝐻1. So, by Hölder inequality and interpolation inequality, we have 𝑧_1,1≤𝐶|𝑡|^{−(3+2𝜁)/4}𝑒^𝛿𝑡𝑣_𝜏(𝑡)²_𝐻1. (159) For 𝑧_1,2, we have 𝑧_1,2=𝐴^(1+𝜁)/2𝑒^𝑡𝐴𝑣²_𝜏(𝑡)𝜕₂𝑣¹_𝜏(𝑡)_𝐻≤𝐴^(1+𝜁)/2𝑒^𝑡𝐴𝐴^1/4𝑣²_𝜏(𝑡)𝐴^1/4𝑣¹_𝜏(𝑡)_𝐻+𝐴^(1+𝜁)/2𝑒^𝑡𝐴𝐴^1/4𝑣¹_𝜏(𝑡)𝐴^1/4𝑣²_𝜏(𝑡)_𝐻+𝐴^(1+𝜁)/2𝑒^𝑡𝐴𝑅_5𝐻, (160) where 𝑅₅=𝐴^1/4𝑣²_𝜏(𝑡)𝐴^1/4𝑣¹_𝜏(𝑡−𝐴_1/4)𝑣¹_𝜏(𝑡),𝐴^1/4𝑣²_𝜏(𝑡)−𝑣²_𝜏(𝑡)𝐴^1/2𝑣¹_𝜏(𝑡). (161) Analogously to estimating 𝑧_1,1, we have 𝑧_1,2≤𝐶|𝑡|^{−(3+2𝜁)/4}𝑒^𝛿𝑡𝑣²_𝜏(𝑡)_𝐿4𝑣¹_𝜏(𝑡)_𝑊1/2,4+𝐶|𝑡|^{−(1+𝜁)/2}𝑒^𝛿𝑡𝑣¹_𝜏(𝑡)_𝑊1/2,4𝑣²_𝜏(𝑡)_𝑊1/2,4≤𝐶|𝑡|^{−(3+2𝜁)/4}+|𝑡|^{−(1+𝜁)/2}𝑒^𝛿𝑡𝑣_𝜏(𝑡)²_𝑊1/2,4≤𝐶|𝑡|^{−(3+2𝜁)/4}+|𝑡|^{−(1+𝜁)/2}𝑒^𝛿𝑡𝑣_𝜏(𝑡)²_𝐻1. (162) Similarly, we can get the same estimates for 𝑧_1,3 and 𝑧_1,4. Therefore, 𝑧₁≤𝐶|𝑡|^{−(3+2𝜁)/4}+|𝑡|^{−(1+𝜁)/2}𝑒^𝛿𝑡𝑣_𝜏(𝑡)²_𝐻1. (163) Analogously to estimating 𝑧₁, we can get for 𝑧₂,𝑧₃, and 𝑧₄ that 𝑧₂≤𝐶|𝑡|^{−(3+2𝜁)/4}+|𝑡|^{−(1+𝜁)/2}×𝑒^𝛿𝑡𝑊_𝐴(𝑡)²_𝐻1+𝑣_𝜏(𝑡)²_𝐻1,𝑧₃≤𝐶|𝑡|^{−(3+2𝜁)/4}+|𝑡|^{−(1+𝜁)/2}×𝑒^𝛿𝑡𝑊_𝐴(𝑡)²_𝐻1+𝑣_𝜏(𝑡)²_𝐻1,𝑧₄≤𝐶|𝑡|^{−(3+2𝜁)/4}+|𝑡|^{−(1+𝜁)/2}×𝑒^𝛿𝑡𝑊_𝐴(𝑡)²_𝐻1. (164) So, by (163)–(164) and (156), we get 𝐴^(1+𝜁)/2𝑒^𝑡𝐴𝐵𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡),𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡)_𝐻≤𝐶|𝑡|^{−(3+2𝜁)/4}+|𝑡|^{−(1+𝜁)/2}×𝑒^𝛿𝑡𝑊_𝐴(𝑡)²_𝐻1+𝑣_𝜏(𝑡)²_𝐻1. (165) For the third term on the right hand side of (154), we obtain 𝐴^(1+𝜁)/2𝑒^𝑡𝐴𝑓𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡)_𝐻≤𝐶|𝑡|^{−(1+𝜁)/2}𝑒^𝛿𝑡𝑣_𝜏(𝑡)³_𝐿6+𝑊_𝐴(𝑡)³_𝐿6≤𝐶|𝑡|^{−(1+𝜁)/2}×𝑒^𝛿𝑡𝑣_𝜏(𝑡)_𝐻⋅𝑣_𝜏(𝑡)²_𝐻1+𝑊_𝐴(𝑡)³_𝐻1≤𝐶|𝑡|^{−(1+𝜁)/2}𝑒^𝛿𝑡×𝑒^𝛿𝑡𝑣_𝜏(𝑡)²_𝐻1+𝑊_𝐴(𝑡)²_𝐻1, (166) since ‖𝑣_𝜏(𝑡)‖_𝐻 and 𝑒^𝛿𝑡‖𝑊_𝐴(𝑡)‖²_𝐻1 are bounded for 𝑡,𝜏∈(−∞,𝑇], the last inequality follows. For the first term on the right hand side of (154), we have 𝑒^𝜏𝐴𝐴^(1+𝜁)/2𝑊_𝐴(−𝜏)_𝐻≤𝑒^−𝛿𝜏𝐴^(1+𝜁)/2𝑊_𝐴(−𝜏)_𝐻. (167) Similar to [26], we can prove that ‖𝐴^(1+𝜁)/2𝑊_𝐴(−𝜏)‖_𝐻 has at most polynomial growth when 𝜏→∞. For the reader convenience, we sketch a proof. By Lemma 2, we know that 𝑊(𝑡)−𝑊(𝑠) is a 𝐷(𝐴^𝜃/2) valued Brownian motion, for 𝑠≤𝑡≤0. So, by the law of iterated logarithm, we have 𝑤_𝑛∶=sup_{𝑛≤𝑠≤𝑡≤𝑛+1}‖𝑊(𝑡)−𝑊(𝑠)‖_𝐻^𝜃|𝑡−𝑠|^{1/2+(𝜁−𝜃)/4}<∞,a.s.𝑛∈ℤ. (168) Obviously, 𝑤_𝑛 is a i.i.d sequence. By the law of large numbers, there exists an integer-valued random variable 𝑛₀(𝑤)>0, when 𝑛≥𝑛₀(𝑤), we have 𝑤_−𝑛𝑛≤𝑤_−𝑛+⋯+𝑤₋₁𝑛≤𝐸𝑤₀+1<∞. (169) This implies that 𝑤_−𝑛≤𝐶₀(𝑤)𝑛, (170) for all 𝑛>0. In other words, ‖𝑊(𝑡)−𝑊(𝑠)‖_𝐻^𝜃≤𝐶₀(𝑤)|[𝑠]|⋅|𝑡−𝑠|^{1/2+(𝜁−𝜃)/4}, (171) when 𝑠≤𝑡≤[𝑠]+1. By the law of iterated logarithm, we have ‖𝑊(𝑡)‖_𝐻^𝜃≤𝐶₁(𝑤)|𝑡|,𝑡∈(−∞,0], (172) for some positive random variable. By Theorem 5.14 in [23], we know that 𝑊_𝐴(𝑡)=^𝑡_−∞𝐴𝑒^{−(𝑡−𝑠)𝐴}(𝑊(𝑡)−𝑊(𝑠))𝑑𝑠. (173) So, we have that 𝐴^(1+𝜁)/2𝑊_𝐴(𝑡)_𝐻≤^𝑡_−∞𝐴^1+1/2+𝜁/2𝑒^{−(𝑡−𝑠)𝐴}(𝑊(𝑡)−𝑊(𝑠))_𝐻=𝑑𝑠_𝑡−∞𝐴^{1+1/2+(𝜁−𝜃)/2}𝑒^{−(𝑡−𝑠)𝐴}𝐴^𝜃/2(𝑊(𝑡)−𝑊(𝑠))_𝐻≤𝑑𝑠_𝑡−∞𝑒^{−𝛿(𝑡−𝑠)}|𝑡−𝑠|^{1+1/2+(𝜁−𝜃)/2}‖𝑊(𝑡)−𝑊(𝑠)‖_𝐻^𝜃≤𝑑𝑠_{𝑡[𝑡]−1}𝑒^{−𝛿(𝑡−𝑠)}|𝑡−𝑠|^{1+(𝜁−𝜃)/4}⋅‖𝑊(𝑡)−𝑊(𝑠)‖_𝐻^𝜃|𝑡−𝑠|^{1/2+(𝜁−𝜃)/4}+^[𝑡]−1_−∞𝑒^{−𝛿(𝑡−𝑠)}|𝑡−𝑠|^{1+(𝜁−𝜃)/4}⋅𝐶₁(𝑤)(|𝑡|+|𝑠|)|𝑡−𝑠|^{1/2+(𝜁−𝜃)/4}≤^𝑡_[𝑡]−1𝑒^{−𝛿(𝑡−𝑠)}|𝑡−𝑠|^{1+(𝜁−𝜃)/4}⋅𝐶₀(𝑤)|[𝑠]|𝑑𝑠+^[𝑡]−1_−∞𝑒^{−𝛿(𝑡−𝑠)}𝐶₁(𝑤𝑡|+|𝑠≤𝐶)(||)₀(𝑤)+𝐶₁(𝑤)(|𝑡|+1), (174) since 𝑠≤[𝑡]−1, the fourth inequality follows. By (167) and (174), we know that sup_𝜏≥0𝑒^𝜏𝐴𝐴^(1+𝜁)/2𝑊_𝐴(−𝜏)_𝐻<∞,a.s. (175) If we let 𝜁=1/2<𝜃, repeating the argument of (174), we can see that ‖𝑊_𝐴(𝑡)‖_𝑊1,4 also has at most polynomial growth, when 𝑡→−∞ a.s., since we have the Sobolev embedding 𝐻^3/2⊂𝑊^1,4. Consider the second term on the right hand side of (154), by (165), ⁰_−𝜏𝐴^(1+𝜁)/2𝑒^𝑡𝐴𝐵𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡),𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡)_𝐻≤𝑑𝑡_0−𝜏𝐶|𝑡|^{−(3+2𝜁)/4}+|𝑡|^{−(1+𝜁)/2}×𝑒^𝛿𝑡𝑊_𝐴(𝑡)²_𝐻1+𝑣_𝜏(𝑡)²_𝐻1𝑑𝑡≤⁰₋₁𝐶|𝑡|^{−(3+2𝜁)/4}+|𝑡|^{−(1+𝜁)/2}×𝑒^𝛿𝑡𝑊_𝐴(𝑡)²_𝐻1+𝑣_𝜏(𝑡)²_𝐻1+𝑑𝑡_−1−∞𝐶𝑒^𝛿𝑡𝑊_𝐴(𝑡)²_𝐻1+𝑣_𝜏(𝑡)²_𝐻1𝑑𝑡<∞, (176) where the last inequality follows by (152). Analogously, we can prove that ⁰_−𝜏𝐴^(1+𝜁)/2𝑒^𝑡𝐴𝑓𝑣_𝜏(𝑡)+𝑊_𝐴(𝑡)_𝐻≤𝑑𝑡_0−𝜏𝐶|𝑡|^{−(1+𝜁)/2}×𝑒^𝛿𝑡𝑣_𝜏(𝑡)_𝐻𝑣_𝜏(𝑡)²_𝐻1+𝑊_𝐴(𝑡)³_𝐻1<∞, (177) where we used (149) and (152) for the last inequality. By (154) and (175)–(177), we get 𝐴^(1+𝜁)/2𝑣_𝜏(0)_𝐻≤𝜉(𝑤),a.s., (178) for some positive random variable 𝜉(𝑤). As 𝐻^1+𝛿⊂𝐻¹ is compact, by Prohorov Theorem, we know that the family of laws for (𝑣_𝜏(0))_𝜏≥0 taking values in 𝐻¹ is tight. Since 𝑣_𝜏(0)=𝑢_𝜏(0)−𝑊_𝐴(0), then so does the law of (𝑢_𝜏(0))_𝜏≥0 taking values in the same space. For 𝑡≥0, set 𝑃_𝑡𝑓(𝑥)=𝐸𝑓(𝑢(𝑡,.;0,𝑥)), (179) where 𝑓∈𝐶_𝑏(𝐻¹₀). Following the arguments in [24], for all 𝑡₀<𝑠<𝑡 and all 𝑢_𝑡0∈𝐻¹₀, by proving 𝐸𝑓𝑢𝑡;𝑡₀,𝑢_𝑡0∣ℱ_𝑠=𝑃_𝑡−𝑠𝑢𝑠;𝑡₀,𝑢_𝑡0, (180) we can show that 𝑢 is a Markov process. Here, ℱ_𝑠 is the 𝜎-algebra generated by 𝑊(𝑟) for 𝑟≤𝑠. So, (𝑃_𝑡)_𝑡≥0 is the Markov semigroup. Define a dual semigroup 𝑃^∗_𝑡 in the space 𝑃(𝐻¹₀) of probability measures on 𝐻¹₀ as follows: 𝑓𝑑𝑃_𝐻10^∗_𝑡𝜇=_𝐻10𝑃_𝑡𝑓𝑑𝜇. (181) Let 𝜇_𝜏 be the law of 𝑢_𝜏(0), which is the solution of (2) with initial condition 𝑢(−𝜏)=0. Then, we have 𝜇_𝜏𝑓=𝐸𝑓𝑢_−𝜏(0)(𝑢(=𝑃=𝐸𝑓𝜏,⋅;0,0))_𝜏𝑓(0)=_𝐻10𝑃_𝜏𝑓𝑑𝛿₀=_𝐻10𝑓𝑑𝑃^∗_𝜏𝛿₀, (182) where we use the fact that 𝑢(𝜏,⋅;0,0) and 𝑢_𝜏(0) have the same law, the second equality follows. Therefore, 𝑃^∗_𝜏1𝜇_𝜏=𝜇_𝜏+𝜏1. (183) Since (𝜇_𝜏)_𝜏≥0 is tight, then by Prokhorov theorem, we know that (𝜇_𝜏)_𝜏≥0 is relatively compact. We can choose a subsequence of (𝜇_𝜏)_𝜏≥0 denoted by (𝜇_𝜏𝑛)_𝑛∈ℕ such that for 𝜇∈𝑃(𝐻^𝜎), _𝐻10𝑃_𝑡𝑓(𝑥)𝜇(𝑑𝑥)=lim_{𝑛→∞𝐻10}𝑃_𝑡𝑓(𝑥)𝜇_𝜏𝑛(𝑑𝑥)=lim_{𝑛→∞𝐻10}𝑓(𝑥)𝑃^∗_𝑡𝜇_𝜏𝑛(𝑑𝑥)=lim_{𝑛→∞𝐻10}𝑓(𝑥)𝜇_{𝜏𝑛+𝑡}(𝑑𝑥)=_𝐻10𝑓(𝑥)𝜇(𝑑𝑥). (184)

5.2. Uniqueness

The main result of this part is as follows.

Theorem 9. Assume 𝜃>1/2 in Lemma 2 and 𝜗>1/4; then,(i)the stochastic Burgers equation (2) has a unique invariant measure 𝜇;(ii)for all 𝑢₀∈𝐻¹₀ and all Borel measurable functions 𝜑, 𝐻¹₀→ℝ, such that ∫_𝐻10|𝜑|𝑑𝜇<∞, lim_𝑇→∞1𝑇^𝑇₀𝜑𝑢𝑡;𝑢₀𝑑𝑡=_𝐻10𝜑𝑑𝜇𝑎.𝑠.; (185)(iii)for every Borel measure 𝜇^∗ on 𝐻¹₀, one has that 𝑃^∗_𝑡𝜇^∗−𝜇_𝑇𝑉⟶0𝑎𝑠𝑡⟶∞, (186)where ‖⋅‖_𝑇𝑉 stands for the total variation of a measure. In particularly, one has that 𝑃^∗_𝑡𝜇^∗(𝐵)⟶𝜇(𝐵),𝑎𝑠𝑡⟶∞, (187) for every Borel set 𝐵∈ℬ(𝐻¹₀)(the Borel 𝜎-algebra of 𝐻¹₀).

In order to prove Theorem 9, we only need Theorem 10 below, see [28, Theorem 4.2.1]. We define 𝑃(𝑡,𝑥,⋅),𝑡>0, 𝑥∈𝐻¹₀, to be the transition probability measure that is, 𝑃(𝑡,𝑥,𝐵)=𝑃^∗_𝑡𝛿_𝑥(𝐵)=𝑃(𝑢(𝑡;𝑥)∈𝐵) (188) for 𝐵∈ℬ(𝐻¹₀).

Theorem 10. Assume that the probability measures 𝑃(𝑡,𝑥,⋅),𝑡>0,𝑥∈𝐻¹₀, are all equivalent, in the sense that they are mutually absolutely continuous. Then, Theorem 9 holds true.

In the following, we will prove the irreducibility and the strong Feller property in 𝐻¹₀ to get the equivalence of the measure 𝑃(𝑡,𝑥,⋅). For the two notations, we outline them below. For 𝑦∈𝐻¹₀,𝜀>0, let 𝐵𝑦,𝜀=𝑥∈𝐻¹₀;𝑥−𝑦_𝐻1<𝜀. (189)(I) For any 𝑥,𝑦∈𝐻¹₀, such that for all 𝜀>0, 𝑃𝑡,𝑥,𝐵𝑦,𝜀>0 (190) for each 𝑡>0.(S)For all 𝑂∈ℬ(𝐻¹₀), every 𝑡>0, and all 𝑥_𝑛,𝑥∈𝐻¹₀ such that 𝑥_𝑛→𝑥 in 𝐻¹₀, it holds that 𝑃𝑡,𝑥_𝑛,𝑂⟶𝑃(𝑡,𝑥,𝑂). (191)

Before checking the condition (I), we need Lemma 11 below. For 𝑥∈𝐻¹₀ and 𝜙∶[0,𝑇]→𝐻¹₀, set 𝑢𝑡,𝑥,𝜙=𝑣𝑡,𝑥,𝜙+𝜙(𝑡), (192) where 𝑣(𝑡,𝑥,𝜙) is solution of the following equation: 𝑑𝑣𝑑𝑡+𝐴𝑣+𝐵𝑣+𝜙,𝑣+𝜙+𝑓𝑣+𝜙=0, (193) for 𝑡∈[0,𝑇], with initial condition 𝑣(0)=𝑥. As it is proved in previously this equation has a unique solution as follows: 𝑣∈𝐶[0,𝑇];𝐻¹₀, (194) when 𝑥∈𝐻¹₀ and 𝜙∈𝐶([0,𝑇];𝐻¹₀).

Lemma 11. Define Ψ(𝜙)=𝑢(⋅,𝑥,𝜙); then, (i)the mapping Ψ∶𝐶₀[0,𝑇];𝐻^3/2⟶𝐶[0,𝑇];𝐻¹₀ (195) is continuous, where 𝐶₀([0,𝑇];𝐵)∶={ℎ∈𝐶([0,𝑇];𝐵);ℎ(0)=0} for Banach space 𝐵;(ii)for every 𝑥,𝑦∈𝐻^3/2 and 𝑇>0 there exists 𝑧∈𝐶₀([0,𝑇];𝐻^3/2) such that 𝑢(𝑇,𝑥,𝑧)=𝑦.

Proof. (i) is proved by (A.30) in the Appendix. To prove (ii), let 𝑥,𝑦,∈𝐻^3/2 and 𝑇>0, define 𝑢 as 𝑢(𝑡)=𝑒^−𝑡𝐴𝑥,𝑡∈0,𝑡₀,𝑢(𝑡)=𝑒^{−(𝑇−𝑡)𝐴}𝑦,𝑡∈𝑡₁,𝑇,𝑢(𝑡)=𝑢𝑡₀+𝑡−𝑡₀𝑡₁−𝑡₀𝑢𝑡₁−𝑢𝑡₀,𝑡∈𝑡₀,𝑡₁. (196) Obviously, 𝑢(𝑡)∈𝐶([0,𝑇];𝐻^3/2). Define 𝑣 as the solution of the following equation: 𝑑𝑑𝑡𝑣+𝐴𝑣+𝐵(𝑢,𝑢)+𝑓(𝑢)=0, (197) with initial condition 𝑣(0)=𝑥; then 𝑣∈𝐶([0,𝑇];𝐻^3/2). Set 𝑧=𝑢−𝑣; then it satisfies all the requirements of the lemma.

Proposition 12. With conditions in Theorem 9, the irreducibility property (I) is satisfied.

Proof. Let 𝑥∈𝐻^3/2 and 𝑧 be the same as (ii) in Lemma 11. By the above lemma, we have that for 𝜀>0, we can find 𝛿>0, such that ‖𝑧−𝑧‖_{𝐶0([0,𝑇];𝐻}^3/2₎<𝛿 (198) implies that ‖𝑢(⋅,𝑥,𝑧)−𝑢(⋅,𝑥,𝑧)‖_{𝐶([0,𝑇];𝐻}¹₎<𝜀. (199) If 𝜃>1/2 in Lemma 2, and denote 𝑧 and 𝑧 the corresponding Ornstein-Uhlenbeck process satisfying conditions in the lemma, then 𝑧,𝑧∈𝐶([0,𝑇];𝐻^3/2). Choose 𝛿₁>0 such that 𝛿₁<𝛿 and 𝑧∈𝑈_𝛿1=∶𝑧∈𝐶₀[0,𝑇];𝐻^3/2;‖𝑧−𝑧‖_{𝐶([0,𝑇];𝐻}^3/2₎<𝛿₁. (200) Then, for 𝑧∈𝑈_𝛿1, we have that 𝑢(𝑇,,𝑥,𝑧)−𝑦_𝐻1<𝜀. (201) Recall now that the solution 𝑢 of the stochastic Burgers equation is equal to Ψ(𝑧), 𝑧 being the Ornstein-Uhlenbeck process. Then, it remains to show that 𝑃𝑧(⋅,𝑤)∈𝑈_𝛿1>0. (202) But this is obviously true. So far, we have proved that for for all 𝑡>0, forall𝑥,𝑦∈𝐻^3/2, for all 𝜀>0, 𝑃𝑡,𝑥,𝐵𝑦,𝜀>0. (203) Next, we will prove for all 𝑥₀∈𝐻¹₀,𝑦₀∈𝐻^3/2, the above inequality also holds. Indeed, for 0<ℎ<𝑡, by Chapman-Kolmogorov equation, we have 𝑃𝑡,𝑥₀,𝐵𝑦₀=,𝜀_𝐻10𝑃𝑡−ℎ,𝑥₀,𝑑𝑦𝑃ℎ,𝑦,𝐵𝑦₀=,𝜀_𝐻3/2𝑃𝑡−ℎ,𝑥₀,𝑑𝑦𝑃ℎ,𝑦,𝐵𝑦₀,𝜀>0. (204) Since 𝑃(𝑡−ℎ,𝑥₀,𝐻^3/2)=1, we will extend (204) to the case for all 𝑥₀∈𝐻¹₀,𝑦₀∈𝐻¹₀. If this is not true, there exists 𝑡₀>0,  𝑥₀,𝑦₀∈𝐻¹₀, 𝜀>0 such that 𝑃𝑡₀,𝑥₀,𝐵𝑦₀,𝜀=0. (205) Then, we can choose 𝑦₁∈𝐻^3/2,  𝜀₁>0 such that 𝐵(𝑦₁,𝜀₁)⊂𝐵(𝑦₀,𝜀). By (204), we have 𝑃𝑡₀,𝑥₀,𝐵𝑦₁,𝜀₁>0, (206) which is contrary to (205).

In this part, it is time to check the condition (S).

We will first obtain the strong Feller property in 𝐻¹₀ for modified Burgers equation (208) below, then let 𝑅→∞ to check the condition (S).

Fix 𝑅>0, let 𝐾_𝑅∶[0,∞[→[0,∞[ satisfy 𝐾_𝑅∈𝐶¹(ℝ₊) such that |𝐾_𝑅|≤1,  |𝐾^′_𝑅|≤2 and 𝐾_𝑅𝐾=1,if𝑥<𝑅,_𝑅=0,if𝑥≥𝑅+1. (207) Consider the following equation: 𝑑𝑢_𝑅(𝑡)+𝐴𝑢_𝑅(𝑡)𝑑𝑡+𝐾_𝑅𝑢_𝑅(𝑡)²_𝐻1𝐵𝑢_𝑅(𝑡),𝑢_𝑅(𝑡)𝑑𝑡+𝐾_𝑅𝑢_𝑅(𝑡)²_𝐻1𝑓𝑢_𝑅(𝑡)=𝑑𝑊(𝑡). (208)

Proposition 13. There exists a unique mild solution 𝑢_𝑅(⋅,𝑤)∈𝐶([0,𝑇];𝐻¹₀) for (208) which is Markov process with the Feller property in 𝐻¹₀, that is for every 𝑅>0, 𝑡>0, there exists a constant 𝐿=𝐿(𝑡,𝑅)>0 such that 𝑃_𝑡^(𝑅)𝜙(𝑥)−𝑃_𝑡^(𝑅)𝜙𝑦≤𝐿𝑥−𝑦_𝐻1 (209) holds for all 𝑥,𝑦∈𝐻¹₀, and all 𝜙∈𝐶_𝑏(𝐻¹₀)≤1, where 𝑃_𝑡^(𝑅)𝜙(𝑥)∶=∫_𝐻10𝜙(𝑦)𝑃_𝑅(𝑡,𝑥,𝑑𝑦),𝑃_𝑅(𝑡,𝑥,⋅) is the transition probabilities corresponding to (204).

Proof. The proof of existence and uniqueness is similar to Section 2. Let 𝜙₁=𝜙₂ in (A.28), by the Gronwall inequality, we know that 𝑢_𝑅 is Lipschitz continuous with respect to initial value. Using the method in Proposition 4.3.3 in [24], we can prove that the solution is a Markov process. To prove the Fell property, we first consider the following Galerkin approximations of (208). Let 𝑃_𝑛 be the orthogonal projection in 𝐻 defined as 𝑃_𝑛𝑥=∑^𝑛_𝑗=1⟨𝑥,𝑒_𝑗⟩𝑒_𝑗,𝑥∈𝐻. Clearly, 𝐻_𝑛∶=𝑃_𝑛𝐻 for every 𝑛. Consider the equation in 𝐻_𝑛 as follows: 𝑑𝑢_𝑛^(𝑅)(𝑡)+𝐴𝑢_𝑛^(𝑅)(𝑡)𝑑𝑡+𝐾_𝑅𝑢_𝑛^(𝑅)(𝑡)²_𝐻1𝑃_𝑛𝐵𝑢_𝑛^(𝑅)(𝑡),𝑢_𝑛^(𝑅)(𝑡)+𝐾_𝑅𝑢_𝑛^(𝑅)(𝑡)²_𝐻1𝑓𝑢_𝑛^(𝑅)(𝑡)=𝑑𝑊(𝑡), (210) with initial condition 𝑢_𝑛^(𝑅)(0)=𝑃_𝑛𝑢₀. This is a finite-dimensional equation with globally Lipschitz nonlinear functions, so it has a unique progressively measurable solution with 𝑃-a.e. trajectory 𝑢_𝑛^(𝑅)(⋅,𝑤)∈𝐶([0,𝑇];𝐻_𝑛), which is also a Markov process in 𝐻_𝑛 with associated semigroup 𝑃^(𝑅)_𝑛,𝑡 defined as 𝑃^(𝑅)_𝑛,𝑡𝜙(𝑥)=𝐸𝜙𝑢_𝑛^(𝑅)(𝑡;𝑥), (211) for all 𝑥∈𝐻_𝑛 and 𝜙∈𝐶_𝑏(𝐻_𝑛). For every 𝑅>0,𝑡>0, we can prove that there exists a constant 𝐿=𝐿(𝑡,𝑅)>0 such that 𝑃^(𝑅)_𝑛,𝑡𝜙(𝑥)−𝑃^(𝑅)_𝑛,𝑡𝜙𝑦≤𝐿𝑥−𝑦_𝐻1 (212) hold for all 𝑛∈ℕ,𝑥,𝑦∈𝐻_𝑛, and all 𝜙∈𝐶_𝑏(𝐻_𝑛) with ‖𝜙‖_𝐻1≤1. Indeed, the following remarkable formula holds true for the differential in 𝑥 of 𝑃^(𝑅)_𝑛,𝑡𝜙 [29]: 𝐷_𝑥𝑃^(𝑅)_𝑛,𝑡𝜙(𝑥)=1⋅ℎ𝑡𝐸𝜙𝑢_𝑛^(𝑅)(𝑡;𝑥)^𝑡₀𝑃_𝑛𝑄𝑄^∗𝑃_𝑛^−1/2𝐷_𝑥𝑢_𝑛^(𝑅)(𝑠;𝑥)⋅ℎ,𝑑𝛽_𝑛(𝑠), (213) for all ℎ∈𝐻_𝑛, where 𝛽_𝑛 is a 𝑛-dimensional standard Wiener process with incremental covariance 𝑃_𝑛𝑄 and 𝑄 is the covariance operator of 𝑊(𝑡). Obviously, 𝑄 is nonnegative, adjoint, Hilbert-Schmidt operator with inverse. Since the eigenvalues 𝛼_𝑛 of the Stokes operator 𝐴, in 2-space dimension, behave like 𝑛, let 𝜃=1/2+𝜀 for some 𝜀>0, in Lemma 2, we have 𝐷(𝐴)⊂ℛ(𝑄)⊂𝐷(𝐴^3/4), where ℛ(𝑄) is the image of 𝑄. Therefore, 𝐷_𝑥𝑃^(𝑅)_𝑛,𝑡𝜙(𝑥)≤1⋅ℎ𝑡𝐸^𝑡₀𝑃_𝑛𝑄𝑄^∗𝑃_𝑛^−1/2𝐷_𝑥𝑢_𝑛^(𝑅)(𝑠;𝑥)⋅ℎ²_𝐻𝑑𝑠^1/2. (214) Since for 𝑦∈𝐻_𝑛𝑃_𝑛𝑄𝑄^∗𝑃_𝑛^−1/2𝑦²_𝐻=𝑃_𝑛𝑄𝑄^∗𝑃_𝑛⁻¹=𝑦,𝑦𝐴𝑃_𝑛𝑄𝑄^∗𝑃_𝑛𝐴⁻¹𝐴𝑦,𝐴𝑦≤𝐶𝑦²_𝐻2, (215) it follows that 𝐷_𝑥𝑃^(𝑅)_𝑛,𝑡𝜙(𝑥)≤1⋅ℎ𝑡𝐶𝐸^𝑡₀𝐷_𝑥𝑢_𝑛^(𝑅)(𝑠;𝑥)⋅ℎ²_𝐻2𝑑𝑠^1/2≤1𝑡𝐶(𝑅)‖ℎ‖_𝐻¹, (216) where the last inequality follows by the Estimate 4 of the Appendix (note that 𝐶(𝑅) is independent of 𝑥∈𝐻_𝑛 and 𝑛∈ℕ). Indeed, 𝑢_𝑛^(𝑅)(𝑡,𝑥) is given by 𝑣_𝑛(𝑡,𝑥)+𝑃_𝑛𝑧(𝑡), where 𝑧 is the Ornstein-Uhlenbeck process, and 𝑣_𝑛 is the solution of (A.2). Therefore, 𝑃^(𝑅)_𝑛,𝑡𝜙(𝑥)−𝑃^(𝑅)_𝑛,𝑡𝜙𝑦≤sup_{‖ℎ‖𝐻1≤1,𝑘∈𝐻𝑛}𝐷_𝑥𝑃^(𝑅)_𝑛,𝑡𝜙(𝑘)⋅ℎ⋅𝑥−𝑦_𝐻1≤1𝑡𝐶(𝑅)𝑥−𝑦_𝐻1. (217) In the following step, we will let 𝑛→∞ to get the Fell property for (208). Let 𝑥∈𝐻¹₀ and 𝜙∈𝐶_𝑏(𝐻¹₀) be given. From the Appendix, Remark A.1, we know that 𝑢_𝑛^(𝑅)(𝑡) converges to 𝑢^(𝑅)(𝑡) strongly in 𝐿²(0,𝑇;𝐻¹₀), 𝑝-a.s.. By the boundedness and continuous of 𝜙 as well as Lebesgue dominated convergence theorem, we have 𝐸^𝑇₀𝜙𝑢_𝑛^(𝑅)(,;𝑥)−𝜙𝑢^(𝑅)(,;𝑥)𝑑𝑡⟶0, (218) which implies that for some subsequence 𝑛_𝑘, 𝐸𝜙𝑢_𝑛^(𝑅)_𝑘(,;𝑥)⟶𝐸𝜙𝑢^(𝑅)(,;𝑥), (219) for a.e. 𝑡∈[0,𝑇]. Take 𝑥,𝑦∈𝐻¹₀, by the previous argument, we can find a subsequence 𝑛_𝑘 such that the previous almost sure convergence in 𝑡∈[0,𝑇] holds true both 𝑥 and 𝑦.
Thus, from (212), we have 𝑃_𝑡^(𝑅)𝜙(𝑥)−𝑃_𝑡^(𝑅)𝜙𝑦≤𝐿𝑥−𝑦_𝐻1, (220) for a.e. 𝑡∈[0,𝑇]. As 𝑢^(𝑅)(𝑡;𝑥) has continuous trajectories with values in 𝐻¹₀, the above inequality holds for all 𝑡∈[0,𝑇].

Proposition 14. Under conditions of Theorem 9, (S) holds true.

Proof. Take 𝑡>0,𝑥_𝑛,𝑥∈𝐻¹₀ satisfying 𝑥_𝑛→𝑥 in 𝐻¹. For every 𝑅>0, we have that 𝑃_𝑅𝑡,𝑥_𝑛,⋅−𝑃_𝑅(𝑡,𝑥,⋅)_𝑇𝑉=sup_{𝜙𝐶𝑏1)(𝐻≤1}𝑃_𝑡^(𝑅)𝜙𝑥_𝑛−𝑃_𝑡^(𝑅)𝜙(𝑥𝑥)≤𝐿_𝑛−𝑥_𝐻1⟶0, (221) as 𝑛→∞ by Proposition 13. Then, 𝑃_𝑅𝑡,𝑥_𝑛,⋅−𝑃𝑡,𝑥_𝑛,⋅_𝑇𝑉+𝑃_𝑅(𝑡,𝑥,⋅)−𝑃(𝑡,𝑥,⋅)_𝑇𝑉=sup_{𝜙𝐶𝑏1)(𝐻≤1}𝑃_𝑡^(𝑅)𝜙𝑥_𝑛−𝑃_𝑡𝜙𝑥_𝑛+sup_{𝜙𝐶𝑏1)(𝐻≤1}𝑃_𝑡^(𝑅)𝜙(𝑥)−𝑃_𝑡𝜙(𝑥)=sup_{𝜙𝐶𝑏1)(𝐻≤1}𝐸𝜙𝑢_𝑅𝑡;𝑥_𝑛−𝐸𝜙𝑢𝑡;𝑥_𝑛+sup_{𝜙𝐶𝑏1)(𝐻≤1}𝐸𝜙𝑢_𝑅(𝑡;𝑥)−𝐸𝜙(𝑢(𝑡;𝑥))≤2_Ω𝐼_{{sup𝑛∈ℕ‖𝑢(𝑡;𝑥𝑛)‖𝐻1>𝑅}}𝑃(𝑑𝑤)+2_Ω𝐼_{{‖𝑢(𝑡;𝑥)‖𝐻1>𝑅}}𝑃(𝑑𝑤)⟶0,as𝑅→∞, (222) where the inequality follows by the consistency of 𝑢(𝑡;𝑥) and 𝑢^(𝑅)(𝑡;𝑥), when ‖𝑢(𝑡;𝑥)‖_𝐻¹≤𝑅, and the limit follows by (A.21). Therefore, 𝑃𝑡,𝑥_𝑛,⋅−𝑃(𝑡,𝑥,⋅)_𝑇𝑉≤𝑃𝑡,𝑥_𝑛,⋅−𝑃_𝑅𝑡,𝑥_𝑛,⋅_𝑇𝑉+𝑃_𝑅𝑡,𝑥_𝑛,⋅−𝑃_𝑅(𝑡,𝑥,⋅)_𝑇𝑉+𝑃_𝑅(𝑡,𝑥,⋅)−𝑃(𝑡,𝑥,⋅)_𝑇𝑉⟶0, (223) as 𝑛→∞.

6. Example

Our theory can be applied to stochastic reaction diffusion equations or stochastic real valued Ginzburg Landau equation in high dimensions as follows: 𝜕𝑢|𝑢|𝜕𝑡−Δ𝑢+²[]𝑢()[]𝑢()𝑢−𝑢=𝑑𝑊,on0,𝑇×𝐷,𝑡,𝑥=0,𝑡∈0,𝑇,𝑥∈𝜕𝐷,0,𝑥=𝑢₀(𝑥),𝑥∈𝐷, (224) where 𝑢(𝑡,𝑥)=(𝑢¹(𝑡,𝑥),𝑢²(𝑡,𝑥)) is the velocity field, Δ denotes the Laplace operator, 𝑊 stands for the 𝑄-Wiener process, and 𝐷 is a regular bounded open domain of ℝ².

Appendix

Fix 𝑅>0 and let 𝐾_𝑅∶[0,∞[→[0,∞[ satisfy 𝐾_𝑅∈𝐶¹(ℝ₊) such that |𝐾_𝑅|≤1,|𝐾^′_𝑅|≤2 and 𝐾_𝑅(𝑥)𝐾=1,if𝑥<𝑅,_𝑅(𝑥)=0,if𝑥≥𝑅+1. (A.1) Consider the following equation: 𝑑𝑣_𝑛𝑑𝑡+𝐴𝑣_𝑛+𝐾_𝑅𝑣_𝑛+𝑃_𝑛𝜙²_𝐻1×𝑃_𝑛𝐵𝑣_𝑛+𝑃_𝑛𝜙,𝑣_𝑛+𝑃_𝑛𝜙+𝐾_𝑅𝑣_𝑛+𝑃_𝑛𝜙²_𝐻1𝑓𝑣_𝑛+𝑃_𝑛𝜙=0, (A.2) where 𝜙∈𝐶([0,𝑇];𝐻^3/2).

Estimate 1. We have the following estimate in 𝐻 for (A.2): 𝑣_{𝑛𝐶([0,𝑇];𝐻)}+𝑣_{𝑛𝐿2([0,𝑇];𝐻1)}≤𝐶‖𝑥‖_𝐻,𝜙_{𝐶([0,𝑇];𝐻3/2)},𝑇, (A.3) where 𝐶(𝑎,𝑏,𝑐) indicates a constant 𝐶 depending on 𝑎,𝑏,𝑐. Analogously to the derivation of (147), we get 𝑑𝑣𝑑𝑡^𝑛2𝐻+𝑣_𝑛²_𝐻1+𝑣_𝑛⁴_𝐿4≤𝐶𝜙⁴_𝐻3/2+1. (A.4) Therefore, for all 𝑡∈[0,𝑇], 𝑣_𝑛(𝑡)²_𝐻+^𝑡₀𝑣_𝑛²_𝐻1𝑑𝑠+^𝑡₀𝑣_𝑛⁴_𝐿4≤‖𝑥‖𝑑𝑠_2𝐻+𝐶^𝑡₀𝜙(𝑠)⁴_𝐻3/2+1, (A.5) Then, we get (A.3).

Estimate 2. We obtain the following estimate in 𝐻¹₀ for (A.2): 𝑣_𝑛(𝑡)²_{𝐶([0,𝑇];𝐻10)}+^𝑇₀𝑣_𝑛(𝑠)²_𝐻2‖𝑥‖𝑑𝑠≤𝐶_𝐻¹,𝜙_{𝐶([0,𝑇];𝐻3/2)},𝑇. (A.6) Since we have 𝑑𝑣𝑑𝑡^𝑛(𝑡)²_𝐻1+𝑣_𝑛(𝑡)²_𝐻2+𝑓𝑣_𝑛(𝑡)+𝑃_𝑛𝜙(𝑡),𝐴𝑣_𝑛(𝑡=𝐵𝑣_𝑛)(𝑡)+𝑃_𝑛𝜙(𝑡),𝑣_𝑛(𝑡)+𝑃_𝑛𝜙(𝑡),𝐴𝑣_𝑛(𝑡), (A.7) the equation is equivalent to 𝑑𝑣𝑑𝑡^𝑛(𝑡)²_𝐻1+𝑣_𝑛(𝑡)²_𝐻2+𝑓𝑣_𝑛(𝑡)+𝑃_𝑛𝜙(𝑡),𝐴𝑣_𝑛(𝑡)+𝑃_𝑛𝜙(𝑡)=𝐵𝑣_𝑛(𝑡)+𝑃_𝑛𝜙(𝑡),𝑣_𝑛(𝑡)+𝑃_𝑛𝜙(𝑡),𝐴𝑣_𝑛(𝑡+𝑓𝑣_𝑛)(𝑡)+𝑃_𝑛𝜙(𝑡),𝐴𝑃_𝑛𝜙(𝑡). (A.8) Denote by 𝑢_𝑛∶=𝑣_𝑛(𝑡)+𝑃_𝑛𝜙(𝑡) and 𝑢_𝑛=(𝑢¹_𝑛,𝑢²_𝑛); then 𝑢_𝑛²𝑢_𝑛,𝐴𝑢_𝑛=3_𝐷𝑢¹_𝑛²𝜕₁𝑢¹_𝑛²𝑑𝑥+3_𝐷𝑢¹_𝑛²𝜕₂𝑢¹_𝑛²𝑑𝑥+3_𝐷𝑢²_𝑛²𝜕₁𝑢²_𝑛²𝑑𝑥+3_𝐷𝑢²_𝑛²𝜕₂𝑢¹_𝑛²+𝑑𝑥_𝐷𝑢²_𝑛²𝜕₁𝑢¹_𝑛²𝑑𝑥+_𝐷(𝑢²_𝑛)²𝜕₂𝑢¹_𝑛²+𝑑𝑥_𝐷𝑢¹_𝑛²𝜕₁𝑢²_𝑛²+𝑑𝑥_𝐷𝑢¹_𝑛²𝜕₂𝑢²_𝑛²𝑑𝑥+4_𝐷𝑢¹_𝑛𝜕₁𝑢¹_𝑛𝑢²_𝑛𝜕₁𝑢²_𝑛𝑑𝑥+4_𝐷𝑢¹_𝑛𝜕₂𝑢¹_𝑛𝑢²_𝑛𝜕₂𝑢²_𝑛𝑑𝑥. (A.9) As 4_𝐷𝑢¹_𝑛𝜕₁𝑢¹_𝑛𝑢²_𝑛𝜕₁𝑢²_𝑛𝑑𝑥≤2_𝐷𝑢¹_𝑛²𝜕₁𝑢¹_𝑛²𝑑𝑥+2_𝐷𝑢²_𝑛²𝜕₁𝑢²_𝑛²4𝑑𝑥,_𝐷𝑢¹_𝑛𝜕₂𝑢¹_𝑛𝑢²_𝑛𝜕₂𝑢²_𝑛𝑑𝑥≤2_𝐷𝑢¹_𝑛²𝜕₂𝑢¹_𝑛²𝑑𝑥+2_𝐷𝑢²_𝑛²𝜕₂𝑢²_𝑛²𝑑𝑥, (A.10) so, we have that 𝑢_𝑛²𝑢_𝑛,𝐴𝑢_𝑛≥_𝐷𝑢¹_𝑛²𝜕₁𝑢¹_𝑛²𝑑𝑥+_𝐷𝑢¹_𝑛²𝜕₂𝑢¹_𝑛²+𝑑𝑥_𝐷𝑢²_𝑛²𝜕₁𝑢²_𝑛²+𝑑𝑥_𝐷𝑢²_𝑛²𝜕₂𝑢²_𝑛²𝑑𝑥+_𝐷𝑢²_𝑛²𝜕₁𝑢¹_𝑛²+𝑑𝑥_𝐷𝑢²_𝑛²𝜕₂𝑢¹_𝑛²+𝑑𝑥_𝐷𝑢¹_𝑛²𝜕₁𝑢²_𝑛²𝑑𝑥+_𝐷𝑢¹_𝑛²𝜕₂𝑢²_𝑛²=𝑑𝑥_𝐷𝑢_𝑛²∇𝑢_𝑛²𝑑𝑥. (A.11) For the first term on the right hand side of (A.3), we have 𝑢_𝑛⋅∇𝑢_𝑛,𝐴𝑣_𝑛(𝑡≤𝑣_𝑛)(𝑡)²_𝐻2+14_𝐷𝑢_𝑛²∇𝑢_𝑛²𝑑𝑥. (A.12) Substitute (A.11) and (A.12) into (A.8), we get 𝑑𝑣𝑑𝑡^𝑛(𝑡)²_𝐻1+𝑣_𝑛(𝑡)²_𝐻2≤𝑓𝑣_𝑛(𝑡)+𝑃_𝑛𝜙(𝑡)(𝑡=𝐴,𝐴𝜙)_1/4𝑓𝑣_𝑛(𝑡)+𝑃_𝑛𝜙(𝑡),𝐴^3/4𝜙(𝑡). (A.13) Denote 𝑢_𝑛(𝑡)=𝑣_𝑛(𝑡)+𝑃_𝑛𝜙(𝑡). (A.14) Then, 𝐴^1/4𝑓𝑣_𝑛(𝑡)+𝑃_𝑛𝜙(𝑡),𝐴^3/4𝜙(𝑡)≤𝜙(𝑡)_𝐻3/2⋅𝐴^1/4𝑢_𝑛(𝑡)²𝑢_𝑛(𝑡)_𝐻≤𝜙(𝑡)_𝐻3/2⋅𝐴^1/4𝑢_𝑛(𝑡)²𝑢_𝑛(𝑡)+𝑢_𝑛(𝑡)²𝐴^1/4𝑢_𝑛(𝑡)+𝑅_𝐻≤𝜙(𝑡)_𝐻3/2⋅𝐴^1/4𝑢_𝑛(𝑡)²𝑢_𝑛(𝑡)_𝐻+𝑢_𝑛(𝑡)²𝐴^1/4𝑢_𝑛(𝑡)_𝐻+‖𝑅‖_𝐻=𝜙(𝑡)_𝐻3/2⋅𝐼₁+𝐼₂+𝐼₃, (A.15) where 𝑅=𝐴^1/4𝑢_𝑛(𝑡)²𝑢_𝑛(𝑡)−𝐴^1/4𝑢_𝑛(𝑡)²𝑢_𝑛(𝑡)−𝑢_𝑛(𝑡)²𝐴^1/4𝑢_𝑛(𝑡). (A.16) For 𝐼₁, we have 𝐼₁≤𝑢_𝑛(𝑡)𝐴^1/4𝑢_𝑛(𝑡)+𝑅₁𝑢_𝑛(𝑡)_𝐻, (A.17) where 𝑅₁=𝐴^1/4𝑢_𝑛(𝑡)²−2𝑢_𝑛(𝑡)𝐴^1/4𝑢_𝑛(𝑡). (A.18) So, 𝐼₁≤𝐶𝑢_𝑛(𝑡)²𝐴^1/4𝑢_𝑛(𝑡)+𝑅₁𝑢_𝑛(𝑡)_𝐻≤𝐶𝑢_𝑛(𝑡)²_𝐿8𝑢_𝑛(𝑡)_𝐻1/2,4+𝑢_𝑛(𝑡)_𝐿4𝑅_1𝐿4≤𝐶𝑢_𝑛(𝑡)²_𝐿8𝑢_𝑛(𝑡)_𝐻1/2,4+𝑢_𝑛(𝑡)_𝐿4𝑢_𝑛(𝑡)_𝐻1/4,8≤𝐶𝑢_𝑛(𝑡)³_𝐻1. (A.19)
Analogously, we can get the same estimate for 𝐼₂ and 𝐼₃. Take advantage of the estimates for 𝐼₁,  𝐼₂, and 𝐼₃, we have 𝑑𝑣𝑑𝑡^𝑛(𝑡)²_𝐻1+𝑣_𝑛(𝑡)²_𝐻2≤𝐶𝜙(𝑡)_𝐻3/2𝑢_𝑛(𝑡)³_𝐻1≤𝐶𝑣_𝑛(𝑡)³_𝐻1+𝜙(𝑡)³_𝐻3/2. (A.20) By the Gronwall inequality and (A.3), we get (A.6).

Remark A.1. It is standard to show that, for 𝑥∈𝐻¹₀ and 𝜙∈𝐶([0,𝑇];𝐻^3/2), there exists a subsequence which converges to some 𝑣, strongly in 𝐿²([0,𝑇];𝐻¹), weekly in 𝐿²([0,𝑇];𝐻²), and weak star in 𝐿^∞([0,𝑇];𝐻¹). Therefore, we have ‖𝑣(𝑡)‖²_{𝐶([0,𝑇];𝐻10)}+^𝑇₀‖𝑣(𝑠)‖²_𝐻2‖𝑥‖𝑑𝑠≤𝐶_𝐻¹,𝜙_{𝐶([0,𝑇];𝐻3/2)},𝑇. (A.21)

Estimate 3. We compare, only in the case 𝑅=∞. Let 𝑣¹_𝑛,𝑣²_𝑛 be two solutions with the same initial condition 𝑥∈𝐻¹ but with different functions 𝜙₁,  𝜙₂, there exists a constant 𝐶(‖𝑥‖_𝐻¹,‖𝜙₁‖_{𝐶([0,𝑇];𝐻3/2)},‖𝜙₂‖_{𝐶([0,𝑇];𝐻3/2)},𝑇), such that 𝑣¹_𝑛−𝑣²_{𝑛𝐶([0,𝑇];𝐻10)}≤𝐶‖𝑥‖_𝐻¹,𝜙_{1𝐶([0,𝑇];𝐻3/2)},𝜙_{2𝐶([0,𝑇];𝐻3/2)}×𝜙,𝑇₁−𝜙_{2𝐶([0,𝑇];𝐻3/2)}, (A.22) for every 𝑛,𝑥∈𝐻¹,𝜙₁,𝜙₂,𝑇. We have 𝑑𝑣^𝑖_𝑛𝑑𝑡+𝐴𝑣^𝑖_𝑛+𝑃_𝑛𝐵𝑣^𝑖_𝑛+𝑃_𝑛𝜙_𝑖,𝑣^𝑖_𝑛+𝑃_𝑛𝜙_𝑖+𝜗𝑣^𝑖_𝑛+𝑃_𝑛𝜙_𝑖²𝑣^𝑖_𝑛+𝑃_𝑛𝜙_𝑖=0, (A.23) with initial condition 𝑣^𝑖_𝑛(0)=𝑃_𝑛𝑥, for 𝑖=1,2. Set 𝜂_𝑛=𝑣¹_𝑛−𝑣²_𝑛,𝜓=𝜙₁−𝜙₂. Then, 𝑑𝜂_𝑛𝑑𝑡+𝐴𝜂_𝑛+𝑃_𝑛𝐵𝑣¹_𝑛+𝑃_𝑛𝜙₁,𝜂_𝑛+𝑃_𝑛𝜓+𝑃_𝑛𝐵𝜂_𝑛+𝑃_𝑛𝜓,𝑣²_𝑛+𝑃_𝑛𝜙₂+𝜗𝑣¹_𝑛+𝜙₁²𝑣¹_𝑛+𝜙₁−𝜗𝑣²_𝑛+𝜙₂²𝑣²_𝑛+𝜙₂=0. (A.24)
Take inner product in 𝐻 with respect to 𝐴𝜂_𝑛, we have 12𝑑𝜂𝑑𝑡^𝑛2𝐻1+𝜂_𝑛²_𝐻2+𝑃_𝑛𝐵𝑣¹_𝑛+𝑃_𝑛𝜙₁,𝜂_𝑛+𝑃_𝑛𝜓,𝐴𝜂_𝑛+𝑃_𝑛𝐵𝜂_𝑛+𝑃_𝑛𝜓,𝑣²_𝑛+𝑃_𝑛𝜙₂,𝐴𝜂_𝑛+𝜗𝑣¹_𝑛+𝜙₁²𝑣¹_𝑛+𝜙₁−𝑣²_𝑛+𝜙₂²𝑣²_𝑛+𝜙₂,𝐴𝜂_𝑛=0. (A.25)
For the third term on the left hand side of (A.23), we have 𝑃_𝑛𝐵𝑣¹_𝑛+𝑃_𝑛𝜙₁,𝜂_𝑛+𝑃_𝑛𝜓,𝐴𝜂_𝑛≤𝜂_𝑛𝐻2𝜂_𝑛+𝑃_𝑛𝜓_𝐻1,4𝑣¹_𝑛+𝑃_𝑛𝜙_1𝐿4≤𝜂_𝑛𝐻2𝜂_𝑛𝐻3/2+𝜓_𝐻3/2𝑣¹_𝑛+𝜙_1𝐻1≤𝑣¹_𝑛+𝜙_1𝐻1𝜂_𝑛𝐻2𝜂_𝑛𝐻^1/2₁𝜂_𝑛𝐻^1/2₂+𝜓_𝐻3/2≤𝜀𝜂_𝑛²_𝐻2+𝐶𝑣¹_𝑛+𝜙₁⁴_𝐻1𝜂_𝑛²_𝐻1+𝐶𝑣¹_𝑛+𝜙₁²_𝐻1𝜓²_𝐻3/2. (A.26)
Similarly, we can get 𝑃_𝑛𝐵𝜂_𝑛+𝑃_𝑛𝜓,𝑣²_𝑛+𝑃_𝑛𝜙₂,𝐴𝜂_𝑛≤𝜀𝜂_𝑛²_𝐻2+𝐶𝑣²_𝑛²_𝐻2𝜂_𝑛²_𝐻1+𝐶𝑣²_𝑛²_𝐻2𝜓²_𝐻1+𝐶𝜙₂²_𝐻3/2𝜂_𝑛²_𝐻1+𝐶𝜙₂²_𝐻3/2𝜓²_𝐻1,𝜗𝑣¹_𝑛+𝜙₁²𝑣¹_𝑛+𝜙₁−𝑣²_𝑛+𝜙₂²𝑣²_𝑛+𝜙₂,𝐴𝜂_𝑛≤𝜀𝜂_𝑛²_𝐻2+𝐶𝜂_𝑛²_𝐻1𝑣¹_𝑛+𝜙₁⁴_𝐻1+𝑣²_𝑛+𝜙₂⁴_𝐻1+𝐶𝜓²_𝐻1𝑣¹_𝑛+𝜙₁⁴_𝐻1+𝑣²_𝑛+𝜙₂⁴_𝐻1. (A.27)(A.28)
By (A.23)–(A.27), we have 𝑑𝜂𝑑𝑡^𝑛2𝐻1+𝜂_𝑛²_𝐻2≤𝐶𝜂_𝑛²_𝐻1𝑣¹_𝑛+𝜙₁⁴_𝐻1+𝑣²_𝑛+𝜙₂⁴_𝐻1+𝑣¹_𝑛+𝜙₁²_𝐻1+𝑣²_𝑛²_𝐻2+𝜙₂²_𝐻3/2+𝐶𝜓²_𝐻3/2𝑣¹_𝑛+𝜙₁⁴_𝐻1+𝑣²_𝑛+𝜙₂⁴_𝐻1+𝑣¹_𝑛+𝜙₁²_𝐻1+𝑣²_𝑛²_𝐻2+𝜙₂²_𝐻3/2. (A.29)
So, by the Gronwall inequality and (A.6), we get (A.21). By (A.6), we know that 𝑣^𝑖_𝑛 converges week star to 𝑣^𝑖 in 𝐶([0,𝑇];𝐻¹₀), for 𝑖=1,2, we have 𝑣¹−𝑣²_{𝐶([0,𝑇];𝐻1)}≤𝐶‖𝑥‖_𝐻¹,𝜙_{1𝐶([0,𝑇];𝐻3/2)},𝜙_{2𝐶([0,𝑇];𝐻3/2)}𝜙,𝑇₁−𝜙_{2𝐶([0,𝑇];𝐻3/2)}. (A.30)

Estimate 4. Let us consider only the case 𝑅∈(0,∞), and denote by 𝑣_𝑛(𝑡) the solution to (A.2). Let 𝜉_𝑛 be the differential mapping 𝑥→𝑣_𝑛 in the direction ℎ at point 𝑥, defined by, for given 𝑥,ℎ∈𝐻 as follows: 𝜉_𝑛(𝑡)=𝐷_𝑥𝑣_𝑛(𝑡;𝑥)⋅ℎ. (A.31)
Set also 𝑢_𝑛(𝑡;𝑥)=𝑣_𝑛(𝑡,𝑥)+𝑃_𝑛𝜙(𝑡), (A.32)
so that 𝜉_𝑛 is also the differential of the mapping 𝑥→𝑢_𝑛(𝑡;𝑥) in the direction ℎ at the point 𝑥. Thus, 𝜉_𝑛 satisfies 𝑑𝜉𝑑𝑡^𝑛+𝐴𝜉_𝑛=2𝐾^′_𝑅𝑢_𝑛²_𝐻1𝐴^1/2𝑢_𝑛,𝐴^1/2𝜉_𝑛𝐵𝑢_𝑛,𝑢_𝑛+𝐾_𝑅𝑢_𝑛²_𝐻1𝐵𝑢_𝑛,𝜉_𝑛+𝐵𝜉_𝑛,𝑢_𝑛+2𝐾^′_𝑅𝑢_𝑛²_𝐻1𝐴^1/2𝑢_𝑛,𝐴^1/2𝜉_𝑛𝑢³_𝑛+3𝐾_𝑅𝑢_𝑛²_𝐻1𝑢_𝑛²𝜉_𝑛. (A.33)
So, 𝑑𝜉𝑑𝑡^𝑛2𝐻1+𝜉_𝑛²_𝐻2=2𝐾^′_𝑅𝑢_𝑛²_𝐻𝐵𝑢_𝑛,𝑢_𝑛,𝐴𝜉_𝑛+𝐾_𝑅𝑢_𝑛²_𝐻𝐵𝑢_𝑛,𝜉_𝑛,𝐴𝜉_𝑛+𝐾_𝑅𝑢_𝑛²_𝐻𝐵𝜉_𝑛,𝑢_𝑛,𝐴𝜉_𝑛+2𝐾^′_𝑅𝑢_𝑛²_𝐻𝐴^1/2𝑢_𝑛,𝐴^1/2𝜉_𝑛𝑢_𝑛²𝑢_𝑛,𝐴𝜉_𝑛+3𝐾_𝑅𝑢_𝑛²_𝐻𝑢_𝑛²𝜉_𝑛,𝐴𝜉_𝑛. (A.34)
Therefore, 𝑑𝜉𝑑𝑡^𝑛2𝐻1+𝜉_𝑛²_𝐻2≤2𝐾^′_𝑅𝑢_𝑛²_𝐻𝑢_𝑛𝐻1𝜉_𝑛𝐻1𝜉_𝑛𝐻2𝑢_𝑛𝐿4𝑢_𝑛𝐻1,4+𝐾_𝑅𝑢_𝑛²_𝐻1𝜉_𝑛𝐻2𝜉_𝑛𝐻1,4𝑢_𝑛𝐿4+𝐾_𝑅𝑢_𝑛²_𝐻1𝜉_𝑛𝐻2𝜉_𝑛𝐿4𝑢_𝑛𝐻1,4+2𝐾^′_𝑅𝑢_𝑛²_𝐻1𝑢_𝑛𝐻1𝜉_𝑛𝐻1𝜉_𝑛𝐻2𝑢_𝑛³_𝐿6+3𝐾_𝑅𝑢_𝑛²_𝐻1𝜉_𝑛𝐻2𝜉_𝑛𝐿4𝑢_𝑛²_𝐿8≤𝐶(𝑅)𝜉_𝑛𝐻1𝜉_𝑛𝐻2𝑣_𝑛𝐻2+𝜙_𝐻3/2+𝐶(𝑅)𝜉_𝑛𝐻^3/2₂𝜉_𝑛𝐻^1/2₁+𝐶(𝑅)𝜉_𝑛𝐻2𝜉_𝑛𝐻1≤𝜀𝜉_𝑛²_𝐻2+𝐶(𝑅)𝜉_𝑛²_𝐻1×1+𝑣_𝑛²_𝐻2+𝜙²_𝐻3/2. (A.35)
By the Gronwall inequality and (A.6), we have 𝜉_𝑛(𝑡)²_𝐻1≤𝐶(𝑅)‖ℎ‖²_𝐻1. (A.36)
And therefore, using again the previous inequality, ^𝑇₀𝜉_𝑛(𝑡)²_𝐻2𝑑𝑡≤𝐶(𝑅)‖ℎ‖²_𝐻1. (A.37)

Acknowledgments

The authors are thankful to the referee for careful reading and insightful comments which led to many improvements of the earlier version. This work was partially supported by the Fundamental Research Funds for the Central Universities (Grant no. CQDXWL-2013-003).