Long-time dynamics for a non-autonomous Navier-Stokes-Voigt equation in Lipschitz domains

This article focuses on the optimal regularity and long-time dynamics of solutions of a Navier-Stoke-Voigt equation with non-autonomous body forces in non-smooth domains. Optimal regularity is considered, since the regularity \begin{document} $H_0^1\cap H^2$ \end{document} cannot be achieved. Given the initial data in certain spaces, it can be shown that the problem generates a well-defined evolutionary process. Then we prove the existence of a uniform attractor consisting of complete trajectories.


1.
Introduction. Navier-Stokes equations (NS for short) are well known and effectively useful in modeling turbulence in fluid phenomena. The mathematical theory on this model, for instance, the well-posedness of strong solutions, has been developed from theoretical and numerical aspects for over 80 years. The analysis of the infinite dimensional dynamical systems from the incompressible Navier-Stokes equations attracts attention as well within physical and mathematical theory. We refer readers to [9,19,20,21,22].
The space L 2 b (R; E) denotes the space of the translation bounded functions. In this paper, we will focus on the existence of the uniform attractor of (1.1) in light of [1,24]. The main features of our work are summarized as follows.
1. Using the background function for the Stokes problem in [1], if the external force f (t, x) is only translation bounded, we prove the existence of global unique solution and its continuous dependence on initial data in V (a function space specified in Preliminaries) for the non-autonomous Navier-Sokes-Voigt equations with nonhomogeneous boundary values. [24] investigated the 2D non-autonomous Navier-Stokes equation but didn't provide the detail proof here, especially the continuous dependence on the initial data of the solution. 2. For the problem (1.1) in smooth domains, the regularity can be improved to W = H 1 0 ∩ H 2 ; however, this regularity argument is impossible for Lipschitz domains, in that higher regularity estimates for the terms (B(ψ, ψ), A σ v) and (νF, A σ v) can not be acquired in function spaces with higher regularity than D(A σ+ 1 4 ) (A is the Stokes operator, F will be specified later in (3.11)). If σ ∈ [0, 1 2 ], we are able to obtain the optimal regularity of solutions in D(A σ+1 2 ) by using the Hardy's inequality. This space is definitely less smooth than W . 3. Under the continuity of the processes generated by solutions, the dissipation and the Uniform Condition-(C) are invoked to achieve uniformly asymptotic compactness of a process. By using the property of the Stokes operator, we establish the existence of uniform attractors in V and D(A σ+1 2 ). This paper is organized as follows: In Section 2, notations and function spaces are stipulated; we also define the background function and provide its related estimates in this part; In Section 3, the global existence of unique solution and the continuous dependence on initial data of our problem are developed; moreover, the optimal regularity in D(A σ+1 2 ) for σ ∈ [0, 1 2 ] is discussed as well. The existence of the uniform attractor for the process in V and D(A σ+1 2 ) are concluded in Section 4. 2. Preliminaries. We shall consider the function space corresponding to our problem with an abstract setting E := {u|u ∈ (C ∞ 0 (Ω)) 2 , divu = 0}. Function space H is the closure of E in (L 2 (Ω)) 2 norm, | · | and (·, ·) denote the norm and inner product in H respectively, i.e., Function space V is the closure of E in (H 1 (Ω)) 2 topology, and · and ((·, ·)) denote, respectively, the norm and inner product in V , i.e., ((u, v) u 2 = ((u, u)), ∀ u ∈ (H 1 0 (Ω)) 2 . H and V are dual spaces of H and V respectively, the injections V → H ≡ H → V are dense and continuous. The notations · * and ·, · denote the norm in V 366 XINGUANG YANG, BAOWEI FENG, THALES MAIER DE SOUZA AND TAIGE WANG and the dual product between V and V , respectively. In later sections, we might use | · | or | · | 2 denoting | · | L 2 and · H denoting the norm of space H. When the integral is outside the integrand, | · | exerted on the integrand denotes the absolute value.
P is the Helmholz-Leray orthogonal projection in (L 2 (Ω)) 2 onto the divergencefree space H. We define A := −P ∆ as the Stokes operator, the sequence {ω j } ∞ j=1 is a series of orthonormal eigenfunctions of A, and D(A s ) denotes the domain of A s and the norm of D(A s ) can be written as · Vs with the inner product The closure of E in the topology D(A) denote as W . Moreover, A s satisfies (see [1]) We define the bilinear and trilinear forms as follows (see [19,21]) where B(u, v) is a linear continuous form and b(u, v, w) satisfies We also need to use some useful inequalities: the Gagliardo-Nirenberg interpolation inequality the Hardy's inequality (2.10) 3. Existence of solutions, uniqueness, and continuity results. We transform (1.1) to a homogeneous boundary problem by v = u − ψ, here the background function ψ need to satisfies This transform was proposed by Miranville and Wang [15,16]. The main idea is to localize the solution of the Stokes problem with boundary data ψ to an εneighborhood of boundary ∂Ω. It aims to reduce the problem into a homogeneous boundary one. Brown, Perry and Shen [1] extended this nonhomogeneous boundary problem for Lipschitz domains.
3.1. The background function for Stokes problem. Let ε ∈ (0, c · diam(Ω)) be a constant to be determined later, and where η ε is defined as the form h( ρ(x) ε ), h is a standard bump function and ρ ∈ C ∞ is a regularized distance bump function to ∂Ω.
here g := x P (−u 2 , u 1 ) · ndx for fixed x ∈ ∂Ω and ϕ · n = 0, where n is the outer unit normal vector of Ω. (3.7) (2) If ψ is defined as (3.14) 3.2. Equivalent homogeneous boundary value problem. Let v = u − ψ, then (1.1) is transformed into the following homogeneous boundary value problem Let v τ ∈ V , the abstract equivalent form of (3.15) reads Similarly as the definition of the bilinear continuous operator is a continuous linear operator from V onto V as well.
3.3. Existence of solutions and uniqueness. Firstly, we define the global weak solution of (1.1).
and dv dt is uniformly bounded in L 4 3 loc (R τ ; V ).
Step 1. We aim to use the standard Faedo-Galerkin procedure to establish the existence of solution to (3.16). Fix n ≥ 1, w j (j ≥ 1) be the normalized eigenfunction basis of the Stokes operator: V → H with the corresponding eigenvalue λ j (j ≥ 0) being 0 < λ 1 ≤ λ 2 ≤ · · · and lim j→∞ λ j = ∞. We define an approximate solution v n to (3.16) as v n (t) = n j=1 a nj (t)w j ∈ V n = span{w 1 , w 2 , · · · , w n } which satisfies the following initial data problem of ordinary differential equations with respect to unknown variables (3.18) By the local existence theory of solutions of ordinary differential equations, there exists a solution in a local interval for problem (3.18).
Step 2. We will get the uniform priori estimate.
Multiplying (3.18) by a nj , integrating by parts, By the Hölder inequality and Young's inequality, it follows that (3.20) and we shall estimate the terms of (3.20) on the right hand side. Using the Hardy's inequality (2.10), and (2.8) and (3.5), choosing suitable ε such that Similarly using the Hölder inequality and Young's inequality for (3.22), we derive By the Cauchy inequality, Hardy's inequality and Young's inequality, via Lemma 3.1, we deduce that By the Poincaré inequality, we have 1 2 The Gronwall inequality leads to Moreover, since the external force f is translation bounded and ϕ 2

LONG-TIME DYNAMICS FOR A NON-AUTONOMOUS NAVIER-STOKES-VOIGT EQU. 371
for an arbitrary fixed f 0 ∈ L 2 b (R; H) which is uniformly bounded in a symbol space Σ defined later. Integrating Step 3. We will establish the global weak solution by compact argument.
Using the technique in [22], considering the equation in distribution sense, we can derive that dv n dt is bounded in L Using the Lions-Aubin Lemma, i.e., the compact argument, passing the limit as n → +∞, we can derive the global weak solution , and ϕ·n = 0 on ∂Ω. Then the problem (1.1) possesses a unique weak solution which continuously depends on the initial data, i.e., u(t, x) ∈ C(R τ ; V ). Proof.
Step 1. The existence of weak solution is established.
From the property of background flows class ψ ε = ψ ∈ C ∞ (Ω) satisfying (3.4) and v = u − ψ and the solution v for (3.15) is obtained in Theorem 3.1, it can be shown easily that u satisfies the conditions (i) (ii) and (iii) in Definition 3.1. Hence u is a weak solution.
Step 2. The uniqueness of the solution and its continuous dependence of the initial data.

XINGUANG YANG, BAOWEI FENG, THALES MAIER DE SOUZA AND TAIGE WANG
The system is rewritten as: Apparently, (3.40) holds for any ω ∈ V . In fact, from the condition (ii) we have Since Integrating (3.42) over [τ, t], using (3.43), we obtain i.e., , neglecting the integrating term on left side of (3.45), we have Using the Gronwall inequality to (3.46), we see Similarly, using the Poincaré inequality, we can derive and Next, we will estimate every term on the right side of (3.51). Using (3.24)  inequality, we obtain estimates: but for the last one, we can not give the appropriate estimate are not necessarily bounded, which means the higher regularity for v, and also u in W can not be obtained. This is the motivation that we will discuss the optimal regularity in D(A σ+1 2 ) in the next subsection.
3.5. Optimal regularity of solutions. In this section, we need to estimate the norms of u as well as some regular norms of v. Since v(t) = U f (t, τ )v τ , then we have a decomposition of the process (3.57) By the Duhamel's principle, v 1 (t) = D(t, τ )v τ and v 2 (t) = Kf (t, τ )v τ are solutions satisfying the sub-problems and

Lemma 3.5. For any ν >
and v τ ∈ W , then there exists k > 0, such that the solution of (3.58) satisfies here Q(·) denotes an increasing nonnegative function.
Proof. Multiplying the equation (3.58) with Av 1 , we have and and Hence, combining (3.62)-(3.65), we have Via the Gronwall inequality and the Poincaré inequality, the theorem is proved.
Lemma 3.6. For the fractional operator A α , we have (a) the generalized Sobolev embedding inequality Proof. See, e.g., [25]. Next, we will estimate the nonlinear nonhomogeneous problem (3.59) with the homogeneous boundary condition.
and f ∈ L 2 loc (R; H), there exist positive constants ) > 0 such that the solution of (3.59) satisfies that for any 0 Proof. Taking inner product of (3.59) with A σ v 2 (t) in H-norm, we have i.e., Next, we shall estimate every terms on the right hand side of (3.73). Using the technique in [25], we have the following estimates.
(1) By the Hölder inequality, for any σ ∈ [0, 1 2 (3.74) (2) By the Hardy inequality, since σ ∈ [0, 1 2 (3.75) (3) By the estimates of background function and the property of trilinear operator, since σ ∈ [0, 1 2 (3.76) (4) From Lemma 3.1 and the Young's inequality, for any σ ∈ [0, 1 2 (3.77) (5) By the Young's inequality and Lemma 4.6, for any σ ∈ [0, 1 2 ], we obtain (3.79) (6) Using the property of the trilinear operator, Lemma 3.5, the Cauchy inequality and Poincaré inequality, we obtain and The proof is finished. (R;H) , v τ W , ϕ 2 L ∞ (∂Ω)) ). Proof. Combining the Lemmas 3.5 and 3.7, we have this theorem immediately. 4. Existence of the uniform attractor. Here we shall use the preliminary theory on attractors of PDEs to find our uniform attractors. The theory can be founded in [4,11,12,24], and we omit the detail in this section. 4.1. Some preliminaries for uniform attractors. We select Σ := H(f 0 ) as our symbol space of the system, and . H). Moreover, if we choose f 0 ∈ L 2 loc (R; H) and fixed, then for every f ∈ H(f 0 ), we have f L 2 b ≤ C f 0 L 2 b . Let {T (·)} t≥0 be the translation semigroup acting on L 2 loc (R; H) which is defined by H). For the non-autonomous system, the solutions generates a two-parameter operator which is called process U f (t, τ ) : E → E which satisfies U f (τ, τ ) = I (the identity). (4.2) and the translation identity holds Definition 4.1. A set B 0 is said to be a uniformly (w.r.t. f ∈ H(f 0 )) absorbing set for the process {U f (t, τ )} (f ∈ H(f 0 )), if for every bounded set B of H and any τ ∈ R, there exists some time t 0 = t 0 (B, τ ) ≥ τ such that   The process {U f (t, τ )} (f ∈ Σ) is called satisfying the uniform (w.r.t. f ∈ Σ) condition-(C) if for all fixed τ ∈ R, B ∈ B(E) and ε ∈ R + , there exist t 0 = t 0 (τ, B, ε) and a subspace E 1 , dim(E 1 ) < ∞, such that where P : E → E 1 is a projection, B(E) is the union of all bounded subsets in E.
Theorem 4.6. The process {U f (t, τ )} (f ∈ Σ) has a uniformly (w.r.t. f ∈ Σ) compact attractor A Σ (K f (0) denotes the nonempty kernel) which satisfies   H) and v τ ∈ V , the process U f (t, τ ) possesses a uniformly absorbing set in V for the system (3.16) which is equivalent to (1.1).
Proof. Since the external forces f ∈ Σ and f 0 ∈ Σ are translation bounded and (4.7) Let D ⊂ V be any bounded set, and v τ ∈ D, then there exists a constant d > 0 such that . From the proof of Theorem 3.3, we see that There exists a fixed time T d > τ such that is a uniformly absorbing ball centered at 0 with radius ρ V in V . Hence we complete the proof.

4.3.
The uniformly asymptotic compactness for the process. In this section, we shall verify the uniform condition-(C) to achieve the uniformly (w. r. t. the symbol f ∈ Σ) asymptotic compactness for the process.  H) and v τ ∈ V , then the process U f (t, τ ) is uniformly asymptotical compactness in V for the system (3.16), which is equivalent to (1.1).
Proof. Step 1. Let D ⊂ V be any bounded set, and v τ ∈ V , then there exists a constant d > 0 such that (4.8) holds, and there exists a uniformly absorbing ball From the existence of global solution and the existence of a uniformly (with respect to f ∈ Σ) absorbing ball, we know v 1 2 ≤ v 2 ≤ ρ 2 V , then we only need to prove the V -norm of v 2 is uniformly small enough.
Step 3. Taking inner product of (3.16) with v 2 , using the orthonormal of v 1 and v 2 , we have Next, we shall estimate every terms at the right-hand side of (4.11).
4.4. The uniform attractor and its structure in V . The main result in the paper is stated as follows: Theorem 4.9. Assume f ∈ L 2 b (R; H) and v τ ∈ V , then the continuous process U f (t, τ ) possesses a uniformly (with respect to the symbol f ∈ Σ) compact attractor A Σ = ω Σ (B 0 ) = f ∈Σ K f (τ ) in V for the system (3.16), which is equivalent to our original problem (1.1). Here B 0 is the uniformly absorbing set in V , and K f (τ ) is the nonempty kernel in V contains almost all bounded completely trajectories.
Proof. Given the uniformly absorbing ball and uniformly asymptotical compactness of the process, from the theory of uniform attractor, it is direct to prove the main result.
4.5. The uniform attractor and its structure in D(A σ+1 2 ) when σ ∈ [0, 1 2 ]. In this final subsection, we reach the existence of the uniform attractor in D(A σ+1 2 ) (σ ∈ [0, 1 2 ]). The proof is similar to subsection 3.5, i.e., take inner product for the equation (3.16) with A σ v or A σ v 2 to achieve the uniformly absorbing ball and then verify the Uniform Condition-(C) for the process by uniform estimates. We just omit the detail here. Proof. Using the same technique in Theorem 4.9, we can prove the result, and we omit the detail.