Longtime behavior for 3D Navier-Stokes equations with constant delays

This paper investigates the longtime behavior of delayed 3D Navier-Stokes equations in terms of attractors. The study will strongly rely on the investigation of the linearized Navier-Stokes system, and the relationship between the discrete dynamical flow for the linearized system and the continuous flow associated to the original system. Assuming the viscosity to be sufficiently large, there exists a unique local attractor for the delayed 3D Navier-Stokes equations. Moreover, the local attractor reduces to a singleton set.


Introduction
The incompressible 3D Navier-Stokes equations are described by time evolution of the velocity u in a bounded or unbounded domain of R 3 and are given by: where ν > 0 is the viscosity of the fluid, p denotes the pressure and u 0 (x) denotes the initial datum. The uniqueness of global weak solutions is a standing open problem. In order to overcome this challenging difficulty, in a previous paper, see [2], we introduced a constant delay µ > 0 into the nonlinear term (u · ∇)u. More precisely, we considered the following modified version of the 3D Navier-Stokes equations: u ′ (t, x) + (u(t − µ, x) · ∇)u(t, x) − ν∆u(t, x) + ∇p(t, x) = f (x), div u(t, x) = 0, u(0, x) = u 0 (x), u(τ, x) = φ(τ, x), τ ∈ [−µ, 0). (1.1) This delay introduces a regularizing effect in the equations and allows to prove the uniqueness of global weak solutions when the initial function (φ, u 0 ) ∈ L 2 (−µ, 0, V 1+α ) × V α with α > 1/2 (for the definition of the spaces V α see Section 2). In particular, when α ≥ 1, then our theory can be extended to include strong solutions. The main ingredient to establish it is to use the regularizing effect of the delay on the convective term by investigating the linearized version of (1.1). This equation comes naturally when investigating the system on the interval [0, µ]. We prove existence and uniqueness of weak solutions, then we establish that these solutions are more regular and are in the spaces V α . Then, we use a concatenation argument by glueing the solutions obtained on each interval [0, µ], [µ, 2µ], . . . and so on. Each solution is obtained from the previous step and uses the linearized construction. As a byproduct, the linearized equation induces a continuous mapping U on the space L 2 (0, µ, V 1+α ) × V α . The nth composition of the map U generates a discrete semigroup U (n) on the same space L 2 (0, µ, V 1+α )×V α . Moreover, thanks to the concatenation argument the solution of system (1.1) generates a continuous semigroup S(t) on the space L 2 (−µ, 0, V 1+α ) × V α for t ≥ 0 given by S(t)(φ, u 0 ) = (u µ t , u µ (t)), where u µ t is the segment function defined by u µ t (s) = u µ (t + s), s ∈ (−µ, 0) (u µ denotes the solution of (1.1)), defined in more details in Section 2. Our goal in this paper is to study the longtime behavior of (1.1) in terms of attractors. Let us point out that the existence of a local attractor for (1.1) is essentially based on getting an invariant ball for the map U . The main ingredient that allows to get the invariance of a bounded ball for the discrete semigroup U (n) and later the semigroup S(t) is the fact that the unique weak global solution is more regular, see Lemma 3.1, 3.2 and 3.3. Combined with compactness embeddings, this allows to prove that the local invariant ball for the discrete semigroup U (n) is compact in the topology of L 2 (0, µ, V 1+α ) × V α . An interesting feature of this model is that we are able to establish that the local attractor A associated to the discrete semigroup U (n) is a single point attractor. These properties are transferred to the original delayed 3D Navier-Stokes equations, due to the key relationship between U and S. In fact, under the same conditions as for the discrete flow U , the continuous flow S is proved to have a local attractor A µ , that reduces to a singleton set and is linked to the local attractor A as S(t) and U are related on the grid points t = nµ, n ∈ N.
The paper is organized as follows. In Section 2 we introduce the abstract setting in which we develop our theory and recall how the construction of the unique weak solution of (1.1) was carried out in the paper [2], by using a suitable linearization of (1.1) on [0, µ]. Section 3 addresses the regularization properties of the solution of (1.1) assuming that the external force is in V α . In Section 4, we first consider a linearized system defined now on any compact interval [0, T ] for a given T > 0 and construct its corresponding unique weak solution. Then we establish a fundamental relationship between the discrete flow U generated by the solution of the linearized system and the continuous flow S generated by the solution of (1.1). Section 5 is devoted to the study of the local attractor for the linearized system and finally, in Section 6, we establish the existence of a unique local attractor for S and we study its inner structure.

Preliminaries: existence and uniqueness of a weak solution
We introduce in this section the functional setting in which our investigations will be carried out and the existence and uniqueness of solutions of the delayed Navier-Stokes equations as well. Consider the torus T 3 L in R 3 of length L given by the set . Let ψ(x) be a L-periodic function that can be expanded into Fourier series denote the Fourier coefficients of ψ. For s ∈ R, we denote by H s (T 3 L ) the Sobolev space of L-periodic functions such thatψ(ζ) =ψ(−ζ) equipped with the norm Whenψ(0) = 0 the corresponding subspace is denoted byḢ s (T 3 L ) with equivalent norm These spaces are Hilbert-spaces with the inner product |ζ| 2s ψ 1 (ζ)ψ 2 (ζ).
We denoteḢ s (T 3 L ) =Ḣ s (T 3 L ) 3 and, for s = −1, 0, 1, we introduce the spaces Then V −1 is the dual space of V 1 and V 1 ⊂ V 0 ⊂ V −1 where the injections are continuous and each space is dense in the following one. We shall denote by (·, ·) the scalar product in V 0 . We introduce the Stokes operator A as in [6], Section 2.2, page 9, with domain given by For the periodic boundary conditions we know that The operator A can be seen as an unbounded positive linear selfadjoint operator on V 0 , and we can define the powers A s , s ∈ R with domain D(A s ). We set V s = D(A s/2 ), that is a closed subspace ofḢ s (T 3 L ), then for any s ∈ R V s = {u ∈Ḣ s (T 3 L ), div u = 0} and the norms A s/2 u 0 and u s are equivalent on V s . The operator A defines an isomorphism from V s to V s−2 , and has a positive countable spectrum of finite multiplicity 0 < λ 1 ≤ λ 2 ≤ · · · , λ j → ∞, where the associated eigenvectors e 1 , e 2 , · · · form a complete orthogonal system in V s . When s 1 < s 2 , the embedding V s2 ⊂ V s1 is compact and dense. The space V −s is the dual space of V s for s ∈ R, see Temam [6], from page 9. We shall denote by ·, · the duality product between V s and V −s no matter the value of s ∈ R.
Let us introduce the trilinear form b given by The following result is essential in our estimates. For the proof, we refer to [2].
Lemma 2.1. The trilinear form b can be continuously extended to V s1 × V s2+1 × V s3 for s i ∈ R if either s i + s j ≥ 0 for i = j, s 1 + s 2 + s 3 > 3/2 or s i + s j > 0 for i = j, s 1 + s 2 + s 3 ≥ 3/2. Therefore, under either of the previous settings, there exists a constant c depending only on s i such that Notice that similar results were proved by Fursikov [3] when considering a bounded domain Ω ⊂ R 3 , ∂Ω ∈ C ∞ with homogeneous Dirichlet conditions, but with more restrictive assumptions. In the periodic boundary setting, for a similar result as Lemma 2.1 above see also Temam [6], Lemma 2.1, page 12, which holds true under the additional assumptions with s 1 , s 2 , s 3 satisfying the conditions of Lemma 2.1.
Finally we mention that for µ > 0 and s ∈ R the spaces L ∞ (0, µ, V s ), L 2 (0, µ, V s ), C([0, µ], V s ) and C β ([0, µ], V s ), β ∈ (0, 1), have the usual meanings. We are interested in studying the dynamics of the following version of the 3D Navier-Stokes equations with constant delay µ: Denote the solution of this equation depending on the time shift by u µ . On account of the Helmholtz-projection, we can formulate the equation as , and, given any v ∈ V α+1 and any test function In order to prove the existence and uniqueness of solutions to (2.4), for t ∈ [0, µ] and ψ ∈ L 2 (0, µ, V 1+α ), we introduce the following 3D linearized Navier-Stokes equations with periodic boundary conditions over the torus These equations are a simpler version of the 3D Navier-Stokes equations, since the term (u, ∇)u has been replaced by (ψ, ∇)u. The existence and uniqueness of solutions to (2.3) and (2.5) can be summarized as follows.
For the sake of readability, the solutions of (2.3) will be denoted by u µ while the corresponding solutions to (2.5) are denoted by u.
Although the proof of this theorem is in the paper [2], for the sake of completeness we would like to give here some explanations of how to prove this result. The existence of a weak solution for the linearized problem (2.5) is obtained thanks to the use of Galerkin approximations, while the uniqueness relies on an energy equality, based on the fact that u ′ ∈ L 2 (0, µ, V −1 ). To prove existence and uniqueness of a weak solution of (2.5) on [−µ, T ], the strategy followed in [2] consists of solving the problem (2.3) step by step, in intervals of length µ, where in each step it is used the fact that for (2.5) there exists a unique weak solution. As a result, a sequence Concatenating the elements of the sequence the global solution of (2.3) is constructed, having the following expression assuming that t ∈ [0, T ], with T ∈ ((k − 1)µ, kµ].

Regularization of weak solutions
In this section, we are going to show that assuming f ∈ V α then the solution u µ to (2.3) is more regular. Thanks to this regularity, we will obtain a suitable compact property that will be further necessary to establish the existence of an attractor for the delayed Navier-Stokes equations.
where we have applied Lemma 2.1 with s 1 = 1 + α, s 2 = α and s 3 = −α (we remind here that in all the paper α > 1/2). As a consequence, and in virtue of Gronwall's lemma, If now t ∈ [µ, 2µ], we can repeat similar steps than before to arrive at d dt It is clear that due to the regularity of the weak solution u µ we can repeat this procedure in any interval. This completes the proof.
We can also establish the following regularity result: Assume that (φ, u 0 ) ∈ Y µ α and f ∈ V α . Then for every ǫ > 0, the solution of (2.3) satisfies Proof. The proof is based on the regularity properties of the weak solution together with the fact that, as a consequence of Lemma 3.1, we know that for any ǫ > 0 Indeed, as in the previous proof, assume first that t ∈ [0, µ]. Then Above, to estimate the trilinear form, we have taken in Lemma 2.1 the parameters s 1 = 1 + α, s 2 = α and s 3 = −α.
Hence, by integration, Reasoning in a similar way, when t ∈ [µ, 2µ], we have Repeating the same argument we conclude the proof.
We can also establish the Hölder regularity of the solution.

Discrete and Continuous Dynamical Flows
As pointed out above in the Introduction, in this paper we are interested in investigating the longtime behavior of the delayed Navier-Stokes equations (2.3). As in the study of the existence and uniqueness of solutions for (2.3), the analysis of its longtime behavior is based on the study of its corresponding linearized system. Hence, we first consider the linearized system on the whole positive real line and introduce its associated discrete flow U . Then, we establish a crucial relationship between U and and the continuous flow S related to (2.3), see (4.5) below. We consider the solution of (2.5) to any compact interval. We can rewrite (2.5) as and consider generalizations of the above problem given for k = 2, 3, · · · by It turns out that we can construct a sequence {(u k )} k∈N ⊂ L 2 (0, µ, V 1+α ) such that, for any k ∈ N, u k (µ) ∈ V α . Concatenating the elements of this sequence we can define the function u given by assuming that t ∈ [0, T ], with T ∈ ((k − 1)µ, kµ]. Therefore, we have constructed u to be the solution of the linearized Navier-Stokes equations (4.1)-(4.2) for t ≥ 0. Due to the above construction, Lemma 3.1, Lemma
It was proven, see [2], that the discrete dynamical system U (n, ·) is a continuous mapping on X µ α while S(t, ·) is continuous on Y µ α .

Longtime behavior for the linearized equation
As we have said in the Introduction, the main goal of this paper is to investigate the existence of an attractor for the delayed Navier-Stokes equations (2.3). To do that, we are going to use the relationship (4.5). To be more precise, in a first step we consider the discrete dynamical system U and look for the existence of an discrete attractor associated to U . The existence of this discrete attractor rests upon the invariance of a ball B ∈ X µ α for U (see Lemma 5.1 below) and suitable compact embeddings of some spaces (see Lemma 5.3).
To simplify the presentation, we identify U (ψ, u 0 ) with U (1, (ψ, u 0 )). Also, since we believe that confusion is not possible, we drop the subindex and represents the solution by u instead of by u 1 .
Lemma 5.1. Consider (ψ, u 0 ) ∈ X µ α and f ∈ V α−1 and assume that the viscosity is large enough. Then for U (ψ, u 0 ) = (u 1 , u 1 (µ)) defined in Section 4 we have , with R and ρ defined by (5.1)-(5.3) below. Proof. By assumption the viscosity is large enough, hence we can find R > 0 such that where c is the positive constant determined in Lemma 2.1 and λ denotes the first eigenvalue of A. Let us define ρ > 0 given by We denote by pr i (·), i = 1, 2, the projection into the corresponding component. We start proving that for which we need to prove that if ψ 2 L2(0,µ,V 1+α ) ≤ R 2 and u 0 2 α ≤ ρ 2 , then u(µ) 2 α ≤ ρ 2 . For t ∈ [0, µ], considering the scalar product with A α u(t) in V 0 , we have , where we have applied Lemma 2.1 taking s 3 = −α, s 1 = 1 + α and s 2 = α. Hence, applying Gronwall's lemma, 1+α dr ds (5.5) and in particular we obtain Hence, thus, if u 0 2 α ≤ ρ 2 with ρ defined by (5.3), we get that u(µ) 2 α ≤ ρ 2 , thus (5.4) is proved. Let us prove now that Notice that from the previous estimates we also obtain and from (5.5) we also know that Therefore, by (5.2), We conclude that choosing ν big enough, we arrive at hence (5.6) is proved and the proof is finished.
Remark 5.2. In the expression (5.7), we take the left hand side to be smaller than R 2 2 but not directly smaller than R 2 . The reason is that this choice will help us later to show the invariance of a ball for the continuous dynamical system S, see Section 6. Anyway, B X µ α (R; ρ) is invariant for U because starting in (ψ, u 0 ) ∈ B X µ α (R; ρ) we know that Now we want to establish the existence of a unique discrete attractor associated to U . To do that, we are using the following lemma, whose proof can be found in Vishik and Fursikov [8] Chapter IV Theorem 4.1.
As a consequence of the previous results, we can establish one of the main theorems of this article: Theorem 5.4. Assume that the viscosity is large enough. Then the discrete dynamical system U associated to the linearized 3D Navier-Stokes equations possesses a local attractor A.
Proof. According to Lemma 5.1 and Remark 5.2, since the viscosity is large enough we can find R such that (5.1) and (5.2) hold true, which imply that U (n, B X µ α (R; ρ)) ⊂ B X µ α (R; ρ), that is, B X µ α (R; ρ) is a forward invariant ball, with ρ given by (5.3).
On the other hand, due to the extra regularity of the solution given by Lemma 3.1, Lemma 3.2 and Lemma 3.3 (for the solution of the linearized equation), in virtue of Lemma 5.3, we know that U (n, B X µ α (R; ρ)) is relatively compact for n ≥ 2. Now defining we have that K is a forward invariant compact set, hence U possesses a unique local attractor A (for a comprehensive presentation of the concept of attractors we refer to the monographs by Babin and Vishik [1], Hale [4] or Temam [7]).
Remark 5.5. Observe that B X µ α (R; ρ) is not an absorbing ball but an invariant ball. Hence, we know that B X µ α (R; ρ) absorbs elements of itself, but not of any bounded set in X µ α . Hence, the localness is related to the fact that the initial condition (ψ, u 0 ) must belong to the ball B X µ α (R; ρ). 5.1. Single point local attractor. Now we are interested in finding sufficient conditions that ensure that the local attractor A associated to the discrete dynamical system U is a single point. Proof. Assume that u 1 and u 2 are two weak solutions to (2.5) with initial conditions given, respectively, by (ψ 1 , u 0,1 ), (ψ 2 , u 0,2 ) ∈ B X µ α (R; ρ), the forward invariant ball of Lemma 5.1. Then, the difference u 1 −u 2 verifies
In other words, due to the invariance property of the attractor A we have obtained sup y1,y2∈A and, as the right-hand side tends to zero, this implies that A is a single point attractor for U .

Longtime behavior for the delayed Navier-Stokes equations
Now we are in position to establish our main result: the existence and uniqueness of a local attractor for the continuous dynamical system S. As mentioned before, the results will be based on the relationship (4.5) and the fact that U has a unique local attractor A. Let us recall that B X µ α (R; ρ) := B L2(0,µ,V 1+α ) (0, R) × B V α (0, ρ), with R and ρ defined by (5.1)-(5.3). Define now B Y µ α (R; ρ) := B L2(−µ,0,V 1+α ) (0, R) × B V α (0, ρ), (6.1) We would like to show that B Y µ α (R; ρ) is a forward invariant ball for S. First of all, we prove the following result for the time interval [0, µ]. Lemma 6.1. Assume the conditions (5.1) and (5.2) and consider (φ, u 0 ) ∈ B Y µ α