Well-posedness and long-time behavior of solutions for two-dimensional Navier-Stokes equations with infinite delay and general hereditary memory

We address the dynamics of two-dimensional Navier-Stokes models with infinite delay and hereditary memory, whose kernels are a much larger class of functions than the one considered in the literature, on a bounded domain. We prove the existence and uniqueness of weak solutions by means of Faedo-Galerkin method. Moreover, we establish the existence of global attractor for the system with the existence of a bounded absorbing set and asymptotic compact property. 2010 Mathematics Subject Classification: 76A10; 35B41; 35D30; 35D35.


Introduction
In this paper, we consider the two-dimensional Navier-Stokes model with past history and infinite delay effect in a bounded domain Ω with smooth boundary ∂Ω of the form , (x, t) ∈ Ω × (0, +∞), u(x, t) = 0, (x, t) ∈ ∂Ω × (0, +∞), where u = u(x, t) = (u 1 , u 2 ) is the unknown velocity vector, p = p(x, t) is the unknown pressure, ν > 0 is the kinematic viscosity coefficient, f = f (x) is a given force field, ϕ is a given function, u t is the function defined by the relation u t (τ ) = u(t + τ ), τ ∈ (−∞, 0]. The convolution term stands for memory effect whose kernel κ is a nonnegative summable function of total mass ∫ ∞ 0 κ(s)ds = 1 having the explicit form κ(s) = ∫ ∞ s µ(r)dr. The Navier-Stokes equations reflect the basic mechanical law of viscous fluid flow, thus are very significant both in a purely mathematical sense and in the fluid applications including physics and biology. Since the first relative paper of Leray [24] published in 1933, the Navier-Stokes equations have been the object of numerous works. Caraballo and Han [12] considered the Navier-Stokes equations with delay effect which shows not only the present state but also its history in both autonomous and non-autonomous cases (see [15,31,39] and references therein). In 2001, Caraballo and Real [13] put forward the possibility of some kind of delay appearing in the Navier-Stokes equations in an open and bounded domain Ω ⊂ R N (N = 2 or 3) with regular boundary Γ: ∇ · u = 0, (1.2) and they proved the existence and uniqueness of solutions. Subsequently, they established the existence of a pullback attractor when N = 2 in [14]. Concerning the Navier-Stokes equations with finite delay in an unbounded domain, Garrido-Atienza and Marín-Rubio [20] addressed the existence and uniqueness of solutions for both the evolutionary and the stationary cases. Especially, in the three-dimensional case, they only proved the existence of solutions. These results were extended in R N with 2 ≤ N ≤ 4 by Niche and Planas [37].
For unbounded delay, the authors investigated the well-posedness and asymptotic behavior of solutions [30,35] and their references. In addition, Liu, Caraballo and Marín-Rubio [27] studied the existence and uniqueness of solutions for two-dimensional Navier-Stokes equations with infinite delay by using the phase space BCL −∞ (H) which will be defined below rather than C γ (H) carried out in [31] for a 2D Navier-Stokes model with infinite delay due to lacking the term ∆(∂ t u). Recently, Anh and Thanh [4] discussed the 3D Navier-Stokes-Voigt equations with infinite delay of the form where α is length-scale parameter describing the elasticity of the fluid. Some results on the existence and uniqueness of weak solutions and the existence of global attractors were proved. Other types of delay including constant, bounded variable delay as well as bounded distributed delay can be found in [10,13] and references therein.
As for memory term, it plays an important role in the description of several phenomena such as non-Newtonian flows, soil mechanics and heat conduction theory and prescribes the effect of past history and provides a more realistic description (see [5] and references therein). So far, there are some authors having studied the asymptotic behavior of solutions for the Navier-Stokes model incorporating memory effects. For example, Gatti, Giorgi and Pata [21] considered the following system Here, ω ∈ (0, 1) is a fixed parameter, the memory kernel is defined by The authors proved the existence of global attractors and constructed a robust family of exponential attractors. Furthermore, Gal and Medjo [19] considered the three-dimensional Navier-Stokes-Voigt model with memory effect and proved the existence of uniform global attractors. In 2018, Di Plinio et al. [17] analyzed Navier-Stokes-Voigt model lacking which assumed the inequality holds for some δ > 0 and achieved the well-posedness and asymptotic properties of solutions.
It is worth pointing out that the condition (1.3) is one of various types of memory conditions scarcely. In fact, the evolution equations, viscoelastic equations have been considered with following memory conditions to study the decay results of solutions. For example, Alabau-Boussouira, Cannarsa and Sforza [2] considered the evolution equations with the assumption µ ′ (s) ≤ −δµ(s) for some s ≥ 0 and p > 2. Subsequently, Messaoudi and Mustefa [33] studied the stability result for the memory type satisfying where ξ is a nonincreasing differentiable function on R + . Moreover, Alabau-Boussouira [1] concerned the memory-dissipative evolution equations in a bounded domain with a positive function satisfying the form where χ is a nonnegative measurable function with χ(0) = χ ′ (0) = 0 and strictly increasing.
In addition, Messaoudi and Mustafa [34] discussed the Timoshenko system where µ is a positive nonincreasing function satisfying and H is a regular and convex function. Moreover, Guesmia [22] considered an abstract linear dissipative integrodifferential equation with infinite memory satisfying (H1) There exists an increasing strictly convex function Φ : For more similar arguments we refer the readers to [3,8,18,25,28,32,36].  [4] due to the presence of memory term ∥η t ∥ 2 1,µ here. For our purpose, we prove the strong convergence of delay term by continuous definition and integral mean value theorem directly. Additionally, to overcome the difficulty brought by the general memory condition (H1), we show the existence of global attractors for the system S(t) by proving the existence of a bounded absorbing set through the use of a new lemma on the generalized differential inequality and the asymptotical compactness of S(t). It should be note that, for the case of three-dimensions, the well-posedness of solutions is still open due to the difficulty of trilinear term estimates where we don't have 'better' exponents in the Sobolev embedding theorem.
The structure of this paper is as follows. In the next section, we recall some basic assumptions and the related lemmas. In Section 3, we prove the existence and uniqueness of weak solutions to problem (2.1) by using the Faedo-Galerkin method. Finally, the existence of a global attractor for the continuous semigroup generated by the weak solutions is studied in Section 4.
Subsequently, we consider the following usual abstract space For simplicity of notation, we will denote by H the closure of V in (L 2 (Ω)) 2 , and by V the In what follows, we denote the dual space of H by H ′ and the dual space of V by V ′ . It follows that V ⊂ H ≡ H ′ ⊂ V ′ , where the injections are dense and continuous. We will use ∥ · ∥ * for the norm in V ′ , and ⟨·, ·⟩ for the duality pairing between where P is the Helmholtz-Leray orthogonal projection in (H 1 0 (Ω)) 2 onto the space V . Let us denote trilinear form We recall that and consequently, And the associated bilinear form B : As for prerequisites, the readers are expected to be familiar with the following lemma used throughout the paper.
Lemma 2.1. [40] We have for all u, v ∈ V , It is required that we choose a suitable phase space. There are several phase spaces which allow us to deal with infinite delays (see e.g. [31]). In this paper, inspired by recent works [27], we will use the following phase space which is a Banach space equipped with the norm We introduce some notations and assumptions on the delay operator. Let X be a Banach space and consider a fixed T > 0. Given u : (−∞, T ) → X, for each t ∈ (0, T ), we define the function u t on τ ∈ (−∞, 0]. Referring to [4,27], we now make the following assumptions: (H2) f ∈ H;

We check that F satisfies the assumptions (H3)(3). It is well defined as a map with values
in H: As in [16], to deal with the memory term, we introduce a new variable which reflects the past history of equation (1.1), that is, Using the fact that µ(s) = −κ ′ (s) and κ(∞) = 0, and applying the Helmholtz-Leray In consideration of memory term, let L 2 µ (R + , H) be the Hilbert space of the functions φ : R + → H endowed with the inner product and norm respectively Similarly, we make the Hilbert spaces L 2 µ (R + , V ) and and endow with the norm ∥φ∥ 1,µ and ∥φ∥ 2,µ respectively.
In addition, we define the space

Existence and uniqueness of weak solutions
In this section, we first give the definition of weak solutions to (2.1) and state the main result. Proof. (i) Existence. We outline the proof of the existence with three steps.

Definition 3.1. Given functions
Step 1 : Galerkin scheme. First of all, let {ω j } ∞ j=1 be a smooth orthonormal basis of H which is also orthogonal in V . Secondly, we select an orthonormal basis is the space of infinitely differentiable X-valued functions with compact support in I ⊂ R, whose dual space is the distribution space on I with values in X * , denoted by D(I, X * ). And one can take a complete set of normalized eigenfunctions for A in V such that Aω j = λ j ω j . Given an integer n, denote by P n and Q n the projections on the subspaces V n = span{ω 1 , ω 2 , . . . , ω n } and L 2 µ (R + , V ) = span{ζ 1 , ζ 2 . . . , ζ n } respectively. Set Therefore, we seek a function (u n , η t n ) satisfying for a.e. t ∈ [0, T ] and every k, j = 0, 1, . . . , n, where ω 0 , ζ 0 are the zero vectors. Then, we take (ω k , ζ 0 ) and (ω 0 , ζ k ) in (3.2), and apply the divergence theorem to the term to get a system about the variable a k (t) and b k (t) of the form with the initial conditions The above system of ordinary differential equations with infinite delay fulfills the conditions for the existence and uniqueness of local solutions in [23], so the appropriate solution (u n , η t n ) exists.
Step 2 : A priori estimates. Multiplying the first equation of (3.3) by a k and the second by b k , then summing in k and adding the result, we get From the Cauchy's inequality and Remark 2.1, it follows that where we have used |u n | ≤ ∥u nt ∥ BCL −∞(H) . Applying integration by parts, we arrive at Adding the (3.5) and (3.6) into (3.4) leads to d dt Choosing ε > 0 small enough such that ν − ε > 0 and integrating (3.7) from 0 to t, we Thus, By the Gronwall inequality, we have which explains that there exists a constant C dependent of L F , f, T, R such that where R > 0 and ∥ϕ∥ BCL −∞ (H) ≤ R. Especially, it's easy to imply that Also, we claim that   We estimate the norm ∥∂ t u n ∥ V * through the estimates of the right-hand side of the above identity. It is easy to get the estimates |⟨νAu n , φ⟩| ≤ |ν⟨∇u n , ∇φ⟩| ≤ ν∥u n ∥∥φ∥, ⟨f, φ⟩ ≤ ∥f ∥ * ∥φ∥. (3.14) For the nonlinear term, by Lemma 2.1, we obtain ⟨B(u, u), φ⟩ ≤ c|u n |∥u n ∥∥φ∥. (3.15) Using the Poincaré inequality, we get Plugging estimates (3.12) − (3.16) into (3.11), we deduce the differential inequality which shows that through the argument in the Theorem 3.2 in [6], for any function ω j in the basis and any continuously differentiable function ψ on [0, T ].
Nevertheless, it is not sufficient to pass to the limit in the delay term F (u t ) because of its discontinuity, we need some kind of strong convergence.
Step 3 : Approximation in BCL −∞ (H) of the initial datum.
In this part, we establish by proving the following: (3.20) Let us check the first convergence of (3.20). Indeed, if not, then there exists ε > 0 and a subsequence (relabeled the same), such that One can assume that θ n → −∞, otherwise if θ n → θ, then P n ϕ(θ n ) → ϕ(θ n ), since which is a contradiction with ( Observing that On the one hand, Since u(·) is continuous at t 0 , then for every ε > 0, there exists a δ > 0 such that On the other hand, using the fact that we are easy to get ∫ t 0 +δ then there is a t ′ ∈ (t 0 − δ, t 0 + δ) such that with the integral mean value theorem. In addition, using the fact that u(t) is continuous (3.28) Combining (3.26) and (3.27), we deduce that which contradicts (3.25). It impies obviously that Hence, we conclude At last, we will show that the limit (u, η t ) is a weak solution of (2.1). Choosing the arbitrary test function where n is a fixed integer, and {a j } n j=1 and {b j } n j=1 are given functions in D((0, T )). We replace (ω k , ζ j ) with (φ, ξ) in (3.2). Then integrating the resulting equation over (0, T ) and passing to the limits, we obtain (3.29) Using a density argument, we have that (u, η t ) satisfies the equation in the weak sense.
we can get (3.31) Briefly, we get Integrating by parts, we obtain In particular, Therefore, Whence applying the Gronwall inequality, we get

Existence of global attractor
For each (u 0 , η 0 ) ∈ H * , satisfying u 0 = ϕ and η 0 = ∫ 0 −s ϕ(x, r)dr, we denote a semigroup S(t) : H * → H * by the formula where (u(t), η t ) is the unique global weak solution of (2.1). In this section, we will prove the existence of a compact global attractor for the semigroup S(t). Proof. Denoting (u i , η t i ) for i = 1, 2, the corresponding solutions to initial datum (u 0i , η 0 i ) ∈ H * . Considering the equations satisfied by u i for i = 1, 2, acting on the element u 1 − u 2 , and taking the difference, we get 1 2 d dt

Continuity of the semigroup in H
As in the proof of Theorem 3.1, using the Ladyzhenskaya's inequality, we get And by the assumptions on delay term, we yields where ϕ 1 , ϕ 2 ∈ BCL −∞ (H) are the initial datum of u. Therefore, then, we conclude that This complete the proof.

Existence of a absorbing set for the semigroup
In this subsection, we state and prove the existence of a bounded absorbing set. Before proving the result, let us first introduce the "modified" energŷ which has the following relation to the energy associated to the problem (2.1) A direct differentiation to (2.1) 1 leads to and then using the Young's inequality, we get ∫ The proof of Lemma 4.1 is based on three lemmas.
Proof. Let us now multiply (2.1) 1 by where , can be estimated in this way by means of the Young's inequality and Lemma 3.2 in [22]. Taking above estimates into account, we are easy to get (4.4).
Proof. Multiplying (2.1) 1 by u and using the Young's inequality, then we can obtain the desired result.
For positive constants M 1 , M 2 , we define a Lyapunov functional Combining the (4.1), Lemma 4.2 and Lemma 4.3, we arrive at Then we choose M 1 large enough such that where Thus, there exist two positive constants m and c such that In addition (see [9]), we can choose M 1 large enough so that i.e., L 1 (t) is equivalent toÊ(t). Here, we introduce an approach to estimateÊ(t) and refer the readers to [22] for more details.

Asymptotical compact of the semigroup
To show that the semigroup S(t) exists a global attractor, it remains to prove that S(t) is asymptotically compact. It is necessary to consider the difficulty caused by the lack of the compactness of L 2 (R + , (H 2 (Ω)) 2 ∩ V ) → L 2 (R + , V ). Now we recall some preliminary results what we need.

The second a priori estimates
Since the presence of the convolution integral in which the embedding L 2 is not compact, we apply the compactness theorem and the construction of the compact subspaces in [5] to overcome the difficulty.
Proof. Multiplying the first equation of (4.15) by Aw ε and using the Cauchy inequality, and using Poincaré inequality then we conclude that By Gronwall inequality, we can find a T > 0 large enough such that Besides, for any ξ 0 ∈ L 2 µ (R + , V ), the Cauchy problem (see [11], [38]) { ∂ t ξ t = −∂ s ξ t + w, t > 0, has a unique solution ξ t ∈ C(R + , L 2 µ (R + , V )) and Then, we have (4.20) Let B 0 be the bounded absorbing set obtained in Lemma 4.1. We have the following results. (2) sup where {S 2 (t)} t≥0 is the solution semigroup of (4.15).
Proof. By (4.20), we have Combining with the Lemma 4.7, we conclude that (1) holds. Afterwards, from (4.20) we can deduce that On account of Lemma 4.7 again, we have that (2) holds.
Moreover, due to (H 2 (Ω)) 2 ∩ V → V compactly, we have that K ε T is relatively compact in L 2 µ (R + , V ) thanks to the following lemma. Lemma 4.9. ( [26,38]) Suppose that µ ∈ C 1 (R + ) ∩ L 1 (R + ) is a nonnegative function and satisfies the following condition: if there exists a s 0 ∈ R + such that µ(s 0 ) = 0, then µ(s) = 0 for all s ≥ s 0 . Let A 0 , A 1 , A 2 be Banach spaces, where A 0 , A 2 are reflexive and satisfy and the embedding A 0 → A 1 is compact. Let C ⊂ L 2 µ (R + , A 1 ) satisfy Then C is relatively compact in L 2 µ (R + , A 1 ).
The semigroup S(t) has a bounded absorbing set in H * and S(t) is asymptotically compact in H * , thus we can get the main result as follows. in H * , we are easy to complete the proof.