Dissipativity of Fractional Navier–Stokes Equations with Variable Delay

: We use classical Galerkin approximations, the generalized Aubin–Lions Lemma as well as the Bellman–Gronwall Lemma to study the asymptotical behavior of a two-dimensional fractional Navier–Stokes equation with variable delay. By modifying the fractional Halanay inequality and the comparison principle, we investigate the dissipativity of the corresponding system, namely, we obtain the existence of global absorbing set. Besides, some available results are improved in this work. The existence of a global attracting set is still an open problem.


Introduction
We study the longtime behavior of the following two-dimensional Navier-Stokes equation of fractional order with variable delay on a bounded domain Ω ⊂ R 2 , D α t u − ν∆u + (u · ∇)u + ∇p = f (t) + g(t, u t ), in (0, T) × Ω, div u ≡ 0, in (0, T) × Ω, where D α t is a fractional derivative of order α ∈ (0, 1), T > 0, Ω ⊂ R 2 is a bounded open set with regular boundary ∂Ω, ν > 0 is the kinematic viscosity, u is the velocity field of the fluid, p is the pressure, φ is the initial datum, h > 0 is a constant, f is an external force field without delay, and g is the external force containing some functional delay. We will refer to (1)-(4) as problem (P).
In fact, hereditary characteristics are ubiquitous in engineering, biology and physics. For example, feedback control problem, immune systems, soft matter with viscoelasticity [1] could all have hereditary properties (including memory, variable delay or distributed delay, constant delay, etc). The delay term is very often denoted by a function u t (·) defined on some interval [−h, 0] (here h could be −∞). The memory effect is modeled by using fractional calculus, which actually has been widely applied in many sciences [2][3][4][5]. We would like to mention that the concept of fractional calculus was raised by L'Hospital, who wrote to Leibniz in the year 1695, seeking the meaning of been introduced [2], but maybe the most commonly used nowadays are the so-called Riemann-Liouville derivative and Caputo derivative. More definitions for Riemann-Liouville and Caputo derivative can be found in [3,6,7].
It is worth pointing out that using a convolution group, Li and Liu [8] introduced a generalized definition of Caputo derivative of order α ∈ (0, 1), and built a convenient framework for studying initial value problems of time fractional differential equations. Compared with Riemann-Liouville derivative, the Caputo derivative defined in [8] removes the singularity at t = 0 and characterizes memory from t = 0 + . It is probably this character that makes the Caputo derivative share many similarities with the corresponding ordinary derivative and then more manageable for Cauchy problems. In this work, we use the Caputo derivative introduced in [8] to investigate the fractional dynamic system (1).
On the other hand, there are many results about time-fractional Navier-Stokes equations, which can be used to simulate anomalous diffusion in fractal media. For instance, applying Laplace and finite Hankel transforms, Chaurasia and Kumar [9] obtained the solution of a time-fractional Navier-Stokes equation. In [10], Zhou and Peng studied the mild solutions of Navier-Stokes equations with a time-fractional derivative, meanwhile Nieto and Planas [11] investigated the existence and uniqueness of mild solutions to the Navier-Stokes equations with time fractional differential operators, and obtained several interesting properties about the solution, such as regularity and decay rate in Lebesgue spaces. Nevertheless, most of the available works including the mentioned ones did not take into account the delay in the external forcing term, and are concerned mainly with the existence of solution/mild solution or the regularity. There is no result on the limit behavior of solutions, even less work about fractional Navier-Stokes equations with delay, such as the existence of weak solution and asymptotical behavior of solutions. Actually, for general fractional PDEs, this discussion is limited due to the lack of tools although some special cases have been studied [12][13][14].
The traditional method used to study solutions of classic nonlinear PDEs is to find some "a priori" estimates of approximate solutions, then to apply some compactness criteria-i.e., the Arzelà-Ascoli theorem, etc. However, this method seems not to work for fractional PDEs with variable delay. Because of the appearance of variable delay term, the generalized fractional Gronwall inequality [15] (Theorem 1) is not enough to find some "a priori" estimates of Lyapunov functions. Even though Ye and Gao [16] obtained the Henry-Gronwall type retarded integral inequalities, this only works for fractional differential equations with constant delay but not for variable delay. Fortunately, Li and Liu [17] (Theorem 4.1-4.2), generalized the classic Aubin-Lions lemma and some convergence theorem to the fractional case, respectively. To our purpose, we first improve [17] (Proposition 3.5) and [8] (Theorem 4.10). Then, under the condition that α ∈ ( 1 2 , 1), we investigate the solutions of our system by combing the Galerkin approximation and the generalized Aubin-Lions lemma as well as the Bellman-Gronwall Lemma.
We would like to mention that Wen, Yu and Wang [18] analyzed the dissipativity of Volterra functional differential equations by using the generalized Halanay inequalities, while Wang and Zou [19] studied the dissipativity and contractivity analysis for fractional functional differential equations and their numerical approximations via a fractional Halanay inequality. However, to analyze the dissipativity of fractional PDEs with variable delay, the fractional Halanay inequality [19] alone is not enough any more, in fact, it cannot be applied directly for our case, either. We modify the fractional Halanay inequality [19] (Lemma 4) to a more general case, and then improve the comparison principle [20] (Lemma 3.4) and combine the fractional Halanay inequality to overcome this difficulty.
Motivated by [19], we study the long time behavior of fractional Navier-Stokes equations with variable delay. More precisely, we first prove the existence and uniqueness of weak solutions by Galerkin approximation, and then analyze the dissipativity of system (P), namely, we obtain the existence of an absorbing set by fractional Halanay inequalities and generalized comparison principle. We would like to mention that similar results about the classic model of problem (P) can be found in [21].
The organization of this work is as follows. In the next Section, we recall some basic concepts about fractional calculus, and present some auxiliary lemmas which will be useful in later study. In Section 3, we focus on the existence and uniqueness of weak solutions, and the dissipativity of the fractional dynamic system (P) is shown in Section 4. Throughout the work, C, c are positive constants, which can be different from line to line, even in the same line.

Preliminaries
In this Section, we first recollect the generalize definitions of fractional calculus to functions valued in general Banach spaces as studied in [8,17]. Then we prefer to recall some notations and abstract spaces for the sake of completeness and to make the reading of the paper easier, although the notations and results included in this section may seem somehow repetitive, since they can be found in several already published monographs or articles [22][23][24]. Besides, two examples of delay are presented and some lemmas, propositions that will be used in our later discussion are stated. Now, we start with the definition of fractional integral, readers are referred to [2,3,8] for more details.

Definition 1.
( [3,17]) The fractional Riemann-Liouville integral of order α ∈ (0, 1) for a function u : R + → R locally integrable is defined by  u(s) − u 0 X ds = 0, then u 0 is called the right limit of u at t = 0, denote as u(0 + ) = u 0 . Similarly, we define u(T − ) = u T to be the left limit of u at t = T-i.e., u T ∈ X such that As pointed out in [8], this fractional integral can be expressed as the convolution between the kernel and H(t)u(t) on R, where is the standard Heaviside step function. By this fact, it is not difficult to verify that the integral operators I α form a semigroup, and I α is a bounded linear operator from L 1 (0, T) to L 1 (0, T). Inspired by [25] (Section 5, Chapter 1), Li and Liu [8] proposed a generalized definition of Caputo derivative. The new definition is consistent with various definitions in the literature while revealing the underlying group structure. The underlying group property makes many properties of the Caputo derivative natural.
Before introducing this generalized Caputo derivative, we need to use the distributions {g α } as the convolution kernels for α ∈ (−1, 1): where δ is the usual Dirac distribution, and D means the distributional derivative. As in [8], the fractional integral operator I α can be expressed as Given f , g ∈ L 1 loc (0, T), we define the convolution between f and g as Now, we introduce the generalized Caputo derivative as 1). Suppose that u ∈ L 1 loc (0, T) has a right limit u(0 + ) = u 0 at t = 0 in the sense of Definition 2. The Caputo derivative of fractional order α of u is a distribution in D (−∞, T) with support in [0, T), given by The right fractional Caputo derivative is defined as To introduce the Caputo derivatives for functions valued in general Banach spaces, for fix T > 0, we present the following sets: which is analogous of the distribution D used in [17]. We would like to point out that D can be understood as the generalization of distribution. In fact, if X = R, then it is reduced to the usual distribution as in [17]. The weak fractional Caputo derivative of the functions valued in Banach spaces is given by 17]) Let X be a Banach space and u ∈ L 1 loc ([0, T); X). Let u 0 ∈ X. We define the weak Caputo derivative of fractional order α of u associated with initial value u 0 to be D α t u ∈ D such that for any test function Next, let us consider the following usual abstract spaces: where the injections are dense and compact. We will use · * for the norm in V , and ·, · for the duality pairing between V and V . Now we define A : Note that the trilinear form B satisfies the following inequalities which will be used later in proofs (see [23] (p. 2015)).

Example 1. A forcing term with bounded variable delay.
Let G : [0, T] × R 2 → R 2 be a measurable function satisfying G(t, 0) = 0 for all t ∈ [0, T], and assume that there exists L G > 0 such that Consider a function ρ(·) : [0, +∞) → [0, h], which plays the role of the variable delay. Assume that ρ(·) is measurable and define g(t, ξ)(x) = G(t, ξ(−ρ(t))(x)) for each ξ ∈ C H , x ∈ Ω and t ∈ [0, T]. Notice that, in this case, the delayed term g in our problem becomes T], and x ∈ Ω. Then, the delayed term g in our problem becomes After introducing the above operators, an equivalent abstract formulation to problem (P) is The definition of weak solution to problem (6) and (7) is defined as Definition 6. ( [17]) Given an initial datum φ ∈ C H , a weak solution u to (6) and (7) where the equation must be understood in the sense of distribution.
The following auxiliary Lemmas will be needed in this work.
(iii) There exists r 2 ∈ [1, ∞), C > 0 such that for any u ∈ W with right limit u 0 at t = 0, it holds that Then, W is relatively compact in L p ((0, T); X).

Proposition 1.
(An improvement in [17] (Proposition 3.5)) Suppose Y is a reflexive Banach space, α ∈ (0, 1) and T > 0. Assume the sequence {u n } converges to u in L p ((0, T); Y), p ≥ 1. If there is an assignment of initial values u 0,n for u n such that the weak Caputo derivatives D α t u n are bounded in L r ((0, T); Y) (r ∈ [1, ∞)), then (i) There is a subsequence such that u 0,n converges weakly to some value u 0 ∈ Y.
(ii) If r > 1, there exists a subsequence such that D α t u nk converges weakly to v and u 0,n k converges weakly to u 0 . Moreover, v is the Caputo derivative of u with initial value u 0 so that Further, if r ≥ 1, then, u(0 + ) = u 0 in Y is the sense of Definition 2.
Proof. We would like to mention that this Proposition is just a slightly improvement of [17] (Proposition 3.5), in which, the final conclusion-i.e., u(0 + ) = u 0 in Y-holds true for r ≥ 1 α . However, this conclusion holds for r ≥ 1.
So, we just need to prove that for r ≥ 1, if D α t u ∈ L 1 loc ([0, T), Y), then u(0 + ) = u 0 in Y under the sense of Definition 2. By a similar argument in [17] (Corollary 2.16) and Young's inequality with the conjugate index p = ∞, q = 1, 1 p + 1 q = 1, we find The proof is finished immediately.
where γ is a positive constant and a + b < 0.
From system (8), there exists a nonnegative function m(t) satisfying According to [2] (Theorem 4.3), the initial value problem (8) has a unique solution that can be represented by where we used that t α−1 and E α,α ((a + b)t α ) are nonnegative and λ * ∈ [a + b, 0], as well as the fact that E α (λt α ) is non-decreasing respect to λ. The proof is complete.

Remark 3.
It turns out that the modified fractional Halanay inequality holds true not only for delay fractional dynamical system but also for the nondelay case, which means that it could be applied to more fractional differential equations. In this sense, we say it improves [19] (Lemma 4).
and the following fractional differential equation where a, b, c are positive constants. Then it holds that u(t) ≤ w(t), for all t ≥ −h.
Proof. Obviously, (12) holds true for any t ∈ [−h, 0]. Hence, we only need to verify that (12) is correct for t ∈ [0, T]. We will prove this through two steps.
Step 2. On the other hand, when (14) and (15) can be rewritten as, respectively, and the following fractional differential equation Then there is a nonnegative function m(t), such that Then, by [28] (Theorem 1), system (16) has a unique solution on [0, h] that can be represented as Similarly, the solution of system (15) can be written as Notice that t α and E α,α (−at α ) are non-negative for a > 0, then we have u(t) ≤ w(t) for all 0 < t ≤ τ(t). In summary, u(t) ≤ w(t) for all t ≥ −h.
Therefore the proof is complete.

Existence and Uniqueness of Weak Solutions
In this section, we prove the existence and uniqueness of weak solutions to problem (6) and (7) by Galerkin approximations. Denote We have the following result: (17) hold true, then for any φ ∈ C H and α ∈ ( 1 2 , 1) system, (6) and (7) has a unique weak solution.
Proof. We split it into several steps.
Step 1. The Galerkin approximation. By the definition of A = −∆ and the classical spectral theory of elliptic operators, it follows that A possesses a sequence of eigenvalues {λ j } j≥1 and a corresponding family of eigenfunctions {w j } j≥1 ⊂ V, which form a Hilbert basis of H, dense on V. We consider the subspace V m = span{w 1 , w 2 , · · · , w m }, and the projector P m : H → V m given by P m u = ∑ m j=1 (u, w j )w j , and define u (m) (t) = ∑ m j=1 γ m,j (t)w j , where the superscript m will be used instead of (m), for short, since no confusion is possible with powers of u, and where the coefficients γ m,j (t) are required to satisfy the Cauchy problem The above system of fractional order functional differential equations with finite delay fulfills the conditions for the existence and uniqueness of a local solution (e.g., cf. [30] (Theorem 3.1)). Hence, we conclude that (18) has a unique local solution defined in [0, t m ) with 0 ≤ t m ≤ T. Next, we will obtain a priori estimates and ensure that the solutions u m do exist in the whole interval [0, ∞). Assume that Step 2. A priori estimates. Multiplying (18) by γ m,j (t), j = 1, . . . , m, summing up, and using Lemma 1, Cauchy-Schwartz and Young's inequalities, we obtain Hence, Multiplying (19) by I α , and let p = 1 + α, q = 1 + 1 α , we find Denote by Therefore, Using the Gronwall Lemma, we obtain that Hence, we conclude that for any T > 0, u m t C H is finite, which means the local solution u m (t; φ) is actually a global one. We also can have that there exists a constant C > 0, depending on some constants of the problem (namely, ν, L g and f ), and on T and M > 0, such that which also implies that {u m } is bounded in L ∞ (−h, T; H). Now it follows from (20) and the above uniform estimates that On one hand, by (26), we have |u(t * )| ≤ lim n→∞ inf |u m (t m )|. Therefore, if we could prove |u m (t m )| → |u(t * )|, then (28) is contradictory, in other words, (27) is obtained immediately. To this end, it is enough to show that lim m→∞ sup |u m (t m )| ≤ |u(t * )|.
On the other hand, for system (18), we have the following energy inequality, Besides, by (24), passing to the limit in (18), we have that u ∈ C([0, T]; H) is a solution of a similar problem to (6)-i.e., which also has the energy inequality, Combing the last convergence in (24) and the dominate convergence theorem, we find Therefore, u also satisfies inequality (30) with the same last term on the right-hand side. Consider now two continuous functions defined as J and J m are non-increasing in t. Moreover, again from (24), we have Assume that t * > 0, consider {t k } ⊂ (0, t * ) with t k → t * , by the continuity of J, Take now m(k ) such that Then, we conclude that for all m ≥ n(k ) which gives (29). Therefore, we find that u m → u in C([0, T]; H).
Thus, we can finally pass to the limit in (18), concluding that u solves (P).
Step 5. Uniqueness of solution. Let u(t; φ), v(t; φ) be two solutions of (P) with the same initial values-i.e., Multiplying above equation by w(t), and integral over Ω, we obtain The above inequality holds true for any t ∈ [0, T], then we have Using the Bellman-Gronwall Lemma 3 and (21), we have Therefore, |w(t)| = 0 on [−h, T]. The proof is finished.

Remark 6.
We prove the existence of solution for a general delay case, namely, g(t, u t ) could be variable delay or distributed delay. In Section 4, we take g(t, u t ) = g(u(t − τ(t)))-i.e., the delay function τ(t) ∈ C(R + ; [0, h]), to study the dissipativity.

Remark 7.
It is worth mentioning that only the existence result is proved under the condition that α ∈ ( 1 2 , 1), which is due to the phase space C([−h, 0]; H). If C([−h, 0]; H) is replaced by some Sobolev space, such as L 2 ((−h, 0); H). Then the existence of solution can be established for any α ∈ (0, 1) and without additional conditions. Theorem 2. Suppose that (g1) − (g3) hold true, then the solutions of system (P) are continuous with respect of initial values-i.e., Proof. Let u(t; φ), v(t; ϕ) be the solutions of (1)-(4) with initial values, φ and ϕ, respectively. Set Multiplying above equation by w(t), and integral over Ω, we obtain Again using the Bellman-Gronwall Lemma 3 and (21), we find that The proof is complete.

Dissipativity
In this section, we derive some uniform estimates of solutions to problem (P) by using Proposition 2. Besides, in this section, we assume that g(t, u t ) = g(u(t − τ(t))). Definition 7. The system (P) is said to be dissipative in C H if there exists a bounded set B ⊂ C H , such that for any given bounded set A ⊂ C H , there is a time t * = t * (A), such that for any given initial function φ ∈ A, for all t ∈ [−h, 0], the values of the corresponding solution u(t) of the problem (P) are contained in B for all t ≥ t * . The set B is called an absorbing set of the system (P).
We assume that Theorem 3. (Existence of absorbing sets in C H ) Assume that (g1) − (g3), (17) and (32) hold. Then there exists T > 0, such that for all t ≥ T, the solution of problem (P) satisfies Proof. Multiplying (1) by u, integrating over Ω, we have Then we obtain Using Proposition 2, we find that |φ(t)| 2 , and the parameter λ * is defined by is strictly negative, namely, there exists some positive constants 0 satisfying −λ 1 ν + 2L 2 g λ 1 ν < − 0 such that λ * ∈ [−λ 1 ν + 2L 2 g λ 1 ν , − 0 ], and the estimate in (9) holds for all t such that t ≥ τ(t) + 1. In other words, for λ * ∈ [−λ 1 ν + For the case of t < τ(t) + 1, in order to analyze the dissipativity of problem (P) in phase space C H by Proposition 3, we first need to consider the following fractional differential equation, Then, by using the method of steps [28] (Theorem 1), we have that the initial value problem (34) has, on the interval [0, kh], a unique solution that can be represented by where c ih is a constant, i = 1, 2, · · · , k.
Since λ * and −λ 1 ν are strictly negative, by the property of Mittag-Leffler function [2], we obtain where C α > 0 is a constant independent of t. Therefore, there exists T > 0 large enough, such that for all t ≥ T, the solution of problem (P) satisfies Denote by B C H = B(0, λ 1 ν f 0 (λ 1 ν) 2 −2L 2 g + 1) the absorbing set in phase space C H , which implies that system (P) is dissipative. The proof is complete.

Discussion
In this work, we prove the existence and uniqueness of solution for fractional Navier-Stokes equations with variable delays for α ∈ ( 1 2 , 1), and we show that this system is dissipative in the phase space C H , namely, there exists a global absorbing set in C H . Different from the classic Navier-Stokes equations with variable delays [22][23][24], in which the existence of pullback absorbing set and pullback attractors were established. Here, we obtained the forward absorbing set, which is more meaningful from the view of applications. Besides, the existence of global attracting set as well as the existence of solution for α ∈ (0, 1) in phase space C H are still open problems. These will be considered in the future.

Conflicts of Interest:
The authors declare no conflict of interest.