Asymptotic Behavior of the Solutions of the Generalized Globally Modified Navier–Stokes Equations

The paper is concerned with the existence and the asymptotic behavior of solutions to a class of generalized Navier–Stokes equations, which generalises the so-called globally modified Navier–Stokes equations. The existence and uniqueness of solutions are proved under different assumptions on the dissipation and modification factors. For the asymptotic behavior of solutions, we prove the existence of global attractors in proper spaces. The results generalize some results derived in our previous work Ann. Polon. Math. 122(2):101–128(2019).

Recently, Dong and Song [23] studied the globally modified Navier-Stokes equations with fractional dissipation in the whole space R 3 : e existence and uniqueness of global solutions was obtained under the assumption α > (3/4), see also [24], for the existence and uniqueness results in a bounded domains. ese results review that the modifying factor F N (‖Λ β u‖) decreases the singularity of the term (u · ∇)u "too much" so that one can control the nonlinear term by using only the fractional dissipation (− Δ) α u, α < 1 rather than Δu in (1).
is inspires us to weaken the modification term and to investigate that how the dissipation and modification terms interact with each other to determine the existence and uniqueness of the solutions. Precisely speaking, we shall consider the following modified Navier-Stokes equations in Ω � [0, L] 3 : with periodic boundary conditions, where the constants ] > 0 and α, β ≥ 0. Assume that the initial data and the forcing term are mean-free functions, i.e., en, the solution is also a mean-free function, and the Poincaré inequality holds. We prove that system (3) admits at least one global weak solution when 4α + 2β > 5, u 0 , f ∈ H. Moreover, if α > βand 4α 2 − 5α + 2β 2 ≥ 0 or 2α + 4β > 5, the weak solution is unique. On the contrary, we prove that when 4α + 2β > 5, u 0 ∈ H s , f ∈ H s− α , and s ≥ β, the system possesses a unique global strong solution and the existence of global attractor A, for the solution semigroup in H s can be proved when s ≥ β. Furthermore, if s ≥ max 1, β , we can give explicit upper bound for the fractal dimension of the attractor A. When β � 1, we prove that the system admits at least one weak solution when 3/4 < α ≤ 1, u 0 , f ∈ H, and a unique global strong solution when 3/4 < α ≤ 1, u 0 ∈ H s , f ∈ H s− α , and s ≥ 1. e existence and uniqueness in [24] are consistent with the results in this study. If β � 0, the modifying factor F N (‖Λ β u‖) is constant 1, and system (3) becomes the well-known generalized Navier-Stokes equation. e standard existence result shows that the system has global regularity when α ≥ 5/4 [25], which are consistent with our result. ese results extend the previous results in [8,23,24] to more general settings. e rest of the paper is organized as follows. In Section 2, we provide some preliminaries about the function spaces and several useful lemmas. en, in Section 3, we prove the existence and uniqueness results of solutions, while in Section 4 and Section 5, we discuss the existence of a global attractor and the upper bound of its fractal dimension.

Preliminaries and Inequalities
Let Ω � [0, L] 3 . e fractional operator Λ 2α � (− Δ) α for any α ∈ R can be defined as for be the space of restrictions to Ω of infinitely differentiable functions that are L-periodic in each direction and with zero mean in Ω. For s ∈ R, we denote by H s (Ω) the closure of Particularly, when s � 0, we denote H 0 by H for short. In this study, for any Banach space X, we denote its norm as ‖ · ‖ X ; particularly, ‖ · ‖ L 2 will be abbreviated as ‖ · ‖. Now, we recall the definitions of the global attractor and the fractal dimension, see [26,27].
{ } t≥0 be a semigroup on a Banach space X. A subset A ⊂ X is called a global attractor for the semigroup if A enjoys the following properties: where dist is the Hausdorff semidistance between sets in X, defined as Definition 2. e fractal dimension of a compact set K in a Banach space X is defined as where N ε (K) is the minimal number of balls of radius ε in X needed to cover K.
e following inequalities may be found in [26,28].

Lemma 1 (Young's inequality).
For any positive constants a, b, and ε and any 1 < p < ∞, it holds that 2 Discrete Dynamics in Nature and Society where p, q, r, n, m, j, and λ satisfy e following product estimates play an essential role in our analysis (see [29]). Lemma 4. Suppose that f, g ∈ S the Schwartz class. en, for s > 0, 1 < p < + ∞, there exist a positive constant C such that with q 1 , p 2 ∈ (1, +∞) satisfying e following lemma will play an important role in the proof of our result. It was first proved by Romito in [30] for the case β � 1. e general case can be proved similarly.

Lemma 5.
For every u, v ∈ H 1 and each N > 0, we have

Existence and Uniqueness Results
We now give the definition of weak solutions to system (3).

Theorem 1. Let α and β be two constants such that
If in addition 4α 2 − 5α + 2β 2 ≥ 0 or 2α + 4β > 5, the weak solution is unique. (ii) On the other hand, if u 0 ∈ H s , f ∈ H s− α , and s ≥ β, then problem (3) admits a unique global solution u satisfying

Remark 2.
e standard existence result for the Navier-Stokes equations shows that system (3) possess a unique global solution, for all β � 0, when α ≥ 5/4, so we only consider the case α < 5/4. However, when α ≤ 1/2, we cannot use the dissipation term of the equations to control the nonlinear term, and the existence results is difficult to prove in this case.
Proof. Let us divide the proof into several steps.
Step 1: we prove the existence of the weak solution by the Galerkin approximation method. Let ϕ j ∞ j�1 be an orthonormal basis of H consisting of eigenfunctions of the Stokes operator A and λ j are the corresponding eigenvalues which are increasing with j. Consider the following ordinary differential system: Discrete Dynamics in Nature and Society j c jm (t)ϕ j , and P m is the orthogonal projection form H onto the space spanned by ϕ 1 , ϕ 2 , . . . ϕ m . By the standard existence theorem for ordinary differential equations, for each m, there exists a local solution u m to system (21) in the interval [0, T m ).
Multiplying (21) by u m (t), using the Poincaré inequality and integrating from 0 to t, we obtain Using Gronwall's inequality, we obtain which implies that Let us perform the estimates for zu m /zt . For any φ ∈ H α , using Hölder's inequality, the product estimates (see Lemma 1), the Gagliardo-Nirenberg inequality, and Young's inequality, we deduce that e last inequality holds since In view of (25), the sequence Using the standard Aubin-Simon-type compactness results [26,27], there exists an element such that up to subsequences, Now, it remains to verify that u is a weak solution to problem (3). We treat the case α > β and α ≤ β separately.
. By the Gagliardo-Nirenberg inequality and Hölder's inequality [27], we deduce that where θ 1 � α − β/α. is, together with (25) and (30), implies that us, up to subsequences, And, hence, anks to (30) and (34), taking φ ∈ H α as a test function in (21) and passing to the limits, we obtain that u is a weak solution to system (3). As the calculations are rather similar to those in [8], we omit the details for concision.
Assume that there exist a positive integer N 0 such that N 0 α ≤ β < (N 0 + 1)α, without lose of generality, we set N 0 � 1. After multiplying equation (21) with Λ 2α u m and integrating, we obtain We can estimate the first term of the right side as follows. Since β + 2α > 5/2, α > 1/2, using the product estimates (see Lemma 1), the Gagliardo-Nirenberg inequality, and Young's inequality, we have the estimate where θ 4 � min 1, (4α + 2β − 5/4α) . Combining with (24), (35), and (36), we obtain For any t ≥ τ ≥ 0, integrating (22) between t and t + τ and using (24), we obtain and denote by |Ω m | the Lebesgue measure of Ω m . We have which implies that ‖Ω m ‖ ≤ τ/2. erefore, for any given By using the Gronwall inequality, we obtain, for all t ≥ ε, Integrating (37) from ε to T and taking (42) into consideration, we deduce that Discrete Dynamics in Nature and Society 5 for all ε ≤ t ≤ T. us, we have Taking Λ 4α u m , Λ 6α u m , . . . , Λ N 0 α u m as test functions and performing similar analysis, we may prove that Denotes α 1 � (N 0 + 1)α; since β < α 1 , we deduce that us, up to subsequences, By the standard diagonal process, we can extract a subsequence of u m (still labeled by u m ) such that en, using (30) and (34), we can take limits in (21) as in Case 1 to obtain that u is a solution of (3).
Step 3: we now prove the second part of the theorem. If u 0 ∈ H s , f ∈ H s− α , and s ≥ β, we multiply (21) by Λ 2s u m to deduce that Using the product estimates and the Gagliardo-Nirenberg inequality, we deduce that Discrete Dynamics in Nature and Society where is, combined with (63), yields that Hence, for all t ≥ τ ≥ 0, us, u m is bounded in L ∞ (0, T; H s ) ∩ L 2 (0, T; H s+α ), ∀T > 0. Passing to the limit, we obtain (20) immediately.

(73)
Hence, according to the standard Sobolev embedding result [26,27], we need only to show that Indeed, for any φ ∈ H α , we have (75) erefore, which implies that Applying the product estimates and the imbedding of fractional Sobolev spaces, we have erefore, Combining (73) and the assumption f ∈ H s− α , we know that Λ s u t ∈ L 2 (0, T; H − α ). e proof is thus complete.

Attractor for Strong Solution.
In this section, we prove the existence of a global attractor for system (3). Proof.
anks to eorem 1, we know that the semigroup is continuous. It remains to prove the existence of an absorbing set and the compactness of the semigroup in H s .
Absorbing set: let u(t) be the solution of system (1). Similar to (24), we have From the above inequality, we can deduce that there exists a T 0 � t(‖u 0 ‖) such that Multiplying (3) by Λ 2s u and integrating, we have for all t > 0. Similar to (65), we deduce that Integrating the both sides from 0 to T 0 and taking (24) into consideration, we obtain Using the Poincaré inequality and (81), we obtain from (83) that Gronwall's inequality then implies that anks to (96), we know that if erefore, there is an absorbing set B 1 for the semigroup S(t) { } t≥0 in H s . Compactness of the semigroup: we show that, for any bounded sequence v n 0 in H s any t > 0, the sequence } has a convergent subsequence in H s . Similar to (66) and (79), we can prove that Using the Aubin-Simon type compactness results [26,27], there exists an element v with such that up to subsequences, i.e., In particular, there exists a τ ∈ (0, 1) such that Recall that the map S(t): H s ⟶ H s is continuous, and we obtain that us, the semigroup S(t) is compact, for any t ≥ 1. anks to the standard existence results on global attractors, we may obtain a global attractor in H s for the solution semigroup S(t).

Finite Dimensionality of the Attractor
In this section, we provide the upper bound for the fractal dimension of the attractor derived in Section 4.
anks to Lemma 8 in [24], we have the following.
Proof. Take the inner product of (50) with Λ 2s 1 w, and we know that w satisfies, for any s − 1 + 2α − ϑ ≤ s 1 ≤ s − 1 + 2α: Since A is bounded in H s 0 +α , we have For L 1 , using Hölder's inequality, the product estimates, Gagliardo-Nirenberg inequality, and Young's inequality, we deduce that Discrete Dynamics in Nature and Society