Quasi-periodic solutions for the incompressible Navier-Stokes equations with nonlocal di ff usion

: This paper studied the incompressible Navier-Stokes (NS) equations with nonlocal di ff u-sion on T d ( d ≥ 2). Driven by a time quasi-periodic force, the existence of time quasi-periodic solutions in the Sobolev space was established. The proof was based on the decomposition of the unknowns into the spatial average part and spatial oscillating one. The former were sought under the Diophantine non-resonance assumption, and the latter by the contraction mapping principle. Moreover, by constructing suitable time weighted function space and using the Banach fixed point theorem, the asymptotic stability of quasi-periodic solutions and the exponential decay of perturbation were proved.


Introduction
Consider the following incompressible Navier-Stokes (NS) system with nonlocal diffusion where the variables x ∈ T d = R d /(2πZ) d (d ≥ 2), t ∈ R. The field u(x, t) and scalar p(x, t) represent the velocity and pressure of fluid at point (x, t), respectively.The nonlocal operator (−∆) α with α ∈ (0, 1) is defined by the Fourier transform on the torus The parameter ε is a small positive constant, and the time-dependent external force f (T ν , T d ) is quasi-periodic with frequency ω = (ω 1 , • • • , ω ν ) ∈ R ν .The incompressible NS system with nonlocal diffusion describes the fluid motion with internal friction interaction [1], and corresponds a stochastic representation given in terms of stochastic differential equations driven by Lévy processes [2,3].It has received increasing attention over the past few decades, and many interesting results have been established in several works [2,[4][5][6][7][8][9][10]. For instance, see [4,5] for the global well-posedness, [6] for the regularity theory of Caffarelli-Kohn-Nirenberg and [8,9] for the ill-posedness of weak solutions.
Concerning the time periodic solutions of system (1.1) with α = 1, we refer the readers to [11][12][13][14][15] for the bounded domain cases, and to [16][17][18][19][20][21][22] for the unbounded domain cases.Serrin [11] proved the existence of time periodic solutions based on the solvability and stability of an initial value problem.Kozono and Nakao [16] converted the original time periodic problem into a mild formulation, and then established the existence and uniqueness of solutions by the L p -L q estimates of an associated semigroup and Kato iteration.Based on the decomposition of unknowns into a steady part and a purely periodic part, Kyed [17,18] derived the maximal regularity of the linearized problem and then established the unique existence of solutions for the nonlinear problem.Galdi, jointly with his collaborators [19][20][21], handled the scenario of flow around a rigid body, which moved or rotated in a prescribed periodic manner.
Recently, Montalto [23] obtained the existence and stability of quasi-periodic solutions for incompressible NS equations on T d (d ≥ 2).Some authors have applied the normal form method and celebrated the Kolmogorov-Arnold-Moser (KAM) theory to the NS and Euler equations (α = 0), and obtained several intriguing results [24][25][26].Baldi and Montalto [24] proved the existence of quasiperiodic solutions for Euler equations on T 3 .Montalto [25] considered the inviscid limit problem for the quasi-periodic solutions to the NS equations on T 2 .Berti, Hassainia and Masmoudi [26] proved the existence of vortex patches close to the Kirchhoff ellipses.In addition, Crouseilles and Faou [27] constructed an explicit quasi-periodic solution with compact support on T 2 , and a generalized version of the higher dimension was established in [28].
However, regarding the time periodic problem for the general case of α ∈ (0, 1), to the best of our knowledge there are few results available in the literature.Motivated by the works above, we would like to extend the results in [23] to system (1.1) and obtain the existence of quasi-periodic solutions near zero for α ∈ [1/2, 1).It is worth mentioning that the remained case of solutions near zero for α ∈ (0, 1/2) is hard to handle, which is due to the lack of the sufficient smoothing effect to control the nonlinear convection type term u • ∇u.Nevertheless, the issues considered in the present paper do not involve the small divisor problem due to the presence of dissipation.
On one hand, to seek the quasi-periodic solutions u ω (t) := U(ωt, x), p ω (t) := P(ωt, x) of system (1.1), it suffices to solve the following equations: then one can decompose system (1.2) as a spatial averaged part and an oscillating one, Here, the terms independent of x appear in the nonlinear term vanishes due to ∇ • U = 0.The first equation (1.3) could be solved under the assumption that the frequency ω is Diophantine, i.e., for some γ ∈ (0, 1), (1.5) The existence for the second equation (1.4) is obtained by means of the contraction mapping principle, provided that ε is suitably small.The main results are stated as follows.
) is a timedependent quasi-periodic function with Diophantine frequency ω and satisfies then there exists a positive constant Moreover, and In particular, if f satisfies then the above conclusion holds for ω ∈ R ν , and Remark 1.2.If we look for the quasi-periodic solutions near a constant vector ζ ∈ R d satisfying a suitable assumption, a similar statement may be obtained for α ∈ (0, 1) by the normal form method and KAM method.Precisely, solutions are written as where 3).In this paper, we choose the zero order term as zero.
On the other hand, let u(t, x) = u ω (t, x) + v(t, x) and p(t, x) = p ω (t, x) + q(t, x) be the perturbed quasi-periodic solutions.The asymptotic stability of quasi-periodic solutions u ω (t, x), p ω (t, x) can be investigated by studying the global existence of solutions for the following initial value problem By constructing time-weighted function space, we prove the global existence of small solutions by the Banach fixed point theorem.As a by product, we present the exponential decay estimates of the perturbation v.
We state the asymptotic stability of quasi-periodic solutions as follows.
Theorem 1.4 (Asymptotic stability).Let α ∈ (1/2, 1), β ∈ (0, 1) and the assumptions of Theorem 1.1 hold.Suppose that u 0 = u(0, x) ∈ H s and then there exists a positive constant δ 1 such that if E 1 ≤ δ 1 , the problems (1.10) and (1.11) have a unique global solution (v, q) ∈ C([0, ∞), H s ).Moreover, The paper is organized as follows.In Section 2, we give some function spaces and useful lemmas.Then, in Section 3, we prove the existence of time quasi-periodic solutions which have the same oscillation frequency with the f (ωt, x).Finally, the asymptotic stability of time quasi-periodic solutions obtained in Section 3 discussed in Section 4.

Preliminaries
This section collects some notations, function space and useful lemmas.

Notations and function spaces
We introduce some notations and function spaces used in this paper.For m ∈ N, let Z m 0 := Z m \{0}.Let P denote the Leray projection operator on solenoidal vector fields.We denote a generic positive constant by C. For any vectors u , j≤d represents the matrix-valued tensor product of u and v.
For any u(x) ∈ L 2 (T d ), one can expand it by the Fourier series [29] where the Fourier coefficients are given by û Let π 0 be the orthogonal projection operator defined by For any s ≥ 0, σ ≥ 0, H s = H s (T d ) is the standard Sobolev space with the norm .
And let Ḣs = Ḣs (T d ) be the corresponding Sobolev space with the norm .
For any u(φ, x) ∈ L 2 (T ν ; L 2 (T d )), one can expand it as where the Fourier coefficients are The function space H σ,s = H σ (T ν ; H s ) is defined by .
For an interval I and a Banach space X, C m (I; X) denotes the space of m-times continuously differentiable functions on I with values in X.One can also define C m (T ν , X) in a similar way.

Existence of quasi-periodic solutions
This section is devoted to establishing the existence of quasi-periodic solutions that have the same oscillation frequency as f for system (1.1).

Proof of Theorem 1.1
Proof.First, for Eq (1.3), by (1.6) we have Clearly, ∇ • U 0 = 0 and From the Diophantine condition (1.5) and the Hölder inequality, it follows that Next, for Eq (1.4), we would justify the solvability by the contraction mapping principle in the setting H σ (T ν ; H s (T d )) with σ > ν/2, s > d/2 + 1.To this end, define the following function space
Owing to the operator L ω as invertible on X σ,s , define the mapping where Now, it suffices to verify that the mapping T is a strict contraction on some closed ball in X σ,s .Define the function space where R is a positive constant that will be determined later.For any u ∈ X σ,s , a direct calculation shows that where we have used the equivalence of ∥u∥ H σ φ H s x and ∥u∥ H σ φ

Ḣs
x , then for any u ∈ X R , it is easy to check that From (3.3), Lemma 2.2 and the Sobolev embedding theorem, it follows that . Moreover, for any u 1 , u 2 ∈ X R , from (3.2), (3.4) and Lemma 2.2, we have
Noting that R = 4C 1 ε and ε ≤ 1/(16C 1 C 2 ), it gives us Combining (3.5) and (3.6), we arrive at T as a strict contraction from X R to X R .Thus, there exists a unique fixed point in X R of the mapping T , which is a unique time quasi-periodic solution U p of (1.4).
Finally, we end the proof with the regularity of solution U = U 0 + U p .By Lemma 2.1, we have For the C 1 regularity with respect to φ, from (1.3) and (1.4), it follows that For the pressure P, On the whole, we conclude that u(ωt, x) = U(φ, x) and p(ωt, x) = P(φ, x) are solutions of system (1.1) and satisfy the desired properties (1.7) and estimates (1.8).For the particular case of f satisfying (1.9), Eq (1.3)only admits a trivial solution U 0 = C ∈ R d and the verification of Eq (1.4) is similar with the above one.We omit here for simplification.The whole proof of Theorem 1.1 is completed.

Stability
In this section, we prove the asymptotic stability of quasi-periodic solutions.
Let 1 ≤ s 0 ≤ s + 1 and β ∈ (0, 1), then for g ∈ Ḣs+1−s 0 satisfying π 0 g = 0, it holds that Proof.We only prove (4.2).From Parseval's equality and the basic property of rapidly decaying functions, one can get The lemma is proved. Denote If t > 1, we decompose the integral as By changing variables, one can treat J 2 similarly with the case 0 < t ≤ 1.For the term J 1 , from the Sobolev imbedding inequality and (4.2) with s 0 = s + 1, it follows that Since e −βt ≤ 1 for any t ≥ 0, the estimates (4.6) can be proved in a similar way.The lemma is proved.Now we turn back to the proof of Theorem 1.4.

Proof. Note that
and, therefore, we mainly focus on the unknown v(t, x) in the following.Due to Duhamel's principle, one can deduce that Therefore, define the following mapping where B( provided that E 1 ≤ 1/(32C 3 ) and ε ≤ min{1/(16C 1 C 2 ), 1/(8C 1 C 3 )}.Combining estimates (4.11) and (4.12), one can see that map M is a strict contraction.We then obtain that there exists a unique fixed point v of the map M in Y R , which is the unique solution of problems (1.10) and (1.11).Therefore, we conclude that the quasi-periodic solution in Theorem 1.1 is asymptotically stable.Moreover, This completes the proof of Theorem 1.4.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
is a Banach algebra whenever s > d/2.The result below readily follows.