Quasi-periodic solutions for the incompressible Navier-Stokes equations with nonlocal diffusion

1.   Introduction

Consider the following incompressible Navier-Stokes (NS) system with nonlocal diffusion

where the variables $x\in \mathbb{T}^d = \mathbb{R}^d/(2\pi \mathbb{Z})^d(d\ge2), t\in \mathbb{R}$. 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 $(-\Delta)^\alpha$ with $\alpha \in (0, 1)$ is defined by the Fourier transform on the torus

The parameter $\varepsilon$ is a small positive constant, and the time-dependent external force $f(\mathbb{T}^\nu, \mathbb{T}^d)$ is quasi-periodic with frequency $\omega = (\omega_1, \cdots, \omega_\nu) \in \mathbb{R}^\nu$.

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 $\alpha = 1$, 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 $\mathbb{T}^d (d\ge2)$. Some authors have applied the normal form method and celebrated the Kolmogorov-Arnold-Moser (KAM) theory to the NS and Euler equations ($\alpha = 0$), and obtained several intriguing results ^[24,25,26]. Baldi and Montalto ^[24] proved the existence of quasi-periodic solutions for Euler equations on $\mathbb{T}^3$. Montalto ^[25] considered the inviscid limit problem for the quasi-periodic solutions to the NS equations on $\mathbb{T}^2$. 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 $\mathbb{T}^2$, and a generalized version of the higher dimension was established in ^[28].

However, regarding the time periodic problem for the general case of $\alpha \in (0, 1)$, 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 $\alpha \in [1/2, 1)$. It is worth mentioning that the remained case of solutions near zero for $\alpha \in (0, 1/2)$ is hard to handle, which is due to the lack of the sufficient smoothing effect to control the nonlinear convection type term $u\cdot \nabla u$. 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 $u_\omega(t) : = U(\omega t, x)$, $p_\omega(t) : = P(\omega t, x)$ 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,

and

Here, the terms independent of $x$ appear in the nonlinear term vanishes due to $\nabla \cdot U = 0$. The first equation (1.3) could be solved under the assumption that the frequency $\omega$ is Diophantine, i.e., for some $\gamma \in (0, 1)$,

The existence for the second equation (1.4) is obtained by means of the contraction mapping principle, provided that $\varepsilon$ is suitably small. The main results are stated as follows.

Theorem 1.1 (Existence). Let $\nu \ge 1$, $d\ge2$ and $\alpha \in [1/2, 1)$, and let $\sigma > \nu/2$, $s > d/2+1$ and $N \ge \max\{\sigma+\nu, \sigma+s-2\alpha\}$ be real numbers. Assume that $f(\omega t, x) \in C^N(\mathbb{T}^\nu \times \mathbb{T}^d; \mathbb{R}^d)$ is a time-dependent quasi-periodic function with Diophantine frequency $\omega$ and satisfies

then there exists a positive constant $\varepsilon_{0}$ such that if $\varepsilon \leq \varepsilon_{0}$, system (1.1) admit time quasi-periodic solutions

Moreover,

and

In particular, if $f$ satisfies

then the above conclusion holds for $\omega \in \mathbb{R}^\nu$, and

Remark 1.2. If we look for the quasi-periodic solutions near a constant vector $\zeta \in \mathbb{R}^d$ satisfying a suitable assumption, a similar statement may be obtained for $\alpha \in (0, 1)$ by the normal form method and KAM method. Precisely, solutions are written as

where $\nabla \cdot w = 0$.

Remark 1.3. It is easy to see that if $U_0(\varphi)$ is the solution of Eq (1.3), then $U_0+{\bf{C}}({\bf{C}}\in\mathbb{R}^d)$ is also a solution of Eq (1.3). In this paper, we choose the zero order term as zero.

On the other hand, let $u(t, x) = u_\omega(t, x) + v(t, x)$ and $p(t, x) = p_\omega(t, x) + q(t, x)$ be the perturbed quasi-periodic solutions. The asymptotic stability of quasi-periodic solutions $u_\omega(t, x)$, $p_\omega(t, x)$ can be investigated by studying the global existence of solutions for the following initial value problem

with initial datum

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 $\alpha \in (1/2, 1)$, $\beta \in (0, 1)$ and the assumptions of Theorem 1.1 hold. Suppose that $u_{0} = u(0, x) \in H^{s}$ and

Put

then there exists a positive constant $\delta_{1}$ such that if $E_{1} \leq \delta_{1}$, the problems (1.10) and (1.11) have a unique global solution $(v, q) \in C([0, \infty), H^{s})$. Moreover,

and

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 $f(\omega t, x)$. Finally, the asymptotic stability of time quasi-periodic solutions obtained in Section 3 discussed in Section 4.

2.   Preliminaries

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

2.1.   Notations and function spaces

We introduce some notations and function spaces used in this paper. For $m\in \mathbb{N}$, let $\mathbb{Z}_0^m : = \mathbb{Z}^m\backslash \{0\}$. Let $\mathbb{P}$ denote the Leray projection operator on solenoidal vector fields. We denote a generic positive constant by $C$. For any vectors $u = (u_1, u_2, \cdots, u_d)$ and $v = (v_1, v_2, \cdots, v_d)$, $u \otimes v = (u_i v_j)_{1\le i, j \le d}$ represents the matrix-valued tensor product of $u$ and $v$.

For any $u(x) \in L^2(\mathbb{T}^d)$, one can expand it by the Fourier series ^[29]

where the Fourier coefficients are given by

Let $\pi_0$ be the orthogonal projection operator defined by

For any $s\ge0, \sigma\ge0$, $H^s = H^s(\mathbb{T}^d)$ is the standard Sobolev space with the norm

And let $\dot{H}^s = \dot{H}^s(\mathbb{T}^d)$ be the corresponding Sobolev space with the norm

For any $u(\varphi, x) \in L^2(\mathbb{T}^\nu; L^2(\mathbb{T}^d))$, one can expand it as

where the Fourier coefficients are

The function space $H^{\sigma, s} = H^\sigma(\mathbb{T}^\nu; H^s)$ is defined by

where

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 $C^m(\mathbb{T}^\nu, X)$ in a similar way.

2.2.   Two lemmas

Lemma 2.1. Let $\sigma > \nu/2$, $\iota \ge 0$, then $H^{\sigma+\iota}(\mathbb{T}^\nu)$ is compactly imbedded in $C^\iota(\mathbb{T}^\nu)$ and

Note that $H^s(\mathbb{T}^d)$ is a Banach algebra whenever $s > d/2$. The result below readily follows.

Lemma 2.2. Let $\sigma > \nu/2$, $s > d/2$, then one can get

for $u(x), v(x) \in H^s(\mathbb{T}^d)$, and

for $u(\varphi, x), v(\varphi, x) \in H^{\sigma}(\mathbb{T}^\nu; H^s(\mathbb{T}^d))$.

3.   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).

3.1.   Proof of Theorem 1.1

Proof. First, for Eq (1.3), by (1.6) we have

Clearly, $\nabla \cdot U_0 = 0$ 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 $H^\sigma(\mathbb{T}^\nu; H^{s}(\mathbb{T}^d))$ with $\sigma > \nu/2$, $s > d/2+1$. To this end, define the following function space

where

Let

Owing to the operator $\mathcal{L}_\omega$ as invertible on $X^{\sigma, s}$, define the mapping

where

Now, it suffices to verify that the mapping $\mathcal{T}$ is a strict contraction on some closed ball in $X^{\sigma, s}$. Define the function space

where $R$ is a positive constant that will be determined later. For any $u\in X^{\sigma, s}$, a direct calculation shows that

where we have used the equivalence of $\|u\|_{H_\varphi^\sigma H_x^s}$ and $\|u\|_{H_\varphi^\sigma \dot{H}_x^s}$, then for any $u \in X_R$, it is easy to check that

From (3.3), Lemma 2.2 and the Sobolev embedding theorem, it follows that

Taking $R = 4C_1\varepsilon$ yields

whenever $\varepsilon \le 1/(16C_1C_2)$. Moreover, for any $u_1, u_2 \in X_R$, from (3.2), (3.4) and Lemma 2.2, we have

Noting that $R = 4C_1\varepsilon$ and $\varepsilon \le 1/(16C_1C_2)$, it gives us

Combining (3.5) and (3.6), we arrive at $\mathcal {T}$ as a strict contraction from $X_R$ to $X_R$. Thus, there exists a unique fixed point in $X_R$ of the mapping $\mathcal {T}$, 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 $\varphi$, from (1.3) and (1.4), it follows that

and

hence,

For the pressure $P$,

On the whole, we conclude that $u(\omega t, x) = U(\varphi, x)$ and $p(\omega t, x) = P(\varphi, x)$ 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 $U_0 = {\bf{C}} \in \mathbb{R}^d$ 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. □

4.   Stability

In this section, we prove the asymptotic stability of quasi-periodic solutions.

Lemma 4.1. Let $\alpha \in [1/2, 1), s\ge0$. Assume $u_0 \in \dot{H}^s$ satisfies $\pi_0 u_0 = 0$, then it holds that

Let $1 \le s_0 \le s+1$ and $\beta \in (0, 1)$, then for $g \in \dot{H}^{s+1-s_0}$ satisfying $\pi_0 g = 0$, 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

Lemma 4.2. Let $\alpha \in (1/2, 1)$, $\beta \in (0, 1)$ and $s > d/2$. There holds that

and

for $u, v \in C([0, \infty); H^s)$ and $\sup_{t\ge0} e^{\beta t} \|v\|_{H^s} < \infty$.

Proof. Since the proof of (4.4) and (4.5) are similar, we only prove the former. If $0 < t \le 1$, by (4.2) with $s_0 = 1$ and Lemma 2.1, we have

If $t > 1$, we decompose the integral as

By changing variables, one can treat $J_2$ similarly with the case $0 < t \le 1$. For the term $J_1$, from the Sobolev imbedding inequality and (4.2) with $s_0 = s+1$, it follows that

Since $e^{-\beta t} \le 1$ for any $t\ge0$, 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 $B(\cdot, \cdot)$ is defined in (4.3). For $0 < \beta < 1$ and $s > d/2$, define the following functional space

where

For the linear term, from (4.1) we get

For the Duhamel integral terms, applying Lemma 4.2, one can get

and

Combining (4.8)–(4.10) yields

Taking $R = 4 E_1$, one can obtain

provided that $E_1 \le 1/(32C_3)$ and $\varepsilon \le \min \{1/(16C_1C_2), 1/(8C_1C_3)\}$. For any $v_1, v_2 \in Y_R$, let $v_\delta = v_1-v_2$, and we get

Applying Lemma 4.2 again, one can get

provided that $E_1 \le 1/(32C_3)$ and $\varepsilon \le \min \{1/(16C_1C_2), 1/(8C_1C_3)\}$. Combining estimates (4.11) and (4.12), one can see that map $\mathcal{M}$ is a strict contraction. We then obtain that there exists a unique fixed point $v$ of the map $\mathcal{M}$ 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.

Acknowledgments

This work is supported partially by NSFC Grants (12225103, 12071065 and 11871140) and the National Key Research and Development Program of China (2020YFA0713602 and 2020YFC1808301).

Conflict of interest

The authors declare there is no conflicts of interest.