Commutators of Hardy-Cesàro operators on Morrey-Herz spaces with variable exponents

1.   Introduction

In this paper, we consider the following Cauchy problem of Navier-Stokes equations with the damping term:

where $u = u(x, t)\in \mathbb{R}^3$, $\pi = \pi(x, t)\in \mathbb{R}$ represent the unknown velocity field and the pressure respectively. $\alpha\ge0$, $\beta\ge1$ are real parameters. $\Lambda : = (- \Delta)^\frac{1}{2}$ is defined in terms of Fourier transform by

Damping originates from the dissipation of energy by resistance, which describes many physical phenomena such as porous media flow, resistance or frictional effects, and some dissipation mechanisms (see ^[1] and references cited therein). When $\alpha = 1$, Cai and Jiu first proved that there exists a weak solution of (1.1)–(1.3) if $\beta > 1$. Furthermore, if $\beta\ge\frac{7}{2}$, the global existence of the strong solution was established. Later, this result was improved by Zhang, Wu and Lu in ^[2], where the lower bound of $\beta$ decreased to $3$. Zhou^[3] proved the lower bound 3 is critical in some sense. For the general case, it is proved that when $\frac{3}{4}\le\alpha < 1$, $\beta\ge\frac{2\alpha+5}{4\alpha-2}$ or $1\le\alpha < \frac{5}{4}$, $\beta\ge1+\frac{10}{4\alpha+1}$, the global existence of the solution was established in ^[4]. For the asymptotic behavior, one can refer to ^[5,6,7] for details.

For the generalized Navier-Stokes equations (our system without damping term) when $\alpha = 1$, there are many regularity criteria to the system (1.1)–(1.3). The classical Prodi-Serrin's-type criteria was given in ^[8,9,10], where it was proved that if a weak solution $u \in L^{p}(0, T; L^{q}(\mathbb{R}^3))$ with $\frac{2}{p}+\frac{3}{q} = 1$, $q\ge3$, then the solution is regular and unique. Beirão da Veiga ^[11] established the analogous result: $\nabla u \in L^{p}(0, T; L^{q}(\mathbb{R}^3))$ with $\frac{2}{p}+\frac{3}{q} = 2$, $q\ge\frac{3}{2}$. For the general case, in ^[12], Jiang and Zhu proved that if $\Lambda^{\theta} u \in L^{p}(0, T; L^{q}(\mathbb{R}^3))$ with $\frac{2\alpha}{p}+\frac{3}{q}\le2\alpha-1+\theta$, $\theta\in [1-\alpha, 1]$, $q > \frac{3}{2\alpha-1+\theta}$, then the solution remains smooth on [0, T]. One can refer to ^[11,13,14] for more classical regularity criteria. For the large time behavior, Jiu and Yu proved the algebraic decay of the solution under specific conditions (see ^[15]).

Our paper devotes to considering the role of damping terms in regularity criteria for the system (1.1)–(1.3). We will explain the role of damping term in the following two questions:

(1) When does the dissipative term work better than the damping term?

(2) How does the damping term work?

For the first question, if $\alpha\ge\frac{5}{4}$, the generalized Navier-Stokes equations (our system without damping term) exists a global strong solution $u \in L^{\infty}(0, T; H^{1}(\mathbb{R}^3))\cap{L^{2}(0, T; H^{1+\alpha}(\mathbb{R}^3))}$. Consequently, we only consider the case when $\frac{1}{2} < \alpha < \frac{5}{4}$.

For the second question, we utilize two structures brought by the damping term: $\Vert \vert u \vert ^{\frac{\beta-1}{2}} \nabla u \Vert^2_{L^2}$ (Theorems 1.1 and 1.2, when $1 < \alpha < \frac{5}{4}$) and $\frac{1}{\beta+1} \frac{d}{dt}\Vert u\Vert_{L^{\beta+1}}^{\beta+1}$ (Theorems 1.3 and 1.4, when $\frac{1}{2} < \alpha < 1$). Actually, $\Vert \vert u \vert ^{\frac{\beta-1}{2}} \nabla u \Vert^2_{L^2}$ works better than $\frac{1}{\beta+1} \frac{d}{dt}\Vert u\Vert_{L^{\beta+1}}^{\beta+1}$, because $\Vert \vert u \vert ^{\frac{\beta-1}{2}} \nabla u \Vert^2_{L^2}$ is a first-order estimate resulting from the damping term while $\frac{1}{\beta+1} \frac{d}{dt}\Vert u\Vert_{L^{\beta+1}}^{\beta+1}$ is a zero-order estimate resulting from the damping term. However, because of the technical limitation, we still use $\frac{1}{\beta+1} \frac{d}{dt}\Vert u\Vert_{L^{\beta+1}}^{\beta+1}$ when $\frac{1}{2} < \alpha < 1$. Consequently, when $\frac{1}{2} < \alpha < 1$, how to utilize $\Vert \vert u \vert ^{\frac{\beta-1}{2}} \nabla u \Vert^2_{L^2}$ may be an insteresting question.

We give our main theorems as follows.

Theorem 1.1. When $1 < \alpha < \frac{5}{4}$, $\beta < 1+\frac{10}{4\alpha+1}$, assume that the initial data $u_0(x)\in H^1(\mathbb{R}^3)$ with $\mathrm{div}\, u_0 = 0$, and $u(x, t)$ is a local strong solution of the system (1.1)–(1.3). If $u(x, t)\in L^{p}(0, T; L^{q}(\mathbb{R}^3))$ with

then, for any $T > 0$, the system (1.1)–(1.3) has a global strong solution satisfying

Remark 1.1. In Theorem 1.1, we roughly combine the regularity criteria brought by the dissipative term and the damping term. In fact, we can verify that if $1 < \alpha < \frac{5}{4}$, $2+\frac{1}{2\alpha-1} < \beta < 1+\frac{10}{4\alpha+1}$, then $\frac{2(\alpha-1)}{3-\beta} > 2\alpha-1$. Consequently, (1.4) becomes

which means that damping the term works better than the dissipative term.

Theorem 1.2. When $1 < \alpha < \frac{5}{4}$, $5-2\alpha < \beta < 1+\frac{10}{4\alpha+1}$, assume that the initial data $u_0(x)\in H^1(\mathbb{R}^3)$ with $\mathrm{div}\, u_0 = 0$, and $u(x, t)$ is a local strong solution of the system (1.1)–(1.3). If $\Lambda^{\alpha} u(x, t)\in L^{p}(0, T; L^{q}(\mathbb{R}^3))$ with

then, for any $T > 0$, the system (1.1)–(1.3) has a global strong solution satisfying

Remark 1.2. In Theorems 1.1 and 1.2, we consider the regularity criteria when $\beta < 1+\frac{10}{4\alpha+1}$, because the global existence was established in ^[4] when $\beta\ge1+\frac{10}{4\alpha+1}$. If $\beta\ge1+\frac{10}{4\alpha+1}$, the regularity criteria in Theorem 1.1 is satisfied naturally, so we recover the result in ^[4] when $1 < \alpha < \frac{5}{4}$.

Theorem 1.3. When $\frac{1}{2} < \alpha < 1$, $\beta < \min\{\frac{2\alpha+5}{4\alpha-2}, \frac{3\alpha+2}{\alpha}\}$, assume that the initial data $u_0(x) \in H^{1}(\mathbb{R}^3)\cap L^{\beta+1}(\mathbb{R}^3)$ with $\mathrm{div}\, u_0 = 0$, and $u(x, t)$ is a local strong solution of the system (1.1)–(1.3). If $u(x, t)\in L^{p}(0, T; L^{q}(\mathbb{R}^3))$ with

then, for any $T > 0$, the system (1.1)–(1.3) has a global strong solution satisfying

Remark 1.3. If $\beta\ge\frac{2\alpha+5}{4\alpha-2}$, the regularity criteria in Theorem 1.3 is satisfied naturally, so we recover the result in ^[4] when $\frac{3}{4}\le\alpha < 1$.

Theorem 1.4. When $\frac{1}{2} < \alpha < 1$, $\beta < \min\{\frac{2\alpha+5}{4\alpha-2}, \frac{3\alpha+2}{\alpha}\}$, assume that the initial data $u_0(x)\in H^1(\mathbb{R}^3)\cap L^{\beta+1}(\mathbb{R}^3)$ with $\mathrm{div}\, u_0 = 0$, and $u(x, t)$ is a local strong solution of the system (1.1)–(1.3). If $\Lambda^{\alpha}u(x, t)\in L^{p}(0, T; L^{q}(\mathbb{R}^3))$ with

then, for any $T > 0$, the system (1.1) has a global strong solution satisfying

2.   Regularity criteria

Proof of the Theorem 1.1. Multiplying (1.1) by $-\triangle u$, after integration by parts and taking the divergence-free property into account, we have

For $\int_{\mathbb{R}^3}(u \cdot \nabla)u \cdot \Delta udx$, we have

where

with $\theta_1 = \frac{2q+9-3\beta}{2\alpha q}$. The conditions in Theorem 1.1 imply $\theta_1\in [\frac{1}{\alpha}, 1)$. By direct calculation, we have

Combining the above estimates, we obtain

A standard Gronwall's inequality shows that

This completes the proof of the Theorem 1.1.□

Proof of the Theorem 1.2. Multiplying (1.1) by $-\triangle u$, after integration by parts and taking the divergence-free property into account, we have

For $\int_{\mathbb{R}^3}(u \cdot \nabla)u \cdot \Delta u \; dx$, we have

where

with $\theta_2 = \frac{q}{(2\alpha+3)q-6}$, $\theta_3 = \frac{5-\beta}{2\alpha}$. The conditions in Theorem 1.2 imply $\theta_2 \in (0, 1]$, $\theta_3\in (\frac{1}{\alpha}, 1)$. By direct calculation, we have

Combining the above estimates, we obtain

A standard Gronwall's inequality shows that

This completes the proof of the Theorem 1.2. □

Proof of the Theorem 1.3. Multiplying (1.1) by $-\triangle u$, $u_{t}$ and adding the two equations, after integration by parts and taking the divergence-free property into account, we have

For $\Vert \nabla u\Vert_{L^3}^{3}$, we have

where

By directly calculating, we have

The conditions in Theorem 1.3 imply $\theta_4 \in [\frac{1}{1+\alpha}, 1)$, $\theta_5 \in [\frac{1}{1+\alpha}, 1)$, $\delta_1 \in (0, 3)$.

For $\Vert u\cdot\nabla u\Vert_{L^2}^{2}$, we have

where

By direct calculation, we have

The conditions in Theorem 1.3 imply $\theta_6 \in [\frac{1}{1+\alpha}, 1)$.

Combining the above estimates, we obtain

A standard Gronwall's inequality shows that

This completes the proof of the Theorem 1.3. □

Proof of the Theorem 1.4. Multiplying (1.1) by $-\triangle u$, $u_{t}$ and adding the two equations, after integration by parts and taking the divergence-free property into account, we have

For $\Vert \nabla u\Vert_{L^3}^{3}$, we have

where

By direct calculation, we have

The conditions in Theorem 1.3 imply $\theta_7 \in [1-\alpha, 1)$, $\theta_5 \in [\frac{1}{1+\alpha}, 1)$, $\delta_3 \in (0, 3)$.

We can estimate $\Vert u\cdot\nabla u\Vert_{L^{2}}^{2}$ similarily.

Combining the above estimates, we obtain

A standard Gronwall's inequality shows that

This completes the proof of the Theorem 1.4. □

3.   Conclusions

In this paper, we have established some regularity criteria for the 3D generalized Navier-Stokes equations with nonlinear damping term. First, we consider the case where the dissipative term is superior to the damping term, which corresponds to when the damping term works. Second, in Remark 1.1, we show that the damping term works better than the dissipative term. Furthermore, we have presented that the damping term has different effects in different cases, which shows the balance and the interaction between the dissipative term and the damping term as well as the role of the damping term in regularity criteria. In fact, considering how the damping term works and the interaction between the dissipative term and the damping term is the main idea of this paper.

Use of AI tools declaration

The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (Grant No.12071439) and Zhejiang Provincial Natural Science Foundation of China (Grant No. LY19A010016).

Conflict of interest

The author declares no conflict of interest.