Liouville-type theorem for the stationary fractional compressible MHD system in anisotropic Lebesgue spaces

1.   Introduction and main results

In this paper, we are interested in the Liouville-type theorem in anisotropic Lebesgue spaces for the following stationary fractional compressible MHD system:

Here, $u = (u_1(x), u_2(x), u_3(x))$, $b = (b_1(x), b_2(x), b_3(x))$ and $\rho$ represent the velocity field, the magnetic field, and the density, respectively. $P(\rho) = a\rho^\gamma$ is the pressure with constant $a > 0$ and the adiabatic exponent $\gamma\geq 1$. $\alpha$ and $\beta$ are positive constants. The fractional Laplacian $(-\Delta)^\alpha$ is defined at the Fourier level by the symbol $\mid \xi \mid^{2\alpha}$.

When $b = 0$, $\alpha = 1$, and $\rho =$constant, the above system (1.1) reduces to the classical 3D stationary Navier-Stokes system

The Liouville problem for (1.2) still remains open: Is zero the only decay solution of (1.2) that verifies the finite Dirichlet integral condition?

There are numerous results on the Liouville problem for (1.2). One of the first results is due to Galdi ^[1], who proved that $u\in L^{\frac{9}{2}}({\mathbb{R}}^3)$ is sufficient to imply that $u = 0$. In ^[2], Chae showed that $\Delta u\in L^{\frac{6}{5}}({\mathbb{R}}^3)$, which with the same scaling as (1.3), implies that $u = 0$. In ^[3], Seregin proved that $u = 0$ if $u\in L^6({\mathbb{R}}^3)\cap BMO^{-1}$. Sufficient conditions involving the head pressure for the triviality of the solution to the Navier–Stokes equations are studied by Chae in ^[4,5,6]. In ^[7], Chae and Wolf proved that the solution $u$ to (1.2) is trivial if the $L^s$ mean oscillation of the potential function $V$ of $u$ has a certain growth condition near infinity. In ^[8], Chae and Yoneda proved that if the solution $u\in\dot{H}^1({\mathbb{R}}^3)$ to (1.2) satisfies additional conditions characterized by the decays near infinity and by the oscillation, then $u = 0$. In ^[9,10], Jarrín and his collaborators studied the Liouville-type theorems in Lorentz and Morrey spaces. Kozono, Terasawa, and Wakasugi proved in ^[11] that $u = 0$ if the vorticity $\omega = o(|x|^{-\frac{5}{3}})$ as $|x|\to \infty$ or $\|u\|_{L^{\frac{9}{2}, \infty}}\leq \delta D(u)^{\frac{1}{3}}$ for a small constant $\delta$. For more studies on the Liouville problem of the stationary Navier–Stokes equations, we refer to ^[12,13,14] and references therein.

For the compressible Navier–Stokes system

Chae ^[15] showed that the (1.4) has only a trivial solution $u = 0, \rho =$constant, provided that

In ^[16], Li and Yu proved several improved Liouville-type theorems for the $d$-dimensional stationary compressible Navier–Stokes system. Particularly, they showed that $\rho\in L^\infty({\mathbb{R}}^d)$ and $u\in\dot{H}^1({\mathbb{R}}^d)$ are sufficient to guarantee $u = 0$ and $\rho =$constant when $d\geq 4$. See ^[17,18,19] and references therein for more studies on the Liouville problem of the stationary compressible Navier–Stokes system.

When $\alpha\in (0, 1)$, $b = 0$ and $\rho =$constant, system (1.1) reduces to the following stationary fractional Navier-Stokes system:

To our knowledge, there are few results on the Liouville problem of such a system. In ^[20], Wang and Xiao proved that the smooth solution $u\in \dot{H}^\alpha({\mathbb{R}}^3)\cap L^{\frac{9}{2}}({\mathbb{R}}^3)$ of (1.5) is trivial for $\alpha\in (0, 1)$. In ^[21], Yang proved the same result for $\frac{5}{6}\leq \alpha < 1$. Recently, Chamorro and Poggi ^[22] proved an almost sharp Liouville's theorem for the stationary fractional Navier–Stokes system.

For the stationary fractional compressible Navier–Stokes system

Wang and Xiao ^[20] proved that $\rho =$constant and $u = 0$ provided that

When $\alpha = \beta = 1$ and $\rho =$constant, system (1.1) reduces to the usual MHD system. There are also many results on the Liouville-type theorems for the stationary MHD system. In ^[23], Chae, Degond, and Liu proved that the solution to the stationary incompressible MHD and Hall-MHD system is trivial if $u, b\in L^{\frac{9}{2}}({\mathbb{R}}^3)\cap L^\infty({\mathbb{R}}^3)$ and ${\nabla} u, {\nabla} b\in L^2({\mathbb{R}}^3)$. Later, Zeng^[24] improved this result by removing the boundedness assumption of $b$ and the finite Dirichlet integral assumption ${\nabla} u, {\nabla} b\in L^2({\mathbb{R}}^3)$. Another interesting result of Chae and Weng ^[25] showed that $u = b = 0$ if $u\in L^3({\mathbb{R}}^3)$ and ${\nabla} u, {\nabla} b\in L^2({\mathbb{R}}^3)$. In ^[26], Chae and Wolf proved Liouville-type theorems for the stationary MHD and the stationary Hall-MHD systems by assuming suitable growth conditions at infinity for the mean oscillations for the potential functions. This work has been generalized in ^[27] by Chae et al.. In ^[28,29], Wang studied the Liouville-type theorems for the planar stationary MHD equations. For more related studies, we refer to ^{[30,31,32,33,34,35]} and references therein.

Recently, many authors have been interested in the Liouville-type theorems for the stationary Navier-Stokes equations and the stationary MHD system in anisotropic Lebesgue spaces. The anisotropic Lebesgue space is defined as follows:

Definition. Let $u = u(x_1, x_2, x_3)$ be a measurable function on ${\mathbb{R}}^3$ and $1\leq p, q, r\leq \infty$. We say that $u$ belongs to the anisotropic Lebesgue space $L^p_{x_1}L^q_{x_2}L^r_{x_3}({\mathbb{R}}^3)$, provided that

Here $\|\cdot\|_{L_{x_i}^p({\mathbb{R}})}$ denotes the $L^p$ norm with respect to the variable $x_i$.

Clearly, $L_{x_1}^pL_{x_2}^pL_{x_3}^p({\mathbb{R}}^3)$ coincides with the usual Lebesgue space $L^p({\mathbb{R}}^3)$. Throughout the paper, for any vector $\vec{p} = (p_1, p_2, p_3)$, we use the notation $\|\cdot\|_{L^{\vec{p}}({\mathbb{R}}^3)}$ to denote $\|\cdot\|_{L_{x_1}^{p_1}L_{x_2}^{p_2}L_{x_3}^{p_3}({\mathbb{R}}^3)}$.

In ^[36], Luo and Yin proved that the bounded smooth solution $u\in \dot{H}^1({\mathbb{R}}^3)$ to (1.2) is trivial if

Note that when $p_i = q_i = r_i = \frac{9}{2}$, this result recovers the classical result of Galdi ^[1]. Moreover, each component $u_j$ of the velocity $u$ may belong to different anisotropic spaces. Phan ^[37] proved that the solution $u\in H^1_{\mathrm{loc}}({\mathbb{R}}^3)$ to (1.2) is trivial if

This result requires all components $u_1, u_2$ and $u_3$ lie in the same anisotropic space. Chae ^[38] proved that the solution $u\in L^6({\mathbb{R}}^3)\cap L^q({\mathbb{R}}^3)$ to (1.2) is trivial if

Note that a different order of integration for different components is allowed. In ^[39], Chae generalized this result to MHD equations. Fan and Wang ^[40] also studied the Liouville problem for the stationary incompressible MHD system; they proved that $u, b\in L_{x_1}^qL^q_{x_2}L^r_{x_3}({\mathbb{R}}^3)$ implies that $u = b = 0$, provided that $q, r\in [3, +\infty)$ and $\frac{2}{q}+\frac{1}{r}\geq\frac{2}{3}$. They also claimed that $u = b = 0$ if $u, b\in L_{x_1}^pL^q_{x_2}L^r_{x_3}({\mathbb{R}}^3)$ with $p, q, r\in [3, \infty)$ and $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}\geq\frac{2}{3}$. For the studies on Liouville-type theorems for the stationary compressible MHD system, we refer to Wu ^[41] and references therein.

Recently, Zeng^[42] studied the Liouville-type theorems for the stationary fractional incompressible MHD system and proved that the solution $(u, b)\in \dot{H}^\alpha({\mathbb{R}}^3)\times \dot{H}^\beta({\mathbb{R}}^3)$ is trivial provided that $u = (u_1, u_2, u_3), b = (b_1, b_2, b_3)$ such that $(u_j, b_j)\in L^{\vec{p}_j}({\mathbb{R}}^3)\times L^{\vec{q}_j}({\mathbb{R}}^3)$ with

Different from the above-mentioned results on the MHD system, which require all components $u_1, u_2, u_3$ and $b_1, b_2, b_3$ to lie in the same space, the result of Zeng ^[42] allows each component $u_i$ and $b_i$ to belong to different anisotropic spaces.

Inspired by the aforementioned results, this paper aims to establish a Liouville-type theorem for the stationary fractional compressible magnetohydrodynamic equations in anisotropic Lebesgue spaces. Our main result is as follows.

Theorem 1. Let $0 < \alpha, \beta < 1$, $(\rho, u, B)\in L^\infty({\mathbb{R}}^3)\times\dot{H}^\alpha({\mathbb{R}}^3)\times \dot{H}^\beta({\mathbb{R}}^3)$ be a smooth solution to (1.1); then $u = b = 0$ provided that

and

where $p_{i, j}, \xi_{i, j}\in [1, \frac{3}{2}]$ and $q_{i, j}, \eta_{i, j}\in [3, +\infty)$ for $i, j = 1, 2, 3$.

Remark 2. The assumption (1.7) can be replaced by the following assumption:

See (3.12) for the estimates of $I_{12}$ and $I_2$ in the proof of Theorem 1 for details. Moreover, by the embedding $\dot{H}^\beta({\mathbb{R}}^3)\hookrightarrow L^{\frac{6}{3-2\beta}}({\mathbb{R}}^3)$ (see [43, Theorem 1.38, p.29] for example) and the fact that $\frac{3-2\beta}{6}\times 3\geq\frac{2}{3}$ when $0 < \beta\leq \frac{5}{6}$, the additional assumption (1.8) $($and also (1.7)$)$ on $b$ can be omitted if $\frac{1}{2}\leq\beta\leq \frac{5}{6}$. Here $\beta\geq \frac{1}{2}$ is needed to ensure that $\frac{6}{3-2\alpha}\geq 3$. To emphasize this observation, we state the following Corollary:

Corollary3. Let $0 < \alpha, \beta < 1$, $(\rho, u, B)\in L^\infty({\mathbb{R}})\times\dot{H}^\alpha({\mathbb{R}})\times \dot{H}^\beta({\mathbb{R}})$ be a smooth solution to (1.1); then $u = b = 0$ provided that one of the following conditions is fulfilled:

(a) $\frac{1}{2}\leq \alpha < 1$, $\beta > \frac{5}{6}$ or $0 < \beta < \frac{1}{2}$, $u_i\in L^{\vec{p_i}}({\mathbb{R}}^3)$, $b_i\in L^{\vec{\xi_i}}({\mathbb{R}}^3)$ with

for $i, j = 1, 2, 3$; or

(b) $\frac{1}{2}\leq \alpha < 1$, $\frac{1}{2}\leq\beta\leq \frac{5}{6}$, $u_i\in L^{\vec{p_i}}({\mathbb{R}}^3)$ with

for $i, j = 1, 2, 3$; or

(c) $0 < \alpha < \frac{1}{2}$, $\beta > \frac{5}{6}$ or $0 < \beta < \frac{1}{2}$, $u_i\in L^{\vec{p_i}}({\mathbb{R}}^3)\cap L^{\vec{q_i}}({\mathbb{R}}^3)$, $b_i\in L^{\vec{\xi_i}}({\mathbb{R}}^3)$ with

for $i, j = 1, 2, 3$; or

(d) $0 < \alpha < \frac{1}{2}$, $\frac{1}{2}\leq\beta\leq\frac{5}{6}$, $u_i\in L^{\vec{p_i}}({\mathbb{R}}^3)\cap L^{\vec{q_i}}({\mathbb{R}}^3)$ with

for $i, j = 1, 2, 3$.

Remark 4. When $b = 0$, Theorem 1 improves the result of Wang and Xiao ^[20] for $d = 3$, since $u\in L^{\frac{3}{2}}({\mathbb{R}}^3)$ and $u\in L^{\frac{9}{2}}({\mathbb{R}}^3)$ satisfy $\frac{2}{3}\times 3 = 2$ and $\frac{2}{9}\times 3 = \frac{2}{3}$, respectively. Indeed, our result strictly covered the result of ^[20] for $d = 3$, $\alpha < \frac{1}{2}$, since their result requires $u\in L^{\frac{3}{2}}({\mathbb{R}}^3)\cap L^{\frac{9}{2}}({\mathbb{R}}^3)$, but our result (case (d) with $b = 0$ in Corollary 3) shows that $u\in L^{\frac{3}{2}}({\mathbb{R}}^3)\cap L^{3}({\mathbb{R}}^3)$ is sufficient.

2.   Preliminaries

2.1.   Caffarelli–Silvestre extension

We first recall the well-known Caffarelli–Silvestre extension for the fractional Laplacian operator $(-\Delta)^\alpha$ with $\alpha \in(0, 1)$ in ^[44]. Throughout this paper, we use $\bar{\nabla}$ and $\overline{\operatorname{ div }}$ to denote the gradient and divergence operators on $\mathbb{R}_{+}^4$, respectively. We say a distribution $u\in\dot{H}^\alpha({\mathbb{R}}^3)$ if $|\xi|^\alpha\hat{u}(\xi)\in L^2({\mathbb{R}}^3)$, where $\hat{u}(\xi)$ denotes the Fourier transform of $u$. Let $u \in \dot{H}^a\left(\mathbb{R}^3\right)$ and set $\lambda = 1-2 \alpha$, according to ^[44], there is an extension in $\mathbb{R}_{+}^4$, denoted by $u^*$ such that

Furthermore, it holds that

and

where $C_\alpha$ is a constant depending only on $\alpha$. This $u^*$ is called the $\alpha$-extension of $u$. The following $L^p$ integrability of such $u^*$ plays a crucial role in our proof.

Lemma 5. (Lemma 2.2 in ^[20]). Let $\alpha \in(0, 1)$ and $u^*$ be the $\alpha$-extension of $u \in L^p\left(\mathbb{R}^3\right)$ given by (2.1); it holds that

By the embedding theorem $\dot{H}^\alpha\left(\mathbb{R}^3\right) \hookrightarrow L^{\frac{6}{3-2 \alpha}}\left(\mathbb{R}^3\right)$, if we choose $p = \frac{6}{3-2 \alpha}$ in Lemma 2.1, it holds that

2.2.   Hölder's inequality and interpolation inequality in anisotropic Lebesgue spaces.

The following Hölder's inequality in anisotropic Lebesgue space (see ^[45] for example) are frequently referred to in the sequel.

Lemma 6. For $\vec{p} = \left(p_1, p_2, p_3\right), \vec{q} = \left(q_1, q_2, q_3\right)$ and $\vec{r} = \left(r_1, r_2, r_3\right)$ with

and $f \in L^{\vec{p}}\left(\mathbb{R}^3\right), g \in L^{\vec{q}}\left(\mathbb{R}^3\right)$, it holds that

We can also prove the following interpolation inequality in anisotropic Lebesgue space.

Lemma 7. For $\vec{p} = \left(p_1, p_2, p_3\right), \vec{q} = \left(q_1, q_2, q_3\right)$, $\vec{r} = \left(r_1, r_2, r_3\right)$ and $\theta\in [0, 1]$ with

and $f \in L^{\vec{p}}\left(\mathbb{R}^3\right)\cap L^{\vec{q}}\left(\mathbb{R}^3\right)$, it holds that

Proof. By successively using the classical interpolation inequality and Hölder's inequality, we have

□

Though the above inequalities are stated for $\mathbb{R}^3$, they hold for any domain $\Omega \subset \mathbb{R}^3$ by a simple zero extension argument.

3.   Proof of Theorem 1

This section is devoted to proving Theorem 1.

For each $R > 0$, we denote the cube in $\mathbb{R}^3$ centered at the origin with radius $R$ by $Q_R = [-R, R]^3$. Let $\psi \in C_0^{\infty}(\mathbb{R})$ be a standard one-dimensional cut-off function such that

For any $R > 0$, we define

Then we have

We also denote $\chi_R(y)$ by a real nonincreasing smooth function in $\mathbb{R}$ such that

and $\left|\chi_R^{\prime}(y)\right| \leq \frac{C}{R}$ for some constant $C$ independent of $y \in \mathbb{R}$ and $R$.

Multiplying (1.1)$_2$ by $\phi_R u$, integrating by parts, and using the divergence-free property of $u$, we have

Similarly, by testing (1.1)$_3$ with $\psi_R b$, we have

On the other hand, by (2.1), we have

Since $\psi_{R}(x)$ is supported in $Q_{2R}$ and $\chi_R(y) = 1$ in $[0, R]$, the divergence theorem gives

Combining (3.3), (3.4) and (2.2), we obtain

Similarly, we have

where $\mu = 1-2 \beta$. Combining (3.1), (3.2), (3.5) and (3.6), we obtain that

Now we estimate $I_1$. Applying Young's inequality, we have

The estimate of $I_{11}$ is divided into the following three cases:

Case 1: $\frac{5}{6} \leq \alpha < 1$.

Since $\frac{5}{6} \leq \alpha < 1$, we have $\frac{3-2\alpha}{6}\leq \frac{2}{9}$. On the other hand, for $p_{i, j}\in [1, \frac{3}{2}]$, we have $\frac{1}{p_{i, j}}\geq\frac{2}{3}$ and $\frac{1}{3p_{i, j}}\geq \frac{2}{9}$. Hence,

It is easily checked that $f_1(x) = \frac{\frac{1}{3x}-\frac{3-2\alpha}{6}}{\frac{1}{x}-\frac{3-2\alpha}{6}}$ is decreasing in $[1, \frac{3}{2}]$ and $f_2(x) = \frac{\frac{1}{3}-\frac{3-2\alpha}{6}}{\frac{1}{x}-\frac{3-2\alpha}{6}}$ is increasing in $[1, \frac{3}{2}]$.

Therefore, for $p_{i, j}\in [1, \frac{3}{2}]$, we have

which is exactly

Therefore, by choosing $\theta = \frac{2\alpha-1}{3+2\alpha}\in (0, 1)$ and defining $r_{i, j}$ such that

we have

by observing (3.9). Therefore,

Moreover, by using Lemma 7, we have

Thus, by letting $s_{i, j}$ be such chat

and

and using Lemma 6, we have

Here we used the fact that

Hence, by (1.6) and (3.10), we have

Case 2: $\frac{1}{2} \leq \alpha < \frac{5}{6}$. By using Lemma 6 and the fractional Sobolev inequality, we have

Case 3: $\alpha < \frac{1}{2}$. From Lemma 6, it follows that

where

Thus, by (1.6) we have

This completes the estimate of $I_{11}$. Similarly, we have $I_{12}\to 0$ as $R\to\infty$. Hence, $I_1\to 0$ as $R\to\infty$.

The estimate of $I_2$ follows from the estimates of $I_{11}$, $I_{12}$, and the use of Young's inequality,

We remark here that we can also get the estimate of $I_{12}$ and then $I_2$ under assumption (1.8) instead of (1.7). Indeed,

where

Now we estimate $I_3$. By the definition of $\psi_R$ and $\chi_R$, we have

It follows by using Hölder's inequality and (2.5) that

Recall the fact that

we immediately get that $I_3 \rightarrow 0$ as $R \rightarrow \infty$. Similarly, $I_4 \rightarrow 0$ as $R \rightarrow \infty$.

It remains to estimate $I_5$. We need a separate treatment for $\gamma > 1$ and $\gamma = 1$.

Case a: $\gamma\in (1, \infty)$. Rewrite

This, along with $\operatorname{div}(\rho u) = 0$, derives

Then it follows from $u_i \in L^{\vec{p}_i}({\mathbb{R}}^3)$ and $\|\rho\|_{L^{\infty}({\mathbb{R}}^3)} < \infty$ that

where

Hence, by (1.6), we have $I_{5} \rightarrow 0$ as $R \rightarrow \infty$.

Case b: $\gamma = 1$. Under this circumstance we have

By using $\operatorname{div}(\rho u) = 0$ again, we obtain

Note that

So

Accordingly, $u_i \in L^{\vec{p}_i}({\mathbb{R}}^3)$ is used to deduce that

where $\vec{t_i}$ is determined by (3.13). Hence, by (1.6), we have $I_{5} \rightarrow 0$ as $R \rightarrow \infty$. Concluding the above two cases, we obtain

Concluding the above estimates for $I_1, I_2, I_3, I_4$, and $I_5$ and letting $R \rightarrow \infty$ in (3.7), we obtain

which implies that $u^* = b^* =$ constant. Hence, $u = u^*(x, 0)$ and $b = b^*(x, 0)$ are both constant vector fields. Since $\left(u_j, b_j\right) \in L^{\vec{p}_j}\left(\mathbb{R}^3\right) \times L^{\vec{q}_j}\left(\mathbb{R}^3\right)$, we conclude that $u = b = 0$. This completes the proof of Theorem 1.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgements

This work was partially supported by the National Natural Science Foundation of China Grant no. 12001069 and the Team Building Project for Graduate Tutors in Chongqing (yds223010).

Conflict of interest

The authors declare there are no conflicts of interest.