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Liouville Type Theorem for the Fractional MHD and Hall-MHD equations in $\mathbb{R}^{3}

Weihua Wang, Zhenyuan Liu

Abstract

In this paper, we are mainly concerned with the Liouville type problem for the stationary fractional magnetohydrodynamics(MHD) and stationary fractional Hall-MHD equations. In addition, we present the results of the Navier-Stokes equation as a byproduct. The key point is to use the Caffarelli-Sivestre extension to overcome the difficulty caused by the non-local operator $(-\triangle)^{s}$ and combined with Yuan and Xiao's method (J. Math. Anal. Appl. 491 (2020) 124343).

Liouville Type Theorem for the Fractional MHD and Hall-MHD equations in $\mathbb{R}^{3}

Abstract

In this paper, we are mainly concerned with the Liouville type problem for the stationary fractional magnetohydrodynamics(MHD) and stationary fractional Hall-MHD equations. In addition, we present the results of the Navier-Stokes equation as a byproduct. The key point is to use the Caffarelli-Sivestre extension to overcome the difficulty caused by the non-local operator and combined with Yuan and Xiao's method (J. Math. Anal. Appl. 491 (2020) 124343).
Paper Structure (9 sections, 7 theorems, 63 equations)

This paper contains 9 sections, 7 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The proof of Theorem \ref{['th1.1']}
  4. $\frac{3}{4}\leq\alpha, ~\beta<1$
  5. Case $3\leq p \leq \frac{9}{2}$
  6. Case $2\leq p \leq \min\{\frac{9}{2}, \frac{18(2\alpha-1)}{14\alpha-9}, \frac{18(2\beta-1)}{14\beta-9}\}$
  7. $\frac{1}{2}\leq\alpha< \frac{3}{4}, ~\frac{1}{2}\leq\beta<1$
  8. The proof of Theorem \ref{['th1.5']}
  9. $\frac{5}{6}\leq\beta<1$

Key Result

Theorem 1.1

Let $\frac{1}{2}\leq\alpha,~\beta<1$ and $\Lambda^\alpha u, \Lambda^\beta b$ in $L^2(\mathbb{R}^3)$. Assume that $(u,b)$ is a smooth solution to the system eq1.1, then $u=b=0$ provided $u, ~b \in L^p(\mathbb{R}^3)$ with $2\leq p\leq\frac{9}{2}$.

Theorems & Definitions (11)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Corollary 1.4
  • Remark 1.5
  • Theorem 1.6
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3: Lemma 2.2 in WX2018
  • Lemma 2.4: Gagliardo-Nirenberg inequality, Lemma 2.2, Wu24 or Nirenberg11
  • ...and 1 more