Dynamics of Partial Differential Equations

Volume 18 (2021)

Number 4

The Liouville type theorem for the stationary magnetohydrodynamic equations in weighted mixed-norm Lebesgue spaces

Pages: 327 – 340

DOI:  https://dx.doi.org/10.4310/DPDE.2021.v18.n4.a4

Authors

Huiying Fan (School of Mathematical Science, Zhejiang University, Hangzhou, China)

Meng Wang (School of Mathematical Science, Zhejiang University, Hangzhou, China)

Abstract

In this paper, we are concentrated on demonstrating the Liouville type theorem for the stationary Magnetohydrodynamic equations in mixednorm Lebesgue spaces and weighted mixed-norm Lebesgue spaces. In particular, we show that, under some sufficient conditions in (weighted) mixed-norm Lebesgue spaces, the solution of stationary MHDs are identically zero. Precisely, we investigate solutions of MHDs that may decay to zero in different rates as $\lvert x \rvert \to \infty$ in different directions. In un-mixed norm case, the result recovers available results. With some additional geometric assumptions on the supports of solutions in weighted mixed-norm Lebesgue spaces, this work also provides several other important Liouville type theorems of solutions in weighted mixed-norm Lebesgue spaces.

Keywords

Liouville type theorem, stationary magnetohydrodynamic equations, weighted mixed-norm Lebesgue spaces

2010 Mathematics Subject Classification

Primary 35Q35, 76W05. Secondary 35B53, 35B65.

The second author was supported by NSFC 11771388.

Received 22 February 2021

Published 2 December 2021