Dynamics of Partial Differential Equations
Volume 18 (2021)
The Liouville type theorem for the stationary magnetohydrodynamic equations in weighted mixed-norm Lebesgue spaces
Pages: 327 – 340
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.
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