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

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.