A Note on the Liouville Type Theorem for the Steady-state Magnetohydrodynamic Equations

Abstract

In this paper, we are concentrated on demonstrating the Liouville type theorem for the stationary incompressible Magnetohydrodynamic equations in the whole space, the half-space, or a periodic slab, and presenting the solution must vanish under the condition that for some 0 ≤ δ ≤ 1 < L and \(q = {{6({3 - \delta})} \over {6 - \delta}},\mathop {\lim \,\inf}\limits_{R \to \infty} {1 \over R}\Vert{({{\bf{u}},{\bf{h}}})}\Vert_{R < \vert x \vert < LR}^{3 - \delta} = 0\). We also deduce sufficient conditions by allowing shrinking ratio L = 1 + R^−α. When in slab with zero boundary condition, stronger decay rate is needed. We do not assume the global bound of the velocity field u and the magnetic field h and investigate the Liouville type theorems by the conditions lim inf rather than lim.