Well-posedness and large time behavior of solutions for the electron inertial Hall-MHD system

Abstract

In this paper, the properties of weak and strong solutions for the Hall-magnetohydrodynamic system augmented by the effect of electron inertia are studied. First, we establish the existence and uniqueness of local-in-time strong solutions; Then, we prove the existence of global strong solutions under the condition that $\|u_0\|_{\dot{H}^{\frac12}}+\|B_0\|_{\dot{H}^{\frac12}} +\|\nabla B_0\|_{\dot{H}^{\frac12}}$ is sufficiently small. Moreover, by applying a cut-off function and generalized energy inequality, we show that the weak solution of electron inertia Hall-MHD system approaches zero as the time $t\rightarrow\infty$. Finally, the algebraic decay rate of the weak solution of electron inertia Hall-MHD system is established by using Fourier splitting method and the properties of decay character.