Global Existence and Optimal Decay Rates of Solutions for Compressible Hall-MHD Equations

Abstract

In this paper, we are concerned with global existence and optimal decay rates of solutions for the three-dimensional compressible Hall-MHD equations. First, we prove the global existence of strong solutions by the standard energy method under the condition that the initial data are close to the constant equilibrium state in $H^2$-framework. Second, optimal decay rates of strong solutions in $L^2$-norm are obtained if the initial data belong to $L^1$ additionally. Finally, we apply Fourier splitting method by Schonbek [Arch.Rational Mech. Anal. 88 (1985)] to establish optimal decay rates for higher order spatial derivatives of classical solutions in $H^3$-framework, which improves the work of Fan et al.[Nonlinear Anal. Real World Appl. 22 (2015)].