Local well‐posedness to the 2D Cauchy problem of nonhomogeneous heat‐conducting Navier–Stokes and magnetohydrodynamic equations with vacuum at infinity

Abstract

This paper concerns the Cauchy problem of nonhomogeneous heat‐conducting magnetohydrodynamic (MHD) equations in ℝ2 with vacuum as far‐field density. By spatial weighted energy method, we derive the local existence and uniqueness of strong solutions provided that the initial density and the initial magnetic decay not too slowly at infinity. As a byproduct, we get the local existence of strong solutions to the 2D Cauchy problem for nonhomogeneous heat‐conducting Navier–Stokes equations with vacuum at infinity.