On the Decay and Stability of Global Solutions to the 3D Inhomogeneous MHD system

Abstract

In this paper, we investigative the large time decay and stability to any given global smooth solutions of the $3$D incompressible inhomogeneous MHD systems. We proved that given a solution $(a,u,B)$ of (\ref{mhd_a}), the velocity field and magnetic field decay to $0$ with an explicit rate, for $u$ which coincide with incompressible inhomogeneous Navier-Stokes equations \cite{zhangping}. In particular, we give the decay rate of higher order derivatives of $u$ and $B$ which is useful to prove our main stability result. For a large solutions of (\ref{mhd_a}) denoted by $(a,u,B)$, we proved that a small perturbation to the initial data still generates a unique global smooth solution and the smooth solution keeps close to the reference solution $(a,u,B)$. Due to the coupling between $u$ and $B$, we used elliptic estimates to get $\Vert (u,B){\Vert }_{L^1({\mathbb{R}}^{+};{\dot{B}}_{2,1}^{5/2})}<C$, which is different to Navier-Stokes equations.

arXiv admin note: text overlap with arXiv:1206.6144 by other authors

References in corpus (2)

• Global Strong solution with vacuum to the 2D nonhomogeneous incompressible MHD system
• Global existence for the MHD system in critical spaces