On the decay and stability of global solutions to the 3D inhomogeneous MHD system

Junxiong Jia; Jigen Peng; Kexue Li

doi:10.3934/cpaa.2017036

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2017, Volume 16, Issue 3: 745-780. Doi: 10.3934/cpaa.2017036

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On the decay and stability of global solutions to the 3D inhomogeneous MHD system

1.
School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China
2.
Beijing Center for Mathematics and Information Interdisciplinary Sciences, China

Received: March 2016

Revised: January 2017

Published: January 2018

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Abstract

In this paper, we investigative the large time decay and stability to any given global smooth solutions of the 3D incompressible inhomogeneous MHD systems. We prove that given a solution $(a, u, B)$ of (2), the velocity field and the magnetic field decay to zero with an explicit rate, for $u$ which coincide with incompressible inhomogeneous Navier-Stokes equations ^[1]. In particular, we give the decay rate of higher order derivatives of $u$ and $B$ which are useful to prove our main stability result. For a large solution of (2) denoted by $(a, u, B)$, we show that a small perturbation of the initial data still generates a unique global smooth solution and the smooth solution keeps close to the reference solution $(a, u, B)$. At last, we should mention that the main results in this paper are concerned with large solutions.

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Mathematics Subject Classification: Primary: 35Q35; Secondary: 76D03, 76W05.

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