Global existence and decay estimates of solutions to the MHD–Boussinesq system with stratification effects*
Xinliang Li, Zhong Tan, Saiguo Xu
Abstract
Abstract In this paper, we consider the inviscid Boussinesq system under the presence of magnetic field in porous media. We firstly proved the global well-posedness and large time behaviour of solutions in the whole space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> </mml:math> . Precisely, when the H 3 norm of initial data is small, but the higher order derivatives can be arbitrary large, the system is globally well-posed by pure energy method. Moreover, by a set of mature negative Sobolev and Besov space interpolation methods, the L p − L 2 (1 ⩽ p ⩽ 2) type of the optimal time decay rates are obtained without any smallness assumption on the L p norm of the initial data. At last, we derive a global weak solution with large initial data and give an explicit decay rate of the solution in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> . Our results mathematically explain the stable phenomenon of the system with stratification effects.