Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum
type: Item
instance of: scholarly article
zbMATH Open document ID: 1278.35195
DOI: 10.1137/120893355
published in: SIAM Journal on Mathematical Analysis
publication date: 2013-08-22T00:00:00Z
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 35A08
Mathematics Subject Classification ID: 35B08
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 76N10
Mathematics Subject Classification ID: 76W05
zbMATH DE Number: 6200933
zbMATH Keywords: compressible magnetohydrodynamic fluids
zbMATH Keywords: global classical solution
zbMATH Keywords: large-time behavior
author: Hai-Liang Li
author: Xinying Xu
author: Jianwen Zhang
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1137/120893355
OpenAlex ID: W2031099254
title: Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum
review text: The authors study the mathematical model to the 3D compressible isentropic MHD flows. These flows are governed by the Cauchy problem NEWLINE\[NEWLINE\begin{aligned} & \frac{\partial \rho}{\partial t}+\text{div}\,(\rho u)=0, \\ & \frac{\partial }{\partial t}(\rho u)+\text{div}\,(\rho u\otimes u)+\nabla P(\rho)= (\nabla \times H)\times H+\mu\Delta u+(\mu+\lambda)\nabla\text{div}\,u, \\ & \frac{\partial H}{\partial t}-\nabla\times(u\times H)=-\nabla\times(\nu\nabla\times H),\quad \text{div}\,H=0 \end{aligned}NEWLINE\]NEWLINE for \(t>0\) and \(x\in \mathbb{R}^3\), NEWLINE\[NEWLINE (\rho,u,H)\rightarrow (\bar{\rho},0,0)\quad \text{as}\;|x|\rightarrow \infty,\;t>0, \tag{infc}NEWLINE\]NEWLINE NEWLINE\[NEWLINE (\rho,u,H)(x,0)=(\rho_0,u_0,H_0)(x),\quad x\in \mathbb{R}^3. \tag{ic}NEWLINE\]NEWLINE Here the density \(\rho\), the velocity \(u\), the pressure \(P\) and the magnetic field \(H\) are unknown unctions. The viscosity coefficients \(\lambda,\mu\) and the resistivity coefficient \(\nu\) are given constants. \(\rho_0,u_0,H_0\) are given functions and \(\bar{\rho}\geq 0\) is the given constant. The pressure \(P\) satisfies to the power law NEWLINE\[NEWLINE P(\rho)=A\rho^\gamma \quad \text{with}\;A>0,\;\gamma>1. NEWLINE\]NEWLINENEWLINENEWLINEIt is proved that if the initial data \((\rho_0,u_0,H_0)\) are small in the sense of energy then the Cauchy problem ({mhd1}), ({infc}), ({ic}) has the unique global solution. The large-time behavior of the global solution is investigated in \(L^q\) spaces. The proof is based on a priory estimates as usual.
reviewed by: Il'ya Sh. Mogilevskii
description: scientific article; zbMATH DE number 6200933