Remarks on the regularity criteria for generalized MHD equations
type: Item
instance of: scholarly article
title: Remarks on the regularity criteria for generalized MHD equations
zbMATH Open document ID: 1211.35230
author: Zujin Zhang
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-01-07T00:00:00Z
review text: The Cauchy problem to the 3D generalized MHD equations is studied in \(\{x\in{\mathbb R}^3,t>0\}\). \[ \begin{aligned} &\frac{\partial u}{\partial t}+(-\Delta)^\alpha u+u\cdot\nabla u -b\cdot\nabla b+\nabla p=0,\quad \text{div}\,u=0,\\ & \frac{\partial b}{\partial t}+(-\Delta)^\beta b +u\cdot\nabla b -b\cdot\nabla u=0,\quad \text{div}\,b=0,\\ &u(x,0)=u_0(x),\quad b(x,0)=b_0(x). \end{aligned} \] Here \(u\) is the velocity field, \(b\) is the magnetic field, \(p\) is the pressure, \(\alpha,\beta>0\), \((-\Delta)^\alpha\) is the fractional power of Laplacian. It is proved that if \(u_0,b_0\in H^1({\mathbb R}^3)\) and \((u,b)\) is a weak solution with \(\nabla u\) integrable in a certain Morrey space then \((u,b)\) is smooth on \((0,T)\).
reviewed by: Il'ya Sh. Mogilevskii
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76W05
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 5833515
zbMATH Keywords: generalized magneto-hydrodynamics equations
zbMATH Keywords: regularity criteria
zbMATH Keywords: Morrey spaces
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.10.017
OpenAlex ID: W2081387829
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DOI: 10.1016/J.JMAA.2010.10.017
description: scientific article; zbMATH DE number 5833515