On the energy decay of a weak solution of the MHD. equations in a three- dimensional exterior domain
type: Item
instance of: scholarly article
title: On the energy decay of a weak solution of the MHD. equations in a three- dimensional exterior domain
zbMATH Open document ID: 0635.76111
author: Hideo Kozono
published in: Hokkaido Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: Let O be a bounded domain in \(R^ 3\) with smooth boundary \(\partial O\). Set \(\Omega =R^ 3-O\), and assume that \(\Omega\) is simply connected. In \(Q:=\Omega \times (0,\infty)\) consider the following magnetohydrodynamic problem: \[ \partial_ tu-\Delta u+(u,\nabla)u+B\times rot B+\nabla \pi =f, \] \[ \partial_ tB-\Delta B+(u,\nabla)B-(B,\nabla)u=0,\quad div u=0,\quad div B=0, \] in Q, with \(u=B\cdot v=rot B\times v=0\) on \(\partial \Omega \times (0,\infty)\) and \(u|_{t=0}=u_ 0\), \(B|_{t=0}=B_ 0\). The author constructs a weak solution on (0,\(\infty)\) of this exterior problem with the property that \[ E(t):=\int_{\Omega}(| u(x,t)|^ 2+| B(x,t)|^ 2)dt \] tends to zero as \(t\to \infty\).
Mathematics Subject Classification ID: 76W05
Mathematics Subject Classification ID: 35Q99
zbMATH DE Number: 4035482
zbMATH Keywords: energy decay
zbMATH Keywords: bounded domain
zbMATH Keywords: smooth boundary
zbMATH Keywords: magnetohydrodynamic problem
zbMATH Keywords: weak solution
zbMATH Keywords: exterior problem
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1381517923
OpenAlex ID: W2094494172
DOI: 10.14492/HOKMJ/1381517923
Recommended article: Q3032017
Recommended article: On the energy equality for very weak solutions to 3D MHD equations
Recommended article: Asymptotic decay of solutions to 3D MHD equations
Recommended article: Q4690356
Recommended article: Energy equality for weak solutions to the 3D magnetohydrodynamic equations in a bounded domain
Recommended article: On the decay and stability of global solutions to the 3D inhomogeneous MHD system
Recommended article: Q5429770
Recommended article: On the regularity criterion of weak solutions for the 3D MHD equations
Recommended article: Q3014893
Recommended article: Decay of solutions to a new Hall-MHD system in \(\mathbb{R}^3\)
description: scientific article; zbMATH DE number 4035482