Global solutions to the Maxwell-Navier-Stokes system in a bounded domain in 2D
type: Item
instance of: scholarly article
title: Global solutions to the Maxwell-Navier-Stokes system in a bounded domain in 2D
zbMATH Open document ID: 1466.35283
author: Jishan Fan
author: Tohru Ozawa
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 2021-05-03T00:00:00Z
review text: Let \(\Omega \subset \mathbb{R}^2\) be a bounded and simply connected domain with smooth boundary \(\partial \Omega\). Consider the system \begin{align*} &\partial_tu+u\cdot\nabla u+ \nabla \pi -\Delta u=(E+u\times b)\times b, \\ &\partial_tE-\operatorname{rot} \ b + E+u\times b=0, \\ &\partial_tb+ \operatorname{rot} \ E=0, \\ &\operatorname{div} \, u=0, \\ &\operatorname{div} \, b=0, \ \ \mbox{in } \Omega \times (0,\infty), \\ &u\cdot n=0, \ \operatorname{rot} \ u=0, \ b\cdot n =0, \ E=0 \ \ \mbox{on } \partial \Omega \times (0, \infty), \\ &(u,E,b)(\cdot , 0)=(u_0, E_0, b_0)(\cdot ) \ \ \mbox{in } \Omega. \end{align*} Here, \(u, \pi, E\) and \(b\) denote the velocity, pressure, electric field and magnetic field, respectively. The authors state the following result: If \(u_0, E_0, b_0\in H^2(\Omega)\) satisfy \(\operatorname{div}\, u_0=\operatorname{div} \, b_0=0\) in \(\Omega\) and \(u_0\cdot n=0\), \(\operatorname{rot}\, u_0=0\), \(b_0\cdot n=0\), and \(E_0=0\) on \(\partial \Omega\), then, for any \(T>0\), there exists a unique solution of the above problem such that \(u,E,b\in L^{\infty}(0,T;H^2(\Omega))\).
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 35Q61
Mathematics Subject Classification ID: 76D03
Mathematics Subject Classification ID: 76W05
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35A02
Mathematics Subject Classification ID: 35D35
zbMATH DE Number: 7342394
zbMATH Keywords: Maxwell-Navier-Stokes system
zbMATH Keywords: two dimensional bounded domain
zbMATH Keywords: Navier boundary condition
zbMATH Keywords: global well-posedness
reviewed by: Gheorghe Moroşanu
MaRDI profile type: Publication
full work available at URL: s00033-020-01364-y
OpenAlex ID: W3045480657
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DOI: 10.1007/S00033-020-01364-Y
description: scientific article; zbMATH DE number 7342394