Local existence and non-relativistic limits of shock solutions to a multidimensional piston problem for the relativistic Euler equations
type: Item
instance of: scholarly article
title: Local existence and non-relativistic limits of shock solutions to a multidimensional piston problem for the relativistic Euler equations
zbMATH Open document ID: 1263.35162
author: Min Ding
author: Yachun Li
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 2013-03-04T00:00:00Z
review text: The authors of this interesting paper consider a multidimensional piston problem with initial-boundary value data. The main goal is the local existence of shock front solutions to the spherically symmetric piston problem for relativistic Euler equations. The basic equations are \[ \partial_t\biggl(\frac{(\rho +p\epsilon )u}{1-u^2\epsilon }\biggr)+\partial_r\biggl(\frac{(\rho +p\epsilon )u^2}{1-u^2\epsilon }+p\biggr)+\frac{(d-1)(\rho +p\epsilon )u^2}{(1-u^2\epsilon )r}=0, \] \[ \partial_t\biggl(\frac{(\rho +p\epsilon )u^2\epsilon }{1-u^2\epsilon }+\rho \biggr)+\partial_r\biggl(\frac{(\rho +p\epsilon )u}{1-u^2\epsilon }\biggr)+\frac{(d-1)(\rho +p\epsilon )u}{(1-u^2\epsilon )r}=0, \] where \(r=|\vec{x}|\), \(u=|\vec{u}|\), \(\epsilon = 1/c^2\). Obviously this system tends to classical non-relativistic case when accept \(c\to\infty \) (\(\epsilon\to 0\)). The path of the piston is described as \(r=b(t)\), and the boundary condition is \(u=b'(t)\), \(r=b(t)\). The state of the gas of the expanding body at the initial time is \(U_0=(\rho_a,0)\). The main result is that for \(b\in C^{\infty }\), \(b(0)=0\), \(b'(0)=b_0>0\), \(b^{(k)}(0)=0\) (\(2\leq k\leq K\)), where \(K\) is a large integer, there is a constant \(T_0>0\), such that there exists a shock front solution \((\rho^{c},u^{c})\) to the problem stated above for \(t\leq T_0\), \(b(t)<r<s(t)\). Moreover, there exists a subsequence \(c_n\to \infty \), as \(n\to\infty \), such that \((\rho^{c_n},u^{c_n})\to (\rho ,u)\) a.e., and the limit \((\rho ,u)\) is the corresponding solution of the classical non-relativistic system; this solution is piecewise smooth.
Mathematics Subject Classification ID: 35L67
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35D30
Mathematics Subject Classification ID: 76Y05
Mathematics Subject Classification ID: 35L04
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 83A05
Mathematics Subject Classification ID: 35Q31
zbMATH DE Number: 6140988
zbMATH Keywords: piston problem
zbMATH Keywords: linearization
zbMATH Keywords: Newton iteration
zbMATH Keywords: energy estimates
reviewed by: Dimitar A. Kolev
MaRDI profile type: Publication
full work available at URL: s00033-012-0227-7
OpenAlex ID: W2019114009
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DOI: 10.1007/S00033-012-0227-7
description: scientific article; zbMATH DE number 6140988