Local smooth solutions of the relativistic Euler equation
type: Item
instance of: scholarly article
title: Local smooth solutions of the relativistic Euler equation
zbMATH Open document ID: 0833.35086
author: Tetu Makino
author: Seiji Ukai
published in: Journal of Mathematics of Kyoto University
publication date: 1995-08-13T00:00:00Z
review text: The authors consider the relativistic Euler equation which governs the motion of a relativistic perfect fluid \[ {\partial\over \partial t} \Biggl({\rho c^2+ p\over c^2- v^2}- {p\over c^2}\Biggr)+ \sum^3_{k= 1} {\partial\over \partial x_k} \Biggl({\rho c^2+ p\over c^2- v^2} v_k\Biggr)= 0, \] \[ {\partial\over \partial t} \Biggl({\rho c^2+ p\over c^2- v^2} v_i\Biggr)+ \sum^3_{k= 1} {\partial\over \partial x_k} \Biggl({\rho c^2+ p\over c^2- v^2} v_i v_k+ p\delta_{ik}\Biggr)= 0,\;i= 1,2,3, \] in flat Minkowski space. Here \(c\) denotes the speed of light, \(p\) the pressure, \(\rho\) the mass density, and \(v\) the velocity field of the fluid. Assuming the equation of state \(p= a^2\rho\) with \(0< a< c\) the authors prove local existence and uniqueness of smooth solutions for initial data in certain uniformly local Sobolev spaces by transforming the system to a symmetric hyperbolic one. The authors also prove that the solutions of the relativistic system converge to the corresponding solutions of the nonrelativistic system as \(c\to \infty\).
reviewed by: Gerhard Rein
Mathematics Subject Classification ID: 35L45
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76Y05
zbMATH DE Number: 784738
zbMATH Keywords: local smooth solutions
zbMATH Keywords: relativistic Euler equation
zbMATH Keywords: relativistic perfect fluid
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250518844
OpenAlex ID: W1555907131
DOI: 10.1215/KJM/1250518844
description: scientific article; zbMATH DE number 784738