Existence of global smooth solution to the relativistic Euler equations
type: Item
instance of: scholarly article
title: Existence of global smooth solution to the relativistic Euler equations
zbMATH Open document ID: 1072.35189
author: Lizhi Ruan
author: Changjiang Zhu
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2005-02-22T00:00:00Z
review text: The authors study the Cauchy problem for the one-dimensional relativistic Euler equations \[ \begin{aligned} &{\partial\over\partial t} \Biggl[{\rho+ Pu^2/c^4\over 1- u^2/c^2}\Biggr]+ {\partial\over\partial x} \Biggl[{(\rho+ P/c^2)u\over 1- u^2/c^2}\Biggr]= 0,\\ & {\partial\over\partial t}\Biggl[{(\rho+ P/c^2)u\over 1- u^2/c^2}\Biggr]+ {\partial\over\partial x}\Biggl[{P+\rho u^2\over 1- u^2/c^2}\Biggr]= 0,\end{aligned} \] with initial conditions \[ \rho(0,x)= \rho_0(x),\quad u(x,0)= u_0(x), \] where \(c\) is the speed of light and \(P\) is a function of \(\rho\) satisfying certain conditions. These equations govern the one-dimensional motion of a perfect gas in Minkowski space-time. The authors prove that, under certain specified conditions, these equations admit a unique global smooth solution on \(t\geq 0\).
reviewed by: Anthony D. Osborne
Mathematics Subject Classification ID: 35Q75
Mathematics Subject Classification ID: 83A05
zbMATH DE Number: 2136982
zbMATH Keywords: maximum principle
zbMATH Keywords: global smooth solution
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: Cauchy problem
zbMATH Keywords: Minkowski space-time
MaRDI profile type: Publication
full work available at URL: j.na.2004.09.019
OpenAlex ID: W2037171229
cites work: Global solutions to the compressible Euler equations with geometrical structure
cites work: Conservation laws for the relativistic P-system
cites work: Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics
cites work: Some existence theorems for hyperbolic systems of partial differential equations in two independent variables
cites work: Solutions in the large for nonlinear hyperbolic systems of equations
cites work: Globally smooth solutions of quasilinear hyperbolic systems in diagonal form
cites work: Global smooth solution of Cauchy problems for a class of quasilinear hyperbolic systems
cites work: On the relativistic Euler equation
cites work: Development of Singularities of Solutions of Nonlinear Hyperbolic Partial Differential Equations
cites work: Development of singularities in the nonlinear waves for quasi-linear hyperbolic partial differential equations
cites work: Global solution for an initial boundary value problem of a quasilinear hyperbolic system
cites work: Global solutions of the relativistic Euler equations
cites work: A functional integral approach to shock wave solutions of Euler equations with spherical symmetry
cites work: Convergence to strong nonlinear diffusion waves for solutions of \(p\)-system with damping
cites work: Global resolvability for a viscoelastic model with relaxation
cites work: Q4387859
DOI: 10.1016/J.NA.2004.09.019
description: scientific article; zbMATH DE number 2136982