Global existence of shock front solutions in 1-dimensional piston problem in the relativistic Euler equations

2007 ◽  
Vol 59 (2) ◽  
pp. 244-263 ◽  
Author(s):  
Yulan Xu ◽  
Yanping Dou
Keyword(s):  
Global Existence ◽  
Shock Front ◽  
Euler Equations ◽  
Piston Problem ◽  
Relativistic Euler Equations ◽  
Front Solutions

GLOBAL EXISTENCE OF SHOCK FRONT SOLUTIONS TO THE AXIALLY SYMMETRIC PISTON PROBLEM FOR COMPRESSIBLE FLUIDS

2004 ◽  
Vol 01 (01) ◽  
pp. 51-84 ◽  
Author(s):  
SHUXING CHEN ◽  
ZEJUN WANG ◽  
YONGQIAN ZHANG
Keyword(s):  
Global Existence ◽  
Shock Front ◽  
Compressible Fluids ◽  
Axially Symmetric ◽  
Piston Problem ◽  
Front Solution ◽  
Glimm Scheme ◽  
Front Solutions ◽  
Shock Front Solution

In this paper, we study the axially symmetric piston problem for compressible fluids when the velocity of the piston is a perturbation of a constant. Under the assumptions that both the velocity of the piston and the density of the gas outside the piston are small, we prove the global existence of a shock front solution by using a modified Glimm scheme.

Global solution to a three-dimensional spherical piston problem for the relativistic Euler equations

2020 ◽  
pp. 1-26
Author(s):  
LAI GENG
Keyword(s):  
Euler Equations ◽  
Wave Motion ◽  
Surrounding Medium ◽  
Three Dimensional ◽  
Spherically Symmetric ◽  
Piston Problem ◽  
Relativistic Euler Equations ◽  
Symmetric Motion ◽  
Characteristics Method ◽  
Spherically Symmetric Motion

The study of spherically symmetric motion is important for the theory of explosion waves. In this paper, we consider a ‘spherical piston’ problem for the relativistic Euler equations, which describes the wave motion produced by a sphere expanding into an infinite surrounding medium. We use the reflected characteristics method to construct a global piecewise smooth solution with a single shock of this spherical piston problem, provided that the speed of the sphere is a small perturbation of a constant speed.

Classical solutions to the relativistic Euler equations for a linearly degenerate equation of state

2017 ◽  
Vol 14 (03) ◽  
pp. 535-563 ◽  
Author(s):  
Changhua Wei
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Classical Solution ◽  
Perfect Fluid ◽  
Euler Equations ◽  
Chaplygin Gas ◽  
Classical Solutions ◽  
Relativistic Euler Equations ◽  
Stiff Matter ◽  
Linearly Degenerate

We are concerned with the global existence and blowup of the classical solutions to the Cauchy problem of one-dimensional isentropic relativistic Euler equations (Chaplygin gas, pressureless perfect fluid and stiff matter) with linearly degenerate characteristics. We at first derive the exact representation formula for all the fluids by the property of linearly degenerate. Then for the Chaplygin gas and the pressureless perfect fluid, we give a classification of the initial data that leads to the global existence and the blowup of the classical solution, respectively. We construct, especially, a class of initial data that contributes to the formation of “cusp-type” singularity and study the evolution of the solution after blowup by introducing a weak solution called delta shock wave. At last, for the stiff matter, we show that this system is indeed a linear system and prove the global existence of the classical solution to this fluid.

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