Global entropy solutions to the relativistic Euler equations for a class of large initial data

2005 ◽  
Vol 56 (2) ◽  
pp. 239-253 ◽  
Author(s):  
Yachun Li ◽  
Dongmei Feng ◽  
Zejun Wang
Keyword(s):  
Initial Data ◽  
Euler Equations ◽  
Entropy Solutions ◽  
Large Initial Data ◽  
Relativistic Euler Equations

scholarly journals A new variation for the relativistic Euler equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mahmoud A. E. Abdelrahman ◽  
Hanan A. Alkhidhr
Keyword(s):  
Euler Equations ◽  
Nonlinear Waves ◽  
Approximation Scheme ◽  
Strength Function ◽  
Convergent Sequence ◽  
Approximate Solutions ◽  
Large Initial Data ◽  
Initial Value ◽  
Relativistic Euler Equations ◽  
Glimm Scheme

Abstract The Glimm scheme is one of the so famous techniques for getting solutions of the general initial value problem by building a convergent sequence of approximate solutions. The approximation scheme is based on the solution of the Riemann problem. In this paper, we use a new strength function in order to present a new kind of total variation of a solution. Based on this new variation, we use the Glimm scheme to prove the global existence of weak solutions for the nonlinear ultra-relativistic Euler equations for a class of large initial data that involve the interaction of nonlinear waves.

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.

scholarly journals Riemann problem with delta initial data for the two-dimensional steady pressureless isentropic relativistic Euler equations

2021 ◽  
Author(s):  
Yu Zhang ◽  
Yanyan Zhang
Keyword(s):  
Initial Data ◽  
Euler Equations ◽  
Riemann Problem ◽  
Contact Discontinuity ◽  
Global Solutions ◽  
Two Dimensional ◽  
Relativistic Euler Equations ◽  
Riemann Solutions ◽  
Isentropic Relativistic Euler Equations ◽  
The Stability

The Riemann problem for the two-dimensional steady pressureless isentropic relativistic Euler equations with delta initial data is studied. First, the perturbed Riemann problem with three pieces constant initial data is solved. Then, via discussing the limits of solutions to the perturbed Riemann problem, the global solutions of Riemann problem with delta initial data are completely constructed under the stability theory of weak solutions. Interestingly, the delta contact discontinuity is found in the Riemann solutions of the two-dimensional steady pressureless isentropic relativistic Euler equations with delta initial data.

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