scholarly journals Global existence of the entropy solutions to the isentropic relativistic Euler equations

2005 ◽  
Vol 4 (4) ◽  
pp. 763-778 ◽  
Author(s):  
Yachun Li ◽  
◽  
Qiufang Shi
Keyword(s):  
Global Existence ◽  
Euler Equations ◽  
Entropy Solutions ◽  
Relativistic Euler Equations ◽  
Isentropic Relativistic Euler Equations

Riemann problem and limits of solutions to the isentropic relativistic Euler equations for isothermal gas with flux approximation

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yu Zhang ◽  
Yanyan Zhang
Keyword(s):  
Shock Wave ◽  
Euler Equations ◽  
Riemann Problem ◽  
Delta Shock Wave ◽  
Relativistic Euler Equations ◽  
Delta Shock ◽  
Isentropic Relativistic Euler Equations ◽  
Flux Approximation ◽  
Two Parameter ◽  
Pressureless Relativistic Euler Equations

Abstract We are concerned with the vanishing flux-approximation limits of solutions to the isentropic relativistic Euler equations governing isothermal perfect fluid flows. The Riemann problem with a two-parameter flux approximation including pressure term is first solved. Then, we study the limits of solutions when the pressure and two-parameter flux approximation vanish, respectively. It is shown that, any two-shock-wave Riemann solution converges to a delta-shock solution of the pressureless relativistic Euler equations, and the intermediate density between these two shocks tends to a weighted δ-measure that forms a delta shock wave. By contract, any two-rarefaction-wave solution tends to a two-contact-discontinuity solution of the pressureless relativistic Euler equations, and the intermediate state in between tends to a vacuum state.

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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