Delta wave formation and vacuum state in vanishing pressure limit for system of conservation laws to relativistic fluid dynamics

Abstract

Key words System of relativistic fluid dynamics, delta shock wave, vacuum, vanishing pressure limit, pressureless relativistic Euler equations.The Riemann solutions to the system of conservation laws to relativistic fluid dynamics with a scaled pressure are shown. Using the method of vanishing pressure limit, it is rigorously proved that the Riemann solution involving two shocks and possibly one contact discontinuity to the system of relativistic fluid dynamics tends to a delta wave solution of pressureless relativistic Euler equations. The intermediate density between the two shocks tends to a weighted δ-measure which forms the delta wave. While the Riemann solution containing two rarefaction waves and perhaps one contact discontinuity tends to a double-contact-discontinuity solution to pressureless relativistic Euler equations. The intermediate state between the two contact discontinuities is a vacuum state.

Concentration and cavitation in the Euler equations for nonisentropic fluids with the flux approximation

“…This point is different from those only in pressure level [7,8,16,19,30,33]. This point is different from those only in pressure level [7,8,16,19,30,33].…”

“…Particularly, Li [16] investigated the zero temperature limit of solutions to the Euler equations for the isothermal case. Then, the results were extended to the relativistic fluid dynamics by Yin and Sheng [33], the perturbed Aw-Rascle model by Shen and Sun [19], the modified Chaplygin gas equations [30], etc. All these works above are only focused on the pressure level.…”

The limits of Riemann solutions to the simplified pressureless Euler system with flux approximation

“…Thus, some effective methods have been developed to deal with this problem, such as the theory of divergence-measure fields in the Radon-measure space established by Chen and Frid 15,16 and the weak asymptotic method introduced by Danilov, Mitrovic, and Shelkovich. 1,9,17 The formation of vacuum state and delta shock wave has been widely considered by using the vanishing pressure limit for the pressureless Euler system [18][19][20][21][22][23][24][25] to explain the physical phenomena of cavitation and concentration, which can be regarded as a particular case of flux function approximation. This method has also been generalized to the Chaplygin gas dynamics system.…”

Vanishing pressure and magnetic field limit of solutions to the nonisentropic magnetogasdynamics

“…Li considered the zero temperature limit for the isothermal case. Then, the method and results were extended to various models, see Shen for the isentropic magnetogasdynamics, Yang and Wang for the modified Chaplygin gas equations, Yin and Sheng for the relativistic fluid dynamics, Yang and Liu for the Euler equations for isentropic fluids and nonisentropic fluids by the flux approximation, etc. One of the main goals of this paper is to analyze rigorously the formation of delta‐shocks and vacuum states in solutions to the nonisentropic magnetogasdynamics as the pressure P and the transverse magnetic field both vanish.…”