Concentration and Cavitation in the Vanishing Pressure Limit of Solutions to a Simplified Isentropic Relativistic Euler
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Journal of Mathematical Fluid Mechanics
Concentration and Cavitation in the Vanishing Pressure Limit of Solutions to a Simplified Isentropic Relativistic Euler Equations Yu Zhang and Yicheng Pang Communicated by G.-Q. G. Chen
Abstract. We identify and analyze the phenomena of concentration and cavitation by studying the vanishing pressure limit of solutions to a simplified isentropic relativistic Euler equations. Firstly, both the explicit expressions and geometric properties of the rarefaction wave curve and shock wave curve based on any left state are given with the help of Lorentz invariance, and the Riemann problem for this system is considered. Then, we rigorously prove that, as pressure vanishes, any two-shock Riemann solution tends to a delta-shock solution of the pressureless relativistic Euler equations, and the intermediate density between the two shocks tends to a weighted δ-measure, which forms the delta shock wave. This describes the phenomenon of mass concentration. On the other hand, any two-rarefaction Riemann solution tends to a two-contact-discontinuity solution of the pressureless relativistic Euler equations and the nonvacuum intermediate state in between tends to a vacuum state, which reveals the phenomenon of cavitation. Both concentration and cavitation are fundamental and physical in fluid dynamics. Mathematics Subject Classification. 35L65, 35L67, 76N10, 76N15. Keywords. Relativistic Euler equations, Riemann problem, Vanishing pressure limit, Delta shock wave, Vacuum, Concentration, Cavitation.
1. Introduction In the one dimensional relativistic fluid dynamics, the Euler system modeling the conservation of baryon number and momentum reads [8,9] ⎧ nv ⎪ ⎪ √ n + √ = 0, ⎨ 2 2 c2 −v 2 x c −v t (1.1) v v2 ⎪ 2 ⎪ + (ρc + ρE + p) + p = 0, ⎩ (ρc2 + ρE + p) 2 c − v2 t c2 − v 2 x where n is the proper number density of baryons, ρ the proper rest mass density, E the internal energy per unit mass, v the classical coordinate velocity, c the speed of light, and p = p(ρ) the pressure. Noticing that, in (1.1), if ni denotes the number density and μi the rest mass of each baryonic species, then n = Σni and ρ = Σni μi (with antiparticles being assigned negative number density). Therefore, (1.1) can be simplified by assuming that neither particles nor rest mass are created or destroyed. That is, in this case we have ρ = n¯ μ with μ ¯ a constant and the system (1.1) simplifies into [1,9,14,15] ⎧ ρv ⎪ ⎪ √ ρ + √ = 0, ⎨ 2 2 c2 −v 2 x c −v t (1.2) v v2 ⎪ 2 ⎪ + (ρc + ρE + p) + p = 0. ⎩ (ρc2 + ρE + p) 2 c − v2 t c2 − v 2 x 0123456789().: V,-vol
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Y. Zhang, Y. Pang
JMFM
Since the solution of conservation laws (1.2) depends strongly on the state equation p = p(ρ), so just for computational convenience, we consider the polytropic gas of the form 5 ργ , 1 0 is a small perturbed parameter. For the more general equation of state p(ρ) = Kργ , the results are valid and the proofs are identical. This equation of state models a class of fluids which are of astrophysical interest, ple
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