Non-relativistic global limits of entropy solutions to the extremely relativistic Euler equations

Abstract

We are concerned in this paper with the non-relativistic global limits of the entropy solutions to the Cauchy problem of 3 × 3 system of relativistic Euler equations modeling the conservation of baryon numbers, momentum, and energy respectively. Based on the detailed geometric properties of nonlinear wave curves in the phase space and the Glimm's method, we obtain, for the isothermal flow, the convergence of the entropy solutions to the solutions of the corresponding classical non-relativistic Euler equations as the speed of light c → +∞. Mathematics Subject Classification (2000). Primary: 35B40, 35A05, 76Y05; Secondary: 35B35, 35L65, 85A05.

Smooth solutions for one‐dimensional relativistic radiation hydrodynamic equations

“…ÂdS D de C pd  1 n à with special entropy S, the state equations for polytropic gas can be given as (cf. [11]) We consider the Cauchy problem for Equations 1.1 and 1.10 with the following data:…”

“…n, , p are the baryon number, the mass-energy density and the pressure, respectively, and they satisfy the following relation: 11) equation in Equation (1.10). This has the effect of deleting the particle rest energy contribution from energy equation and makes the passage to the non-relativistic limit more straightforward.…”

Special relativistic effects revealed in the Riemann problem for three-dimensional relativistic Euler equations

“…We know that (5.1) is the formal limit of (1.1) as c → +∞, which can be derived similarly as in [11]. We need to take a look at the elementary waves to the Riemann problem of (5.1) and (1.8).…”

“…T dS = de + p d 1 n with specific entropy S, the equations of state for polytropic gas can be given as [11] p = (γ − 1)c 2 (ρ − n), (1.4) and p = n γ exp (γ − 1)(S − S 0 ) R . (1.5) For simplicity, hereinafter we will take the speed of light c = 1.…”

“…Much progress has been made to the mathematical theory of relativistic fluid dynamics, yet mainly for one-dimensional system or spherically symmetric three-dimensional system (see [2][3][4][5][6][7]9,11,[15][16][17][18][19][20]22,27,28,34,37,44,45] and references cited therein). For multi-dimensional cases, local existence results of smooth solutions were obtained in [24,25], the singularity formation of smooth solutions was studied in [12,26,30].…”