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

Abstract

We consider the Riemann problem of three-dimensional relativistic Euler equations with two discontinuous initial states separated by a planar hypersurface. Based on the detailed analysis on the Riemann solutions, special relativistic effects are revealed, which are the variations of limiting relative normal velocities and intermediate states and thus the smooth transition of wave patterns when the tangential velocities in the initial states are suitably varied. While in the corresponding non-relativistic fluid, these special relativistic effects will not occur. (2000). Primary 35B40 · 35A05 · 76Y05; Secondary 35B35 · 35L65 · 85A05. Mathematics Subject Classification

On relativistic gasdynamics: invariance under a class of reciprocal-type transformations and integrable Heisenberg spin connections

“…The differential equations descriptive of 3+1-dimensional relativistic gasdynamics may be formulated as a system of conservation laws (e.g. [17]) and reduce to the classical Eulerian gasdynamics system in the limit c → ∞. In §3 of the present work, the two-dimensional, steady reduction of this nonlinear system is shown to be invariant under a novel four-parameter class of reciprocal-type transformations.…”

Non-relativistic limit analysis of the Chandrasekhar-Thorne relativistic Euler equations with physical vacuum

“…Recently, there have been made important progress on the mathematical theory on these topics. For instance, the global existence of Riemann solutions, BV solutions and related nonrelativistic limits have been obtained in [1,4,5,6,20,23,25,29,31,32] respectively for either the relativistic system Eq. (2.42) or the relativistic Euler equations consisting of the momentum equation (2.42) 2 and energy equation (1.8).…”

Global radial solutions to 3D relativistic Euler equations for non-isentropic Chaplygin gases

“…Recently, Ding et al [12] considered the full Euler equations (1.11) in two space dimensions, they proved the global existence of smooth spherically symmetric solutions by applying Alinhac's ghost weight energy and the idea of Godin [14], in which he proved the smooth spherically symmetric solutions for 3D full compressible Euler equations (1.11) of Chaplygin gases. For relativistic Euler equations (1.10) of Chaplygin gases, known results are mainly on the Riemann problems, we refer to [7,8,16]. Due to the complexity of relativistic system (1.8) or (1.10), there are no global existence results for more than one space dimensions.…”