Riemann problem with delta initial data for the two-dimensional steady pressureless isentropic relativistic Euler equations

Yu Zhang,Yanyan Zhang

doi:10.1051/mmnp/2021011

Abstract

The Riemann problem for the two-dimensional steady pressureless isentropic relativistic Euler equations with delta initial data is studied. First, the perturbed Riemann problem with three pieces constant initial data is solved. Then, via discussing the limits of solutions to the perturbed Riemann problem, the global solutions of Riemann problem with delta initial data are completely constructed under the stability theory of weak solutions. Interestingly, the delta contact discontinuity is found in the Riemann solutions of the two-dimensional steady pressureless isentropic relativistic Euler equations with delta initial data.

Highlights

IntroductionKeywords and phrases: Relativistic Euler equations, steady flow, Riemann problem, delta shock wave, delta contact discontinuity
Definition 2.1. ([24]) A triple distribution (n, u, v) consists of a solution of (1.3) in the sense of measures if it satisfies (a) n ∈ L∞ [0, ∞), BM (R1) ∩ C [0, ∞), H−s(R1), (b) u ∈ L∞ [0, ∞), L∞(R1) ∩ C [0, ∞), H−s(R1), (c) v ∈ L∞ [0, ∞), L∞(R1) ∩ C [0, ∞), H−s(R1), (d) u and v are measurable with respect to n at almost for all x ≥ 0, RIEMANN PROBLEM WITH DELTA INITIAL DATA
A two-dimensional weighted delta function w(s)δS supported on a smooth curve L parameterized as x = x(s), y = y(s) (a ≤ s ≤ b) is defined by b w(s)δL, φ(x, y) = w(s)φ(x(s), y(s))ds a for all φ ∈ C0∞(R2)

Summary

Introduction

Keywords and phrases: Relativistic Euler equations, steady flow, Riemann problem, delta shock wave, delta contact discontinuity. The Riemann problem for the 3 × 3 full relativistic Euler system with generalized Chaplygin proper energy density-pressure relation has been studied by Shao [15], and the delta shock waves on which both two state variables simultaneously contain the Dirac delta function are obtained. By analyzing the limit behavior of solutions to the perturbed Riemann problem (1.3) and (1.5) as → 0+, the delta contact discontinuity, on which the state variable n contains the Dirac delta function, is obtained in the solutions of the steady pressureless isentropic relativistic Euler equations (1.3) with delta initial data (1.4).

The Riemann problem with constant initial data

Riemann problem with delta initial data