The Non-selfsimilar Riemann Problem for 2-D Zero-Pressure Flow in Gas Dynamics*

Abstract

The non-selfsimilar Riemann problem for two-dimensional zero-pressure flow in gas dynamics with two constant states separated by a convex curve is considered. By means of the generalized Rankine-Hugoniot relation and the generalized characteristic analysis method, the global solution involving delta shock wave and vacuum is constructed. The explicit solution for a special case is also given.

The Riemann problem for nonlinear degenerate wave equations

“…Noting (12), (14), and (16), we find that, when γ > 1, the characteristic regions v < 0 and v > 0 are genuinely nonlinear. In the next, we begin to seek the smooth solutions.…”

“…Hence, by introducing the degenerate shock under the generalized shock condition as [3], the global solutions to (1) and (2) are constructively obtained case by case. For the characteristic analysis method, we can refer to [10][11][12][13][14]. The paper is arranged as follows.…”

Non‐selfsimilar global solutions to a two‐dimensional system of conservation laws

“…In the two-dimensional systems of conservation laws, Chen et al [16] investigated the simplified Euler equations, and the triple-shock pattern of the non-selfsimilar solution has been discovered. Sun and Sheng [17] studied the two-dimensional zero-pressure flow in gas dynamics, and the initial data are separated by a closed curve, where all of the non-selfsimilar solutions are showed. However, many problems are still open in this direction [16,18].…”

Construction of the 2d Riemann Solutions for a Nonstrictly Hyperbolic Conservation Law

“…Thus we should deal with wave interactions including the delta shock wave for its resonant wave structure. About the delta shock wave solution in the multi-dimensional hyperbolic conservation laws, we can see [4,7,9,15,18,20,21] and the reference therein.…”