Wave interactions and stability of the Riemann solution for a strictly hyperbolic system of conservation laws

Abstract

In this article, we study the interaction of delta shock waves for the one-dimensional strictly hyperbolic system of conservation laws with split delta function. We prove that Riemann solutions are stable under local small perturbations of the Riemann initial data. The global structure and large time asymptotic behaviour of the perturbed Riemann solutions are constructed and analyzed case by case.

Interaction of delta shock waves for the Chaplygin Euler equations of compressible fluid flow with split delta functions

“…For example, Shen and Sun 32 discussed the interaction of delta shock waves and vacuum states for the transport equations, which corresponds to the pressureless case of (1) when p = 0. Guo et al 33 studied the interactions of delta shock waves for the Chaplygin gas equations for isentropic fluids (see also other works [34][35][36][37][38] ).…”

Delta shock wave and wave interactions in a thin film of a perfectly soluble anti-surfactant solution

“…The Riemann problem of (1) cannot be solved for all possible Riemann initial data in terms of classical elementary waves. There are some nonclassical situations [13,23,22] when for few cases of initial data the Riemann problem fails to contain a weak L ∞solution. For that when we solve the Cauchy problem in this nonclassical situation, we have to introduce delta shock type singularities, which are solutions of the system of conservation laws.…”

Structural stability of the Riemann solution for a strictly hyperbolic system of conservation laws with flux approximation

“…Tan et al [15] considered the Riemann problem of (1) and they proved the form of the standard Dirac delta function supported on a shock wave that was used as a part in their Riemann solution for certain initial data. Wave interactions and stability of the Riemann solution for the system (1) have been discussed by Anupam et al [11]. A delta shock wave is a generalization of an ordinary shock wave; speaking informally, it consists of a discontinuity line x = x(t) plus a distributed Dirac delta function with the discontinuity line as its support.…”