On a Nonsymmetric Keyfitz-Kranzer System of Conservation Laws with Generalized and Modified Chaplygin Gas Pressure Law

This paper is devoted to the study of a nonsymmetric Keyfitz-Kranzer system of conservation laws with the generalized and modified Chaplygin gas pressure law, which may admit delta shock waves, a topic of interest. Firstly, we solve the Riemann problems with piecewise constant data having a single discontinuity. For the generalized Chaplygin gas pressure law, the solution consists of three different structures: , , and . Existence and uniqueness of delta shock solution are established under the generalized Rankine-Hugoniot relation and entropy condition. For the modified Chaplygin gas pressure law, the structures of solution are and . Secondly, we discuss the limits of Riemann solutions for the modified Chaplygin gas pressure law as the pressure law tends to the generalized Chaplygin gas one. In particular, for some cases, the solution tends to a delta shock wave, and it is different from the delta shock wave for the generalized Chaplygin gas pressure law with the same initial data. Thirdly, we simulate the Riemann solutions and examine the formation process of delta shock wave by employing the Nessyahu-Tadmor scheme. The numerical results are coincident with the theoretical analysis. 1. Introduction Nonlinear hyperbolic conservation laws are a fundamental principle in building mathematical models for many natural processes. For them, there exists an important kind of nonclassical solution, that is, delta shock wave. It is a generalization of an ordinary shock. From the mathematical point of view, it is a kind of discontinuity, on which at least one of the state variables contains Dirac delta function with a shock as its support. From the physical point of view, it represents the process of concentration of the mass. The theory of nonlinear hyperbolic conservation laws admitting delta shock waves is interesting and has been extensively developed in the last several years; see [1–13] and the references cited therein. Consider the hyperbolic system of conservation laws: where and . It belongs to the nonsymmetric Keyfitz-Kranzer system (see [14, 15]): which is of interest because it arises in such areas as elasticity theory, magnetohydrodynamics, and enhanced oil recovery. For delta shock waves, the nonsymmetric form is more convenient than the symmetric form (see [14]). Model (1) is also a transformation of the traffic flow model introduced by Aw and Rascle [16], where and are the density and velocity of cars on the roadway and is the velocity offset. Let us recall the linear degeneracy and genuine nonlinearity of characteristic fields for quasilinear