%0 Journal Article
%T Delta Shock Waves for a Linearly Degenerate Hyperbolic System of Conservation Laws of Keyfitz-Kranzer Type
%A Hongjun Cheng
%J Advances in Mathematical Physics
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/958120
%X This paper is devoted to the study of delta shock waves for a hyperbolic system of conservation laws of Keyfitz-Kranzer type with two linearly degenerate characteristics. The Riemann problem is solved constructively. The Riemann solutions include exactly two kinds. One consists of two (or just one) contact discontinuities, while the other contains a delta shock wave. Under suitable generalized Rankine-Hugoniot relation and entropy condition, the existence and uniqueness of delta shock solution are established. These analytical results match well the numerical ones. Finally, two kinds of interactions of elementary waves are discussed. 1. Introduction Consider the following hyperbolic system of conservation laws: where with . Model (1) belongs to the nonsymmetric system of Keyfitz-Kranzer type [1, 2] as System (1) was also introduced as a macroscopic model for traffic flow by Aw and Rascle [3], where and are the density and the velocity of cars on the roadway, and the function is smooth and strictly increasing and it satisfies Model (1) was also studied by Lu [2]. By using the compensated compactness method, he established the existence of global bounded weak solutions of the Cauchy problem under the following two assumptions on , respectively: One can find that system (1) has two characteristics; one is always linearly degenerate while the other is linearly degenerate or genuinely nonlinear depending on the behaviors of . In [2, 3], the condition (3) is required, which implies that the second characteristic is genuinely nonlinear. Thus, one interesting topic is the case when both characteristics are linearly degenerate. These motivate us to consider (1) with which is the prototype function satisfying For system (1) with (5) or (6), a distinctive feature is that both eigenvalues are linearly degenerate; that is, this is a fully linearly degenerate system. Thus, all the classical elementary waves only consist of contact discontinuities. Moreover, the overlapping of linearly degenerate characteristics may result in the formation of delta shock wave. A delta shock wave is a generalization of an ordinary shock wave. Speaking informally, it is a kind of discontinuity, on which at least one of the state variables may develop an extreme concentration in the form of a Dirac delta function with the discontinuity as its support. It is more compressive than an ordinary shock wave in the sense that more characteristics enter the discontinuity line. From the physical point of view, a delta shock wave represents the process of concentration of the mass and formation of
%U http://www.hindawi.com/journals/amp/2013/958120/