Delta Shock Waves for a Linearly Degenerate Hyperbolic System of Conservation Laws of Keyfitz-Kranzer Type

Abstract

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.

Delta shock waves for a system of Keyfitz‐Kranzer type

“…In this case, P satisfies the condition and it is not possible to solve the Riemann problem to using only classical waves. In this paper, we generalize the result of for an arbitrary strictly increasing function . More precisely, we solve the Riemann problem for the following system, where , with for all , and .…”

“…In , he consider the case when is not a convex function (in this case, the system describes how the addition of a polymer affects the flow of water and oil in a reservoir) and prove the existence of a global weak solution to the Cauchy problem. Lu showed the existence of a global weak solution of the Cauchy problem for when f is a non negative convex function and P satisfies the following condition: Using delta‐shock waves, H. Cheng solved the Riemann problem to the non symmetric system of Keyfitz‐Kranzer type when is the function , , and is the indentity function, i.e. for all .…”

“…Using delta-shock waves, H. Cheng [3] solved the Riemann problem to the non symmetric system of Keyfitz-Kranzer type (KK) when ( ) is the function ( ) = − 1 , > 0, and ( ) is the indentity function, i.e. ( ) = for all ∈ ℝ.…”

Global weak solutions for a 2 × 2 balance non-symmetric system of Keyfitz-Kranzer type / Soluciones débiles globales para un sistema 2x2 balanceado no simétrico de tipo Keyfitz-Kranzer

“…Another system of the type (1) was considered in [3] as a generalization to the scalar Buckley-Leverett equations describing two phase flow in porous media. The system (1), recently, has been object of constant studies, in [4] the author considered the particular case in which Φ(w) = w, P(ρ) = 1 ρ , in this case the two characteristics of the system (1) are linear degenerate, solving the Riemann problem the existence and uniqueness of delta shock solution were established. In this line in [5] the authors considered the case Φ(w) = w and P(ρ) = B ρ α with α ∈ (0, 1), the existence and uniqueness of solutions to the the Riemann problem was got by solving the Generalize Rankine-Hugoniot condition.…”

The Perturbed Riemann Problem for Special Keyfitz-Kranzer System with Three Piecewise Constant States

“…has also been proposed and widely investigated in [6][7][8]; also see [9,10] about the Riemann problem for some special forms of system (2). Recently, it has been assumed in [11], where is a function of = + V, that system (1) can be simplified into the form + ( ( )) = 0, + ( ( )) = 0.…”