Singular solutions to the Riemann problem for a macroscopic production model

Abstract

Key words Delta standing wave, composite wave, vacuum state, wave interaction, Riemann problem, hyperbolic conservation law. MSC (2000) 35L65, 35L67, 35B30, 76N10This paper is devoted to the Riemann problem for the data-fitted second-order macroscopic production model when one of the Riemann initial data is at a vacuum state or at rest. The Riemann solutions are constructed completely by using the continuous method with respect to the initial data, which include some non-classical waves such as the delta standing wave and the superposition of a shock wave and a contact discontinuity etc. In addition, an example is given to explain the wave interaction problem when the vacuum state is involved.

The singular limits of Riemann solutions to a chemotaxis model with flux perturbation

“…Remark 2.4. On the other hand, if − = 0, then we have 1 = − and = ( − − ) for > 0 from (33) and (34). That is to say, the 1-shock wave curve is a parabola which intersects the -axis at the origin (0,0) in the phase plane.…”

“…(4) If − < 0 < + , then consists of a jump function plus a weighted Dirac delta function supported on the line = 0, which is called as the delta standing wave and denoted by DSW here (see Figure 4(e)). In order to define a delta standing wave, we adopt the similar definition of weighted Dirac delta function in the framework, [6,[32][33][34] also see [15,18,23,26] for the related concepts of measure-valued solutions.…”

The Convergence of Riemann Solutions to the Modified Chaplygin Gas Equations with a Coulomb-Like Friction Term as the Pressure Vanishes

“…About the definition of delta shock wave, please refer to [31,34,39,40], for example. The delta shock wave satisfies the generalized Rankine-Hugoniot conditions:…”

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

“…Furthermore, we can see that the solutions of the Riemann problem (4) and (6) are stable under the particular small perturbation (5) of Riemann initial data (6) when the limit 0 → 0 is taken in the solutions. In addition, it should be pointed out that the wave interaction problem for the Temple class has been widely investigated recently, such as for the Aw-Rascle model [18,19], the Chaplygin gas model [20][21][22][23], and the chromatography model [24][25][26][27][28].…”