Abstract
The main object of the paper is finding a unique solution to Riemann problem for generalized Chaplygin gas model. That is a model of the dark energy in Universe introduced in the last decade. It permits an infinite mass concentration so one has to consider solutions containing the Dirac delta function. Although it was easy to construct solution to any Riemann problem, the usual admissibility conditions, overcompressiveness, do not exclude unwanted delta-type waves when a classical solution exists. We are using Shadow Wave approach in order to solve that uniqueness problem since they are well adopted for using Lax entropy–entropy flux conditions and there is a rich family of convex entropies.
Highlights
A generalized Chaplygin gas appears in a number cosmology theories and it is a model for a compressible fluid with a pressure inversely proportional to a gas energy density, p = −C/ρα, C > 0, 0 < α < 1, see [2] for the first model, and [9] for some more advanced models
Let us note that there is a solution to system (1) with the Riemann data containing a delta shock solution constructed in [21]
Our aim was to prove that a simple shadow wave solution (SDW for short) to our problem is admissible for all convex entropy pairs only at the points that can not be connected by two shock solution
Summary
Let us note that there is a solution to system (1) with the Riemann data containing a delta shock solution constructed in [21] (and the one for the system with friction in [20]) In both cases, the authors just claimed and proved that there is an overcompressive delta shock solution in a region without classical ones. It is interesting that in this case a lot of conditions are good enough to obtain a unique solution: overcompressibility, and entropy condition with only one convex entropy functional (mechanical energy, for example). Our aim was to prove that a simple shadow wave solution (SDW for short) to our problem is admissible for all convex entropy pairs only at the points that can not be connected by two shock solution.
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