On the uniqueness of solution to generalized Chaplygin gas

Marko Nedeljkov; Sanja Ružičić

doi:10.3934/dcds.2017190

Article Contents

2017, Volume 37, Issue 8: 4439-4460. Doi: 10.3934/dcds.2017190

This issue Previous Article Existence of heterodimensional cycles near Shilnikov loops in systems with a $\mathbb{Z}_2$ symmetry Next Article Normalization in Banach scale Lie algebras via mould calculus and applications

On the uniqueness of solution to generalized Chaplygin gas

Marko Nedeljkov^, and
Sanja Ružičić

Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia

^* Corresponding author: Marko Nedeljkov
^* Corresponding author: Marko Nedeljkov

Received: May 31, 2016

Revised: April 30, 2017

Published: May 2017

The first author is partially supported by the projects OI174024 and III44006, Serbian Ministry of Science and by the Project 114-451-2098, APV Secretariat for Science.

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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.

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Mathematics Subject Classification: Primary: 35L65, 35L67; Secondary: 76N10.

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Figure 1. Classical waves

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Figure 2. Overcompressive SDW vs. S1+S2

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Figure 3. Energy entropy condition

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Figure 4. Inequalities (15) and (16)

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Figure 5. Inequalities (15) and (16) are not enough to prove non-positivity of $\hat{E}_{\lambda}^1$

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Figure 6. Entropies at $\Gamma_{ss}$ curve – the first entropy pair

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Figure 7. Entropies at $\Gamma_{ss}$ curve – the second entropy pair

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