The Exact Riemann Solutions to the Generalized Pressureless Euler Equations with Dissipation

The Riemann solutions for the generalized pressureless Euler equations with a dissipation term are constructed explicitly. It is shown that the delta shock wave appears in Riemann solutions in some situations. The generalized Rankine–Hugoniot conditions of the delta shock wave are established, and t...

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Bibliographic Details
Published inBulletin of the Malaysian Mathematical Sciences Society Vol. 43; no. 6; pp. 4361 - 4374
Main Authors Zhang, Qingling, He, Fen
Format Journal Article
LanguageEnglish
Published Singapore Springer Singapore 01.11.2020
Springer Nature B.V
Subjects
Applications of Mathematics
Euler-Lagrange equation
Eulers equations
Mathematical analysis
Mathematics
Mathematics and Statistics
Self-similarity
Shock wave propagation
Shock waves
Delta shock wave
Riemann problem
Generalized pressureless Euler equations
Dissipation
35B30
35L67
35L65
76N10
Vacuum state
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Summary:The Riemann solutions for the generalized pressureless Euler equations with a dissipation term are constructed explicitly. It is shown that the delta shock wave appears in Riemann solutions in some situations. The generalized Rankine–Hugoniot conditions of the delta shock wave are established, and the exact position, propagation speed and strength of the delta shock wave are given explicitly. Unlike the homogeneous case, it is shown that the dissipation term makes contact discontinuities and delta shock waves bend into curves and the Riemann solutions are not self-similar anymore. Moreover, as the dissipation term vanishes, the Riemann solutions converge to the corresponding ones of the generalized pressureless Euler equations. Finally, we give the application of our results on two typical examples.
ISSN:0126-6705
2180-4206
DOI:10.1007/s40840-020-00926-7