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Hypersonic Limit for $C^1$ Solution of One Dimensional Isentropic Euler Equations

Hypersonic Limit for $C^1$ Solution of One Dimensional Isentropic Euler Equations

Year:    2024

Author:    Wenjian Peng, Tian-Yi Wang, Wei Xiang

Communications in Mathematical Analysis and Applications, Vol. 3 (2024), Iss. 4 : pp. 558–581

Abstract

In this article, we study the hypersonic limit problem to the 1-D isentropic Euler equations. For uniformly bounded density and velocity, it can be formulated as the behavior of solution as $\gamma→1.$ First we study and clarify the mechanism of singularity formation for two case: Only derivatives blow up when $\gamma>1,$ the derivatives blow up with mass concentrates when $\gamma=1.$ Then we showed as $\gamma→1,$ the classic solutions of the isentropic Euler equations converge to the solutions of the pressureless Euler equations. We proved that $u$ converges in $C^1$ and $ρ$ converges in $C^0.$ By a level set argument, the convergence rate is proved to be $\sqrt{\gamma-1}$ on any fixed level set. Furthermore, we show that the time that singularity forms for $\gamma>1$ converges to the time of singularity forms for $\gamma=1.$

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cmaa.2024-0024

Communications in Mathematical Analysis and Applications, Vol. 3 (2024), Iss. 4 : pp. 558–581

Published online:    2024-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    24

Keywords:    Compressible Euler equations hypersonic limit mass concentration asymptotic behavior convergence rate.

Author Details

Wenjian Peng

Tian-Yi Wang

Wei Xiang