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.