TY - JOUR T1 - Hypersonic Limit for $C^1$ Solution of One Dimensional Isentropic Euler Equations AU - Peng , Wenjian AU - Wang , Tian-Yi AU - Xiang , Wei JO - Communications in Mathematical Analysis and Applications VL - 4 SP - 558 EP - 581 PY - 2024 DA - 2024/12 SN - 3 DO - http://doi.org/10.4208/cmaa.2024-0024 UR - https://global-sci.org/intro/article_detail/cmaa/23619.html KW - Compressible Euler equations, hypersonic limit, mass concentration, asymptotic behavior, convergence rate. AB -

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 $\gamma=1.$