Hypersonic Limit for C¹ Solution of One Dimensional Isentropic Euler Equations

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 γ→1. First we study and clarify the mechanism of singularity formation for two case: Only derivatives blow up when γ>1, the derivatives blow up with mass concentrates when γ=1. Then we showed as γ→1, the classic solutions of the isentropic Euler equations converge to the solutions of the pressureless Euler equations. We proved that u converges in C¹ and $ρ$ converges in C⁰. By a level set argument, the convergence rate is proved to be √γ-1 on any fixed level set. Furthermore, we show that the time that singularity forms for γ>1 converges to the time of singularity forms for γ=1.