$L^{1}$ convergences and convergence rates of approximate solutions for compressible Euler equations near vacuum

Abstract

In this paper, we study the rarefaction wave case of the regularized Riemann problem proposed by Chu, Hong and Lee in SIMA MMS, 2020, for compressible Euler equations with a small parameter $\nu$. The solutions $\rho _\nu$ and $v_\nu$ of such problems stand for the density and velocity of gas flow near vacuum, respectively. We show that as $\nu$ approaches 0, the solutions $\rho _\nu$ and $v_\nu$ converge to the solutions $\rho$ and v, respectively, of pressureless compressible Euler equations in $L^1$ sense. In addition, the $L^1$ convergence rates of these physical quantities in terms of $\nu$ are also investigated. The $L^1$ convergences and convergence rates are proved by two facts. One is to invent an a priori estimate coupled with the iteration method to the high-order derivatives of Riemann invariants so that we obtain the uniform boundedness of $\partial _{x}^{i} \rho _{\nu }$ ($i=0,1,2$) and $\partial _{x}^{j} v_{\nu }$ ($j=0,1,2,3$) on the requisite regions. The other is about convexity of characteristic curves, which is used to estimate the distances among characteristic curves in terms of $\nu$. These theoretic results are also supported by numerical simulations.