L1 convergences and convergence rates of approximate solutions for compressible Euler equations near vacuum

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. approaches 0, the solutions rho(nu) and v(nu) converge to the solutions rho and nu 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 L1 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 ix (i = 0, 1, 2) and (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.