Approximation and Existence of Vacuum States in the Multiscale Flows of Compressible Euler Equations

Abstract

In this paper, we study the approximation and existence of vacuum states in the multiscale gas flows governed by the Cauchy problem of compressible Euler equations containing a small parameter $\eta$ in the initial density. The system of compressible Euler equations is reduced to a hyperbolic resonant system at the vacuum so that the weak solution of the Riemann problem is not suitable as the building block of the Glimm (or Godunov) scheme to establish the existence of vacuum states. We construct a new type of approximate solutions, the weak solutions of the regularized Riemann problem for the leading-order system derived from asymptotic expansions around vacuum states. Such an approximate solution obtained by solving the pressureless Euler equations with generalized Riemann data consists of constant states separated by a composite hyperbolic wave. We show the stability of the regularized Riemann solution, together with the numerical simulations, under the small perturbations of initial data. Adopting the approximate solution as the building block of the generalized Glimm scheme, we prove the existence of the vacuum states by showing the stability and consistency of the scheme as $\eta \rightarrow 0$. The numerical simulation indicates that for any small fixed $t>0$, the approximate solutions converge to the exact solutions of the Cauchy problem in $L^1$ as $\eta \rightarrow 0$. The results of this paper can be applied to some hyperbolic resonant systems of balance laws.