On Exact Solutions of Rarefaction-Rarefaction Interactions in Compressible Isentropic Flow

Abstract

Consider the interaction of two centered rarefaction waves in one-dimensional, compressible gas flow with pressure function \(p(\rho )=a^2\rho ^\gamma \) with \(\gamma >1\). The classic hodograph approach of Riemann provides linear 2nd order equations for the time and space variables t, x as functions of the Riemann invariants r, s within the interaction region. It is well known that t(r, s) can be given explicitly in terms of the hypergeometric function. We present a direct calculation (based on works by Darboux and Martin) of this formula, and show how the same approach provides an explicit formula for x(r, s) in terms of Appell functions (two-variable hypergeometric functions). Motivated by the issue of vacuum and total variation estimates for 1-d Euler flows, we then use the explicit t-solution to monitor the density field and its spatial variation in interactions of two centered rarefaction waves. It is found that the variation is always non-monotone, and that there is an overall increase in density variation if and only if \(\gamma >3\). We show that infinite duration of the interaction is characterized by approach toward vacuum in the interaction region, and that this occurs if and only if the Riemann problem defined by the extreme initial states generates a vacuum. Finally, it is verified that the minimal density in such interactions decays at rate O(1)/t.