Large Time, Weak Solutions to Reacting Euler Equations

Abstract

We prove the large time existence of weak solutions to the Euler equations for one dimensional flow of an ideal gas, which undergoes a simple, one-step exothermic chemical reaction under Arrhenius type kinetics. For such kinetics, the Arrhenius function does not vanish except at absolute sero temperature. We assume that the initial temperature is bounded away from sero and that the total variation of the initial data is sufficiently small. As a consequence, we obtain uniform decay of the reactant to zero as t approaches infinity. This allows us to estimate the increase in total variation that results from the chemical reaction. Numerical calculations show that this increase can be very significant; there remains an interesting challenge to obtain large time existence where uniform decay of the reactant does not occur.