A mixed type system of three equations modelling reacting flows

Abstract:

In this paper we contrast two approaches for proving the validity of relaxation limits $\alpha \rightarrow \infty$ of systems of balance laws \begin{equation*} u_t +{f(u)}_x = \alpha g(u) \quad . \end{equation*} In one approach this is proven under some suitable stability condition; in the other approach, one adds artificial viscosity to the system \begin{equation*} u_t +{f(u)}_x = \alpha g(u) + \epsilon u_{xx} \end{equation*} and lets $\alpha \rightarrow \infty$ and $\epsilon \rightarrow 0$ together with $M \alpha \leq \epsilon$ for a suitable large constant $M$. We illustrate the usefulness of this latter approach by proving the convergence of a relaxation limit for a system of mixed type, where a subcharacteristic condition is not available.