Viscosity and relaxation approximations for a hyperbolic-elliptic mixed type system

Abstract:

To a given system of conservation laws \begin{equation*}\left \{ \begin {array}{l} u_t + f(u,v,h(u,v))_x =0 v_t + g(u,v,h(u,v))_x =0 \end{array}\right . \end{equation*} we associate the system \begin{equation*}\left \{ \begin {array}{l} u_t + f(u,v,s)_x = \epsilon u_{xx} v_t + g(u,v,s)_x = \epsilon v_{xx} s_t + {s - h(u,v) \over \tau } = \epsilon s_{xx}, \end{array}\right . \end{equation*} which is of mixed type. Under certain conditions, convergence of this latter system for $\epsilon \rightarrow 0$ with $\tau = o(\epsilon )$ is established without the need of stability criteria or hyperbolicity of the left-hand sides of the equations.