The Cauchy problem for hyperbolic conservation laws with three equations (Q1922991)

The authors consider the Cauchy problem for the nonlinear system \[ v_t-u_x=0,\quad u_t-\sigma(v,s)_x+\alpha u=0,\quad s_\sigma+ {\beta\cdot\{s-f(v)\}\over\tau}=0 \] with bounded \(L^2\) measurable initial data \((v,u,s)|_{t=0}= (v_0(x),u_0(x),s_0(x))\). They prove existence of the global generalized solution for the case \(\beta=0\). Moreover, they show that the solution of the equilibrium system \[ v_t-u_x=\sigma,\quad u_t-\sigma(v,f(v))_x+\alpha u=0 \] is given by the limit of the solutions of viscous approximations as the dissipation and the reaction time \(\tau\) go to zero in the case \(\beta>0\).