A system of conservation laws with a relaxation term

The Cauchy problem for the following system of conservation laws with relaxation time $delta$ is discussed: $(ast)$ $(u+v)_t+f(u)_x=0$, $delta v_t=A(u)-v$. A theorem on the well-posedness of the problem is given in the class of functions with bounded total variation. Then the behaviour of solutions to $(ast)$ as $delta o 0$ is treated and convergence of a certain finite-difference scheme to the solution of an equilibrium model $(astast)$ $(w+A(w))_t+f(w)_x=0$ is proved. It is shown that the $L_1$-difference between an equilibrium solution and a nonequilibrium one is bounded by $O(delta^{1/3})$. Detailed proofs are given in related papers by the authors.