A class of initial value problems for 2×2 hyperbolic systems with relaxation

The global existence and uniform BV estimates of weak solutions to a class of initial value problems of general 2×2 genuinely nonlinear strictly hyperbolic conservation laws with relaxation are proved, which also yield the convergence to weak solutions of the equilibrium equation as the relaxation parameter tends to zero. The results obtained in this paper extend the results in Luo and Yang (2004) [17], for a flood wave equation to the general 2×2 system. Also, the constraint as required in Luo and Yang (2004) [17] on the end states at x=±∞, which says that the end states at x=±∞ are on the equilibrium, is removed.