Abstract: The authors investigated the singular limits of stiff relaxation and dominant diffusion for general 2×2nonlinear systems of balance laws, that is, τ=o(ε), ε→ 0, the relaxation time τtends to zero faster than the diffusion parameter ε. If there exists a priori L^∞ bound that is uniformly with respect to ε for the solutionsof a system, then the solution sequence converges to the corresponding equilibrium solution of the system. This framework can be applied to some important nonlinear systems with relaxation terms and inhomogeneous terms, such as the system of quadratic flux, the LeRoux system, the system of elasticity and the extended models of traffic flows.

Key words: hyperbolic balance system, singular limits of stiff relaxation, inhomogeneous terms, weak solution, compensated compactness method