Singular Limit of Stiff Relaxation and Dominant Diffusion for the Cauchy Problem Associated With the System of Quadratic Flux

We study the singular limit of stiff relaxation and dominant diffusion for the Cauchy problem associated with the diffusion system of quadratic flux with relaxation vt+uvx=ϵvxxut+1/23u2+v2x+1/ταv,uu−hv=ϵuxx and bounded measurable initial data. This problem was previously studied for the case where αv,u=1. In that context, the authors noted that the same conclusions could be obtained when the relaxation term is given by 1/ταv,uu−hv, provided that αv,u>0 is Lipschitz continuous. In this paper, we show that this condition on αv,u can be weakened as follows. Assuming only the continuity of the function αv,u>0, we prove the existence of a subsequence of solutions to the viscous equations that converges to the equilibrium solution.