Weakly Nonlinear Limit for the Relaxation Approximation of Conservation Laws in Several Space Dimensions

In this paper, the weakly nonlinear limit for the relaxation approximation of conservation laws in several space dimensions is derived through asymptotic expansions and justified by employing the energy estimates. Compared with the work of G. Q. Chen, C. D. Levermore, and T. P. Liu (1994,Comm. Pure Appl. Math.47, 787–830), the main difficulty we confront in our analyzes lies in the fact that the problem of the existence of an (convex) entropy, which was essential in the proof of their estimate (5.29), for the system under our consideration is still an open problem since their Theorem 3.2 was proved only for the 2×2 case.