Weakly nonlinear limit for the relaxation approximation of conservation laws in several space dimensions
The author considers the two-dimensional scalar conservation law \[ u_t+f(u)_x+g(u)_y=0, \quad u\in \mathbb{R} \] and the corresponding relaxation system \[ u_t+v_x+w_y=0, \quad v_t+au_x+\frac{v-f(u)}{\varepsilon}=0,\quad w_t+bu_y+\frac{w-g(u)}{\varepsilon}=0. \] The weakly nonlinear limit for the relaxation approximation of the conservation law is derived through asymptotic equations and justified by employing energy estimates. Although two-dimensional results are present, and the main results hold for arbitrary space dimensions. Compared with the former works, the main difficulty lies in the problem of the existence of an (convex) entropy. To overcome this difficulty, the author obtains an estimate that is quite different from the previous estimates.
scientific article; zbMATH DE number 1271368