On a limit of perturbed conservation laws with saturating diffusion and non-positive dispersion

Abstract

We consider a conservation law with convex flux, perturbed by a saturating diffusion and non-positive dispersion of the form $u_t + f(u)_x = ε(u_x/\sqrt{1+u_x^2})_x − δ(|u_xx|^n)_x$. We prove the convergence of the solutions $u^{ε,δ}$ to the entropy weak solution of the hyperbolic conservation law, $u_t + f(u)_x = 0$, for all real number $1 ≤ n ≤ 2$ provided $δ = o(ε^{n(n+1)/2};ε^{n+1/n})$.