The regularized Chapman-Enskog expansion for scalar conservation laws

Rosenau [R] has recently proposed a regularized version of the Chapman-Enskog expansion of hydrodynamics. This regularized expansion resembles the usual Navier-Stokes viscosity terms at low wave numbers, but unlike the latter, it has the advantage of being a bounded macroscopic approximation to the linearized collision operator. In this paper we study the behavior of the Rosenau regularization of the Chapman-Enskog expansion (R-C-E) in the context of scalar conservation laws. We show that this R-C-E model retains the essential properties of the usual viscosity approximation, e.g., existence of travelling waves, monotonicity, upper-Lipschitz continuity, etc., and at the same time, it sharpens the standard viscous shock layers. We prove that the regularized R-C-E approximation converges to the underlying inviscid entropy solution as its mean-free-path e{open} ↓ 0, and we estimate the convergence rate.