Nonlinear diffusive-dispersive limits for multidimensional conservation laws

TL;DR

This paper investigates the behavior of solutions to multidimensional conservation laws with small diffusion and dispersion, proving convergence to entropy solutions under certain conditions, extending previous one-dimensional results.

Contribution

It extends the analysis of diffusive-dispersive limits from one-dimensional to multidimensional conservation laws, establishing uniform bounds and convergence results.

Findings

Solutions are uniformly bounded in Lp for large p.

Solutions converge to entropy solutions as diffusion and dispersion vanish.

The approach uses DiPerna's uniqueness theorem for measure-valued solutions.

Abstract

We consider a class of multidimensional conservation laws with vanishing nonlinear diffusion and dispersion terms. Under a condition on the relative size of the diffusion and dispersion coefficients, we establish that the diffusive-dispersive solutions are uniformly bounded in a space Lp ($p$ arbitrary large, depending on the nonlinearity of the diffusion) and converge to the classical, entropy solution of the corresponding multidimensional, hyperbolic conservation law. Previous results were restricted to one-dimensional equations and specific spaces Lp. Our proof is based on DiPerna's uniqueness theorem in the class of entropy measure-valued solutions.