Nonlinear diffusive-dispersive limits for multidimensional conservation laws
This provides a rigorous justification for the vanishing viscosity-dispersion limit in multidimensional conservation laws, which is a fundamental problem in PDE theory.
The authors prove that solutions of multidimensional conservation laws with vanishing nonlinear diffusion and dispersion converge to the entropy solution of the hyperbolic conservation law, under a condition on the relative size of coefficients. The result extends previous one-dimensional results to multiple dimensions and arbitrary Lp spaces.
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.