Relative entropy in diffusive relaxation for a class of discrete velocities BGK models

TL;DR

This paper develops a relative entropy framework for analyzing diffusive relaxation systems with discrete velocities, demonstrating convergence to parabolic limits in smooth regimes and applying it to models related to Navier-Stokes equations.

Contribution

It extends the relative entropy method to discrete velocity relaxation systems, providing a unified approach for convergence analysis in these models.

Findings

Proves convergence of discrete velocity models to parabolic equations in smooth regimes.

Applies the method to 1D Jin-Xin model and 2D vector-BGK model.

Establishes a direct proof of convergence using relative entropy.

Abstract

We provide a framework to extend the relative entropy method to a class of diffusive relaxation systems with discrete velocities. The methodology is detailed in the toy case of the 1D Jin-Xin model under the diffusive scaling, and provides a direct proof of convergence to the limit parabolic equation in any interval of time, in the regime where the solutions are smooth. Recently, the same approach has been successfully used to show the strong convergence of a vector-BGK model to the 2D incompressible Navier-Stokes equations.