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

We provide a general 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.

Keywords

relaxation, diffusive scaling, vector-BGK models, discrete velocities, relative entropy method

2010 Mathematics Subject Classification

35A35, 35L40, 35Q20, 35Q35

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This paper was partially funded by the GNAMPA (INdAM) project Partially dissipative hyperbolic systems with applications to biological models 2019.

This manuscript has been written while the author was a postdoc at LJLL, Sorbonne Université, Paris, funded by the FSMP. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program Grant agreement No 637653, project BLOC “Mathematical Study of Boundary Layers in Oceanic Motion. This work was supported by the SingFlows project, grant ANR-18-CE40-0027 of the French National Research Agency (ANR).

Received 9 January 2020

Accepted 22 July 2020

Published 24 March 2021