Strong convergence of a vector-BGK model to the incompressible Navier-Stokes equations via the relative entropy method

TL;DR

This paper proves the strong convergence of a vector-BGK model to the incompressible Navier-Stokes equations on a 2D torus, using relative entropy methods and Sobolev space estimates, valid for any finite time interval.

Contribution

It establishes the strong convergence of the vector-BGK model to the Navier-Stokes equations with global uniform bounds, extending previous local results.

Findings

Strong convergence of the model to Navier-Stokes equations

Global in time uniform boundedness of solutions

Use of relative entropy and Sobolev estimates

Abstract

The aim of this paper is to prove the strong convergence of the solutions to a vector-BGK model under the diffusive scaling to the incompressible Navier-Stokes equations on the two-dimensional torus. This result holds in any interval of time $[0, T]$, with $T>0$. We also provide the global in time uniform boundedness of the solutions to the approximating system. Our argument is based on the use of local in time $H^s$-estimates for the model, established in a previous work, combined with the $L^2$-relative entropy estimate and the interpolation properties of the Sobolev spaces.