# Convergence of a vector BGK approximation for the incompressible Navier-Stokes equations

## TL;DR

This paper proves that a vector BGK approximation model converges to the incompressible Navier-Stokes equations, providing a rigorous mathematical foundation for this approximation in Sobolev spaces.

## Contribution

It introduces a new convergence proof for a vector BGK model approximating the incompressible Navier-Stokes equations using a symmetrizer-based energy estimate approach.

## Key findings

- Convergence of the vector BGK model to Navier-Stokes solutions in Sobolev spaces.
- Use of a constant right symmetrizer for uniform energy estimates.
- Establishment of a conservative-dissipative form enabling compactness arguments.

## Abstract

We present a rigorous convergence result for the smooth solutions to a singular semilinear hyperbolic approximation, a vector BGK model, to the solutions to the incompressible Navier-Stokes equations in Sobolev spaces. Our proof is based on the use of a constant right symmetrizer, weighted with respect to the parameter of the singular pertubation system. This symmetrizer provides a conservative-dissipative form for the system and this allow us to perform uniform energy estimates and to get the convergence by compactness.