# American Institute of Mathematical Sciences

February  2019, 12(1): 133-158. doi: 10.3934/krm.2019006

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

 1 Ecole Normale Supérieure de Lyon, UMPA, ENS-Lyon, 46, allée d'Italie, 69364-Lyon Cedex 07, France 2 Istituto per le Applicazioni del Calcolo "Mauro Picone", Consiglio Nazionale delle Ricerche, via dei Taurini 19, I-00185 Rome, Italy

* Corresponding author: Roberto Natalini

Received  July 2017 Published  July 2018

Fund Project: The first author was supported by a Ph. D. grant of University of Rome Tor Vergata

We present a rigorous convergence result for smooth solutions to a singular semilinear hyperbolic approximation, called vector-BGK model, to the solutions to the incompressible Navier-Stokes equations in Sobolev spaces. Our proof deeply relies on the dissipative properties of the system and on the use of an energy which is provided by a symmetrizer, whose entries are weighted in a suitable way with respect to the singular perturbation parameter. This strategy allows us to perform uniform energy estimates and to prove the convergence by compactness.

Citation: Roberta Bianchini, Roberto Natalini. Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations. Kinetic & Related Models, 2019, 12 (1) : 133-158. doi: 10.3934/krm.2019006
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