A lattice gas model for the incompressible Navier-Stokes equation (Q731705)

The departure point for the present analysis is a research by \textit{R. Esposito, R. Marra} and \textit{H. T. Yau} [Rev. Math. Phys. 6, No.~5a, 1233--1267 (1994; Zbl 0841.60082) and Commun. Math. Phys. 182, No.~2, 395--455 (1996; Zbl 0868.60079)] on the hydrodynamical limits for stochastic particle systems on a lattice, as a possible microscopic foundation of the incompressible Navier-Stokes equation. The long-range asymmetric exclusion process on a lattice is combined with a collision process exchanging particle velocities at the same site. Properly tuning the size and rates of the particle jumps, it is possible to prove that in the hydrodynamic limit a small perturbation of constant density and momentum profile evolves in a diffusive time scale as the solution of the incompressible Navier-Stokes equation, in a fixed time interval. The presented method extends as well to lower than 3 dimensions. The technical drawback of the method is elucidated: to set a bound on the spectral gap of the full dynamics, no general results are obtained. Only under specific conditions upon the collision process the bound is established. The proof relies on the relative entropy method introduced by \textit{H. T. Yau} [Lett. Math. Phys. 22, No.~1, 63--80 (1991; Zbl 0725.60120)]. Spectral gap estimates are established by means of the Gronwall inequality for the relative entropy. No non-gradient methods or the multiscale analysis are involved.