2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/80506We establish the incompressible Navier--Stokes limit for the discrete velocity model of the Boltzmann equation in any dimension of the physical space, for densities which remain in a suitable small neighborhood of the global Maxwellian. Appropriately scaled families solutions of discrete Boltzmann equation are shown to have fluctuations that locally in time converge strongly to a limit governed by a solution of Incompressible Navier--Stokes provided that the initial fluctuation is smooth, and converges to appropriate initial data. As applications of our results, we study the Carleman model and the one-dimensional Broadwell model.arxiv version is already officialExactly Solvable and Integrable Systemstext