From the Boltzmann equation to the incompressible Navier–Stokes equations on the torus: A quantitative error estimate

Marc Briant (1)

1 Cambridge Center for Analysis

We investigate the Boltzmann equation, depending on the Knudsen number, in the Navier-Stokes perturbative setting on the torus. Using hypocoer-civity, we derive a new proof of existence and exponential decay for solutions close to a global equilibrium, with explicit regularity bounds and rates of convergence. These results are uniform in the Knudsen number and thus allow us to obtain a strong derivation of the incompressible Navier-Stokes equations as the Knudsen number tends to 0. Moreover, our method is also used to deal with other kinetic models. Finally, we show that the study of the hydrodynamical limit is rather different on the torus than the one already proved in the whole space as it requires averaging in time, unless the initial layer conditions are satisfied.

Mots clés: Boltzmann equation on the Torus; Explicit trend to equilibrium; Incompressible Navier-Stokes hydrodynamical limit; Knudsen number; Hypocoer- civity; Kinetic Models

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Marc Briant. From the Boltzmann equation to the incompressible Navier–Stokes equations on the torus: A quantitative error estimate. Journal of Differential Equations, 2015, 259 (11), pp.6072-6141. ⟨10.1016/j.jde.2015.07.022⟩. ⟨hal-01491834⟩