1207.4379 (Marc Briant)

From Boltzmann to Navier-Stokes equations on the torus    [PDF]

Marc Briant

In this article we study the Boltzmann Equation, depending on the Knudsen number, in the Navier-Stokes perturbative setting on the torus in di- mension N. Using the tools of the hypocoercivity theory, we prove existence and exponential decay results for the solution of this linearized equation, with explicit regularity bounds and rates. These results are uniform in the Knudsen number and then allow us to obtain a strong derivation of the incompressible Navier-Stokes equations as the Knudsen number tends to 0, in the case of Maxwellian molecules. Moreover, our method shows that the smaller the Knudsen number, the less con- trol on the v-derivatives on the initial perturbation is needed to have existence and decay and also to deal with other kinetic models. Finally, we show that the study of this hydrodynamical limit is rather different on the torus than the already proven convergences in the whole space as it requires averaging in time, unless the initial layer conditions are satisfied.

View original: http://arxiv.org/abs/1207.4379