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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:259

Article

Briant, Marc¹ 

[1] Univ Cambridge, Ctr Math Sci, CCA, Cambridge CB3 0WA, England

关键词: Boltzmann equation on the torus;    Explicit trend to equilibrium;    Incompressible Navier-Stokes hydrodynamical limit;    Knudsen number;    Hypocoercivity;    Kinetic models;   

DOI  :  10.1016/j.jde.2015.07.022

来源: Elsevier

【 摘 要 】

We investigate the Boltzmann equation, depending on the Knudsen number, in the Navier-Stokes perturbative setting on the torus. Using hypocoercivity, 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. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free