Diffusive Limit of the Boltzmann Equation with Fluid Initial Layer in the Periodic Domain

Abstract

We justify the global-in-time diffusive limit of the Boltzmann equation inside a periodic domain $\mathbb{T}^3$. We only assume that in the initial expansion $F^{\epsilon}=\mu+\sqrt{\mu} f^{\epsilon}$ at $t=0$, the kinetic parts are well-prepared, but the fluid parts could be general, i.e., the fluids parts are not required to satisfy the incompressibility and Boussinesq relations. For this case, the fluid initial layers are created and preserved in the periodic domain. Employing the method of time averaging, we analyze the propagation of the initial layers, and thus extend Guo's results in [Comm. Pure Appl. Math., 59 (2006), pp. 626--687] in which both fluid and kinetic parts are required to be well-prepared.