The 3D Kinetic Couette Flow Via The Boltzmann Equation In The Diffusive Limit

This talk is about the study of the Boltzmann equation in the diffusive limit in a channel domain 𝕋2×(−1,1) nearby the 3D kinetic Couette flow.  We will begin the talk with a substantial introduction for non-experts.  Our result demonstrates that the first-order approximation of a solution to the Boltzmann equation is governed by the perturbed incompressible Navier-Stokes-Fourier system around the fluid Couette flow.  Moverover, in the absence of external forces, we show that the 3D kinetic Couette flow  converges in large time to the 1D steady planar kinetic Couette flow.  Our method relies on three ingredients: (i) the Fourier transform on 𝕋2 allows us to reduce the 3D problem to 1D, (ii) anisotropic Chemin-Lerner type function spaces (using the Wiener algebra) are used to bound the nonlinear terms and to control the singularity associated with a small Knudsen number in the diffusive limit, and (iii) the Caflisch decomposition, combined with the L2∩L∞

technique, to manage the growth of large velocities.  This is a joint work in collaboration with Renjun Duan, Shuangqian Liu, and Anita Yang.