Some recent progress on the 3D kinetic shear flow for the Boltzmann equation in the hydrodynamic limit

In this talk, I will present our recent studies on the hydrodynamic limits of the Boltzmann equation, focusing on the rigorous derivation of Couette flow from the Boltzmann equation to the incompressible Navier-Stokes system. Central to our approach are novel anisotropic estimates within the Wiener algebra function space, which enable a precise treatment of the kinetic-to-fluid transition. Additionally, I will discuss extensions to more general shear flows, including the Kolmogorov flow.