| Abstract: |
| In this talk, I will present our recent study on the Boltzmann equation in the diffusive limit for 3D kinetic shear flow. Our results show that the first-order approximation of the solutions is governed by the perturbed incompressible Navier-Stokes-Fourier system around the fluid shear flow. The proof is based on: (i) applying the Fourier transform on $\T^2$ to effectively reduce the 3D problem to a one-dimensional one; (ii) using anisotropic Chemin-Lerner type function spaces, incorporating the Wiener algebra, to control nonlinear terms and address the singularities arising from the small Knudsen number in the diffusive limit; and (iii) employing Caflisch`s decomposition, together with the $L^2 \cap L^\infty$ interplay technique, to manage the growth of large velocities. |
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