Nonlinear Evolution Equations and Related Topics

Speaker: Lingbing He　(Tsinghua University)
Title: Sharp bounds for Boltzmann and Landau Collision Operators
Abstract: Motivated by the justification of the asymptotics from Boltzmann equation to Landau equation in the so-called grazing collisions limit, in this talk, we will give several sharp bounds for Boltzmann and Landau operators in weighted Sobolev spaces and in anisotropic spaces. Here the sharpness means the lower and upper bounds for the operators are almost as the same as those for the linearized operators. As a direct consequence, we complete the entropy dissipation estimates of the Boltzmann collision operator in the sense that the fractional Laplacian-Beltrami operator on the unit sphere can be captured explicitly. The main strategy of the proof is the introduction of two types of dyadic decompositions performed in both phase and frequency spaces according to the Bobylev's formula of the Boltzmann collision operator and the new observation of the geometric structure of the collision.