601, Pao Yue-Kong Library
We present a unified framework to prove the well-posedness for the inhomogeneous Boltzmann equation with and without angular cutoff. The results and proofs have the following main features and innovations.
(i). We solve the Boltzmann equation with general initial data in weighted Sobolev spaces with polynomial weights. We do not assume that the solution initially is a small perturbation of some background solution, for instance, the equilibrium.
(ii). The quasi-linear method instead of the standard linearization method is used to prove the existence and the non-negativity of the solution. The key point of the method is constructing the suitable energy space, which depends heavily on the initial data, to prove the propagation of the regularity.
(iii). We create new tools for the equation such as the sharp Povnzer inequality for $L^1$ moment, splitting of the collision operator to catch the full dissipation and the localization of the equation in the frequency space(with respect to the velocity variable).
(iv). Our method is robust such that it can be combined with the entropy method to study the global dynamics of the inhomogeneous Boltzmann equation.
Lingbing He received his Ph.D in 2007 from Institute of Mathematics of AMSS, China. He joined the Department of Mathematical Sciences of Tsinghua University in 2009. His research is mainly focused on kinetic equations, such as Boltzmann and Landau equations from statistical physics, and the equations, such as Navier-Stokes, Euler equations and Magneohydrodynamics equations, from fluid mechanics and plasma physics.