This mini-workshop aims at exchange and collaboration between researchers on kinetic equations. The topics include wellposedness theory of Boltzmann equation, and the hydrodynamic limit of kinetic theory, etc. The mini-workshop is supported by Institute of Natural Sciences, and School of Mathematical Sciences in Shanghai Jiao Tong University.
September 11, 2018
Room 601, Pao Yue-Kong Library, Minhang Campus, Shanghai Jiao Tong University
|Sep 11||14:00-15:00||Two-Fluid Incompressible Navier-Stokes-Maxwell System with Ohm’s Law Limit from Vlasov-Maxwell-Boltzmann System||Ning Jiang|
|Sep 11||15:10-16:10||Global Dynamics of Inhomogeneous Boltzmann Equation with Hard Potentials and Maxwellian Molecules||Lingbing He|
Ning Jiang, Wuhan University
We prove a global-in-time classical solution limit from the two-species Vlasov-Maxwell-Boltmann system to the two-fluid incompressible Navier- Stokes-Fourier-Maxwell system with solenoidal Ohm’s law. Besides the techniques developed for the classical solutions to the Vlasov-Maxwell-Boltzmann equations in the past years, such as the nonlinear energy method and micro-macro decomposition are employed, key roles are played by the decay properties of both the electric field and the wave equation with linear damping of the divergence free magnetic field. This work is a collaboration with Yi-Long Luo and Teng-fei Zhang.
Lingbing He,Tshinghua University
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 L1 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.