Dynamical Billiard and a long-time behavior of the Boltzmann equation in general 3D toroidal domains

Establishing global well-posedness and convergence toward equilibrium of the Boltzmann equation with specular reflection boundary condition has been one of the central questions in the subject of kinetic theory. Despite recent significant progress in this question when domains are strictly convex, as shown by Guo and Kim-Lee, the same question without the strict convexity of domains is still totally open in 3D. The major difficulty arises when a billiard map has an infinite number of bounces in a finite time interval or when the map fails to be Lipschitz continuous, both of which happen generically when the domain is non-convex. In this paper, we develop a new method to control a billiard map on a surface of revolution generated by revolving any planar analytic convex closed curve (e.g., typical shape of tokamak reactors' chamber). In particular, we classify and measure the size (to be small) of a pullback set (along the billiard trajectory) of the infinite-bouncing and singular-bouncing cases. As a consequence, we solve the open question affirmatively in such domains. To the best of our knowledge, this work is the first construction of global solutions to the hard-sphere Boltzmann equation in generic non-convex 3-dimensional domains.Comment: 97 pages, 11 figure

Large amplitude problem of BGK model: Relaxation to quadratic nonlinearity

Bhatnagar-Gross-Krook (BGK) equation is a relaxation model of the Boltzmann equation which is widely used in place of the Boltzmann equation for the simulation of various kinetic flow problems. In this work, we study the asymptotic stability of the BGK model when the initial data is not necessarily close to the global equilibrium pointwisely. Due to the highly nonlinear structure of the relaxation operator, the argument developed to derive the bootstrap estimate for the Boltzmann equation leads to a weaker estimate in the case of the BGK model, which does not exclude the possible blow-up of the perturbation. To overcome this issue, we carry out a refined analysis of the macroscopic fields to guarantee that the system transits from a highly nonlinear regime into a quadratic nonlinear regime after a long but finite time, in which the highly nonlinear perturbative term relaxes to essentially quadratic nonlinearity.Comment: 34 pages, 1 figure

On C2 solution of the free-transport equation in a disk

The free transport operator of probability density function f (t, x, v) is one the most fundamental operator which is widely used in many areas of PDE theory including kinetic theory, in particular. When it comes to general boundary problems in kinetic theory, however, it is well-known that high or-der regularity is very hard to obtain in general. In this paper, we study the free transport equation in a disk with the specular reflection boundary condi-tion. We obtain initial-boundary compatibility conditions for C1t,x,v and C2t,x,v regularity of the solution. We also provide regularity estimates.11Nsci

The large amplitude solution of the Boltzmann equation with soft potential

In this paper, we deal with the (angular cut-off) Boltzmann equation with soft potential (-3 < gamma < 0). In particular, we construct a unique global solution in L-x,v(infinity) which converges to global equilibrium asymptot-ically provided that initial data has a large amplitude but with sufficiently small relative entropy. Because frequency multiplier is not uniformly positive anymore, unlike hard potential case, time-involved veloc-ity weight will be used to derive sub-exponential decay of the solution. Motivated by recent development of L-2-L-infinity approach also, we introduce some modified estimates of quadratic nonlinear terms. Linearized collision kernel will be treated in a subtle manner to control singularity of soft potential kernel. (C) 2021 Elsevier Inc. All rights reserved.11Nsciescopu