On $C^2$ solution of the free-transport equation in a disk

Abstract

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 order 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 condition. We obtain initial-boundary compatibility conditions for C 1 t,x,v and C 2 t,x,v regularity of the solution. We also provide regularity estimates.of Theorem 1.2 4. Initial-boundary compatibility condition for C 2 t,x,v 4.1. Condition for ∇ xv 4.2. Condition for ∇ vv 4.3. Condition for ∇ xx 4.4. Condition for ∇ vx 4.5. Compatibility conditions for transpose : ∇ T xv = ∇ vx and ∇ T xx = ∇ xx 4.6. Conditions including ∂ t 4.7. Proof of Theorem 1.3 5. Regularity estimate of f 5.1. First order estimates of characteristics 5.2. Second order estimates of characteristics 5.3. Proof of Theorem 1.6 References