Regularity Estimates for the Non-cutoff Soft Potential Boltzmann Equation with Typical Rough and Slowly Decaying Data
Time: 2023-04-16
Published By: He Liu
Speaker(s): Lingbing He (Tsinghua University)
Time: 15:45-16:45 April 17, 2023
Venue: Room 1114, Sciences Building No. 1
For the non-cutoff soft potential Boltzmann equation, if the Boltzmann collision operator is strictly elliptic in $v$ variable, it is conjectured that the solution to the equation will become immediately $C^\infty$ smoothness for both spatial and velocity variables for any positive time even though the initial datum has only polynomial decay at large velocity regime. The conjecture is quite important because it is related closely to the partial regularity problem on the weak solution. In this talk, we show that the conjecture might not hold for general case due to the degenerate property and the non-local property of the collision operator. We demonstrate it via three steps: (i). Construct so-called {\it typical rough and slowly decaying data}; (ii). Prove that such kind of the data will induce the finite smoothing effect in Sobolev spaces; (iii). Prove that this finite smoothing property will induce the local properties for any positive time: Leibniz rule does not hold for high derivatives on the collision operator(even in the weak sense) and the discontinuity in $x$ variable of average of the solution on some certain domain in $\mathbb{R}^3_v$.
