Global Stability and Scattering Theory for Boltzmann Equation with Soft Potentials in the Whole Space: Weak Collision Regime
Time: 2023-09-08
Published By: He Liu
Speaker(s): Lingbing He(Tsinghua University)
Time: 16:00-17:00 September 11, 2023
Venue: Room 77201, Jingchunyuan 78, BICMR
A Traveling Maxwellian $\mathcal{M} = \mathcal{M}(t, x, v)$ represents a traveling wave solution to the Boltzmann equation in the whole space $\R^3$(for the spatial variable $x$). The primary objective of this talk is to investigate the global-in-time stability of $\mathcal{M}$ and its associated scattering theory in $L^1_{x,v}$ space for the Boltzmann equation with soft potentials when the dissipative effects induced by collisions are {\it weak}. We demonstrate the following results: (i) $\mathcal{M}$ exhibits Lyapunov stability; (ii) The perturbed solution, which is assumed to satisfy the same conservation law as $\mathcal{M}$, scatters in the $L^1_{x,v}$ space towards a particular traveling wave (with an explicit convergence rate), which may not necessarily be $\mathcal{M}$. The key elements in the proofs involve the formulation of the {\it Strichartz-Scaled Boltzmann equation}(achieved through the Strichartz scaling applied to the original equation) and the propagation of analytic smoothness.
