Decay estimates and continuation for the non-cutoff Boltzmann equation

Abstract

We consider the non-cutoff Boltzmann equation in the spatially inhomogeneous, soft potentials regime, and establish decay estimates for large velocity. In particular, we prove that pointwise algebraically decaying upper bounds in the velocity variable are propagated forward in time whenever the solution has finite weighted $L^\infty_{t,x} L^p_v$-norms for certain $p$. The main novelty is that these estimates hold for any decay exponent above $\max\{2,3 + \gamma\} +2s$, where $\gamma$ and $s$ are standard physical parameters such that $\gamma \in (-3,0)$ and $s\in (0,1)$. Our results are useful even for solutions with mild decay. As an application, we combine our decay estimates with recent short-time existence results to derive a continuation criterion for large-data solutions. Compared to past results, this extends the range of allowable parameters and weakens the requirements on smoothness and decay in velocity of solutions.