The Boltzmann equation, Besov spaces, and optimal time decay rates in the whole space

TL;DR

This paper establishes optimal decay rates for derivatives of solutions to the Boltzmann equation in the whole space, using Besov space techniques, for both hard and soft potentials without angular cutoff.

Contribution

It introduces new decay estimates in Besov spaces for the Boltzmann equation, extending previous results to include non-cutoff potentials and higher derivatives.

Findings

Derives optimal decay rates in $L^r_x(L^2_v)$ norms for solutions.

Shows faster decay in the hard potential case under certain initial conditions.

Provides decay estimates for derivatives of all orders in the perturbative regime.

Abstract

We prove that $k$-th order derivatives of perturbative classical solutions to the hard and soft potential Boltzmann equation (without the angular cut-off assumption) in the whole space, ${\mathbb R}^{n}_x$ with $n \ge 3$, converge in large-time to the global Maxwellian with the optimal decay rate of $O(t^{-1/2(k+\varrho+\frac{n}{2}-\frac{n}{r})})$ in the $L^r_x(L^2_{v})$-norm for any $2\leq r\leq \infty$. These results hold for any $\varrho \in [0, n/2]$ as long as initially $\| f_0|_{\dot{B}^{-\varrho,\infty}_2 L^2_{v}} < \infty$. In the hard potential case, we prove faster decay results in the sense that if $|\mathbf{P} f_0\|_{\dot{B}^{-\varrho,\infty}_2 L^2_{v}} < \infty$ and $|({\mathbf{I} - \mathbf{P}}) f_0|_{\dot{B}^{-\varrho+1,\infty}_2 L^2_{v}} < \infty$ for $\varrho \in (n/2, (n+2)/2]$ then the solution decays to zero in $L^2_v(L^2_x)$ with the optimal large time decay rate of $O(t^{-1/2\varrho})$.