An $L^1_k\cap L^p_k$ approach for the non-cutoff Boltzmann equation in $\mathbb{R}^3$

TL;DR

This paper introduces an $L^1_k igcap L^p_k$ analytical approach to prove the global existence of low-regularity solutions for the non-cutoff Boltzmann equation in $R^3$, avoiding Sobolev embedding constraints.

Contribution

It develops a novel $L^1_k igcap L^p_k$ framework for global solutions near equilibrium without Sobolev embedding, highlighting the importance of $L^p_k$ norms in whole space analysis.

Findings

First global existence result for low-regularity solutions in $R^3$

Demonstrates decay rate $(1+t)^{-rac{3}{2}(1-rac{1}{p})_+}$ for solutions

Shows the crucial role of $L^p_k$ norms in controlling nonlinear collision terms.

Abstract

In the paper, we develop an $L^1_k\cap L^p_k$ approach to construct global solutions to the Cauchy problem on the non-cutoff Boltzmann equation near equilibrium in $\mathbb{R}^3$. In particular, only smallness of $\|\mathcal{F}_x{f}_0\|_{L^1\cap L^p (\mathbb{R}^3_k;L^2(\mathbb{R}^3_v))}$ with $3/2<p\leq \infty$ is imposed on initial data $f_0(x,v)$, where $\mathcal{F}_x{f}_0(k,v)$ is the Fourier transform in space variable. This provides the first result on the global existence of such low-regularity solutions without relying on Sobolev embedding $H^2(\mathbb{R}^3_x)\subset L^\infty(\mathbb{R}^3_x)$ in case of the whole space. Different from the use of sufficiently smooth Sobolev spaces in those classical results by Gressman-Strain and AMUXY, there is a crucial difference between the torus case and the whole space case for low regularity solutions under consideration. In fact, for the former, it is enough to take the only $L^1_k$ norm corresponding to the Weiner space as studied in Duan-Liu-Sakamoto-Strain. In contrast, for the latter, the extra interplay with the $L^p_k$ norm plays a vital role in controlling the nonlinear collision term due to the degenerate dissipation of the macroscopic component. Indeed, the propagation of $L^p_k$ norm helps gain an almost optimal decay rate $ (1+t)^{-\frac{3}{2} (1-\frac{1}{p})_+}$ of the $L^1_k$ norm via the time-weighted energy estimates in the spirit of the idea of Kawashima-Nishibata-Nishikawa and in turn, this is necessarily used for establishing the global existence.