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

Abstract

In the paper, we develop an $L_k^1\cap L_k^p$ 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 $\Vert {\mathcal{F}}_x{f}_0{\Vert }_{L^1\cap L^p({\mathbb{R}}_k^3;L^2({\mathbb{R}}_v^3))}$ with $3/2<p\le \infty$ 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 $H^2({\mathbb{R}}_x^3)\subset L^{\infty }({\mathbb{R}}_x^3)$ 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 $L_k^1$ norm corresponding to the Weiner space as studied in Duan-Liu-Sakamoto-Strain. In contrast, for the latter, the extra interplay with the $L_k^p$ norm plays a vital role in controlling the nonlinear collision term due to the degenerate dissipation of the macroscopic component. Indeed, the propagation of $L_k^p$ norm helps gain an almost optimal decay rate $(1+t{)}^{-\frac{3}{2}(1-\frac{1}{p}{)}_{+}}$ of the $L_k^1$ 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.

38 pages