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

Duan, Renjun
Sakamoto, Shota
Ueda, Yoshihiro

Publication date

September 2022

Language

English

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 AMUX...

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

Abstract

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

Abstract

Extracted data

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