Abstract:

For $\frac{3}{2} < p \leq \infty$, when the norm of the initial data $\|\mathcal{F}_x f_0\|_{L^1 \cap L^p \cap \mathcal{X}^{-p}(\mathbb{R}^3_{\xi}; L^2(\mathbb{R}^3_v))}$ is sufficiently small, we construct global solutions to the Cauchy problem for the non-cutoff Boltzmann equation near equilibrium in the whole space $\mathbb{R}^3$. Here, $\mathcal{F}_x f_0(\xi, v)$ denotes the Fourier transform of $f_0(x, v)$ with respect to the spatial variable $x$, and $\mathcal{X}^{-p}$ is the $L^p$ space incorporating a Hardy potential.Compared to the $L^1_{\xi} \cap L^p_{\xi}$ space used in [15], we consider the low-regularity Sobolev space $L^1_{\xi} \cap L^p_{\xi} \cap \mathcal{X}^{-p}_{\xi}$ in the whole-space framework. Under the energy method framework, we establish a priori estimates to close the argument, thereby obtaining global solutions. In particular, we also derive the decay estimate, for any arbitrarily small $\delta>0,$

Key words: Boltzmann equation, global solution, energy method, $(L^1_{\xi}\cap L^p_{\xi}\cap\mathcal{X}^{-p}_{\xi})L^2_v$ space.