论文标题
$ l^1_k \ cap l^p_k $用于$ \ mathbb {r}^3 $中的非cutoff boltzmann方程的方法
An $L^1_k\cap L^p_k$ approach for the non-cutoff Boltzmann equation in $\mathbb{R}^3$
论文作者
论文摘要
在论文中，我们开发了一个$ l^1_k \ cap l^p_k $方法，以在$ \ mathbb {r}^3 $中的均衡附近的非切割Boltzmann方程式上构建Cauchy问题的全局解决方案。特别是，仅$ \ | \ Mathcal {f} _x {f} _0 \ | _ {l^1 \ cap l^p（\ Mathbb {r}^3_k; l^2（\ Mathbb {r} $ f_0（x，v）$，其中$ \ Mathcal {f} _x {f} _0（k，v）$是空间变量中的傅立叶变换。这为全局存在这种低指标解决方案提供了第一个结果，而不依赖于嵌入$ h^2（\ Mathbb {r}^3_x）\ subset l^\ infty（\ Mathbb {r}^3_x）$在整个空间的情况下。与Gressman-Strain和Amuxy在这些经典结果中使用足够平滑的Sobolev空间不同，圆环案例与整个空间案例之间存在至关重要的差异，以实现正在考虑的低规律性解决方案。实际上，对于前者而言，要采用与Duan-Liu-Sakamoto-Strain所研究的Weiner空间相对应的唯一$ l^1_k $ norm。相比之下，对于后者，与$ l^p_k $ norm的额外相互作用在控制非线性碰撞项中起着至关重要的作用，这是由于宏观分量的退化耗散。确实，$ l^p_k $ norm的传播有助于获得几乎最佳的衰减率$（1+t）^{ - \ frac {3} {2} {2}（1- \ frac {1} {1} {p} {p}）_+} $ n n of of the kawa和kawa的精神的$ l^1_k $ norm的$ l^1_k $ norma这必然用于建立全球存在。
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.
