论文标题
de Giorgi加权$ l^2 \ cap l^\ infty $解决方案的解决方案boltzmann方程
De Giorgi argument for weighted $L^2 \cap L^\infty$ solutions to the non-cutoff Boltzmann equation
论文作者
论文摘要
本文为$ l^\ infty $ setting中没有角截止的全球存在方程的全球存在问题提供了第一个肯定的答案。特别是，我们表明，当初始数据接近平衡，并且扰动在$ l^2 \ cap l^\ inty the the Tolymial Decay尾巴中，Boltzmann方程在加权$ L^2 \ 2 \ cap l^\ cap l^\ infty $ -space中具有全球解决方案。为了克服奇异的横截面和低规律性引起的困难，在平均引理的帮助下，在动力学环境中制定了de giorgi类型的参数。更具体地说，我们使用强大的平均引理来获得合适的$ l^p $估计用于级别集合功能。这些估计对于构建适当的能量功能以执行De Giorgi论点至关重要。与\ cite {amsy}相似，我们使用线性化玻尔兹曼操作员的光谱间隙将本地解决方案扩展到全局解决方案。然后，获得与均衡状态的融合作为副产品，并在$ l^2 $和$ l^\ infty $ spaces中显示放松。
This paper gives the first affirmative answer to the question of the global existence of Boltzmann equations without angular cutoff in the $L^\infty$-setting. In particular, we show that when the initial data is close to equilibrium and the perturbation is small in $L^2 \cap L^\infty$ with a polynomial decay tail, the Boltzmann equation has a global solution in the weighted $L^2\cap L^\infty$-space. In order to overcome the difficulties arising from the singular cross-section and the low regularity, a De Giorgi type argument is crafted in the kinetic context with the help of the averaging lemma. More specifically, we use a strong averaging lemma to obtain suitable $L^p$-estimates for level-set functions. These estimates are crucial for constructing an appropriate energy functional to carry out the De Giorgi argument. Similar as in \cite{AMSY}, we extend local solutions to global ones by using the spectral gap of the linearised Boltzmann operator. The convergence to the equilibrium state is then obtained as a byproduct with relaxations shown in both $L^2$ and $L^\infty$-spaces.
