Global Mild Solutions of the Landau and Non‐Cutoff Boltzmann Equations
Global Mild Solutions of the Landau and Non‐Cutoff Boltzmann Equations
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DOI：
10.1002/cpa.21920
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发表时间：
2019-04
影响因子：
3
通讯作者：
Renjun Duan;Shuangqiang Liu;Shota Sakamoto;Robert M. Strain
中科院分区：
文献类型：
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作者：
Renjun Duan;Shuangqiang Liu;Shota Sakamoto;Robert M. Strain
本文证明了含库仑势的朗道方程和无角截断的Boltzmann方程的小振幅整体时间唯一温和解的存在性.由于著名的工作[45]和[3，43]在光滑Sobolev空间中构造经典解，特别是在空间变量上是正则的，因此在Lx，v∞框架中获得整体解仍然是一个开放的问题，类似于[49]，在一般有界区域中具有截断假设的Boltzmann方程。一个主要的困难来自于非截断玻尔兹曼和朗道方程中的输运算子和速度扩散型碰撞算子之间的相互作用;另一个主要的困难是边值问题解的奇点的潜在形成。
This paper proves the existence of small‐amplitude global‐in‐time unique mild solutions to both the Landau equation including the Coulomb potential and the Boltzmann equation without angular cutoff. Since the well‐known works [45] and [3, 43] on the construction of classical solutions in smooth Sobolev spaces which in particular are regular in the spatial variables, it still remains an open problem to obtain global solutions in an Lx,v∞ framework, similar to that in [49], for the Boltzmann equation with the cutoff assumption in general bounded domains. One main difficulty arises from the interaction between the transport operator and the velocity‐diffusion‐type collision operator in the non‐cutoff Boltzmann and Landau equations; another major difficulty is the potential formation of singularities for solutions to the boundary value problem.