论文标题
抛物线进化方程的最大规律性简介
An introduction to maximal regularity for parabolic evolution equations
论文作者
论文摘要
在本说明中，我们介绍了最大$ l^p $ regularity的概念，作为解决非线性偏微分方程的方法。我们首先为自主和非自主问题定义最大规律性，并描述与傅立叶乘数和$ \ Mathcal r $ bundedness的连接。抽象结果适用于整个空间中的一大类抛物线系统，并将其用于一般的抛物线边界价值问题。为此，讨论了解决边界价值问题的解决方案运算符的构建和Sobolev空间的痕迹空间的表征。对于非线性方程式，我们在适当选择的Sobolev空间中获得时间良好的局部序列。该手稿基于已知结果，由有关此主题的讲义的扩展版本组成。
In this note, we give an introduction to the concept of maximal $L^p$-regularity as a method to solve nonlinear partial differential equations. We first define maximal regularity for autonomous and non-autonomous problems and describe the connection to Fourier multipliers and $\mathcal R$-boundedness. The abstract results are applied to a large class of parabolic systems in the whole space and to general parabolic boundary value problems. For this, both the construction of solution operators for boundary value problems and a characterization of trace spaces of Sobolev spaces are discussed. For the nonlinear equation, we obtain local in time well-posedness in appropriately chosen Sobolev spaces. This manuscript is based on known results and consists of an extended version of lecture notes on this topic.
