AI 中文总结
该讲座从傅里叶变换和索伯列夫空间结果出发，先以非线性为多项式的非线性热方程为玩具模型，最终证明了临界空间中纳维 - 斯托克斯方程温和解的存在性与唯一性，讲稿还包含练习及考试题目。
AI 中文摘要
以下内容是2026年第一学期在澳大利亚国立大学举办的一个专题讲座的一部分。这是一次很棒的经历，我很享受这为期12周（每周2小时）的讲座。讲座内容自成体系，从傅里叶变换和索伯列夫空间的结果开始。作为一个玩具模型，在处理纳维 - 斯托克斯系统之前，我们专注于非线性为多项式的非线性热方程。最终，我们证明了临界空间中纳维 - 斯托克斯方程温和解的存在性和唯一性。这个讲稿包含一些练习、期中考试和期末考试的题目。
英文摘要
The content of the following pages was part of a special topics lecture given at the Australian National University during the first semester of 2026. That was a very nice expericence, I really enjoyed giving this 12 weeks (2 hours a week) lecture. It is meant to be self contained, starting with results on Fourier transform and Sobolev spaces. As a toy model, before treating the Navier-Stokes system, we focus on the non linear heat equation where the non linearity is polynomial. Ultimately, we prove existence and uniqueness of mild solutions of the Navier-Stokes equations in critical spaces. This script contains some exercises and the text of the mid-semester exam as well as the final exam.
