Beijing Institute of Mathematical Sciences and Applications

The focus of this course will be on evolution equations of both Hamiltonian (conservative, such as wave, Klein-Gordon, or Schroedinger) and dissipative kinds (heat equation)on $\mathbb{R}^{d},d\geq 2$, but the main focus will essentially be $d=3$. Our goal is to provide an introduction to the \textbf{concentration compactness} technique (or profile decomposition to capture the defect of compactness), which is a core method in modern large data theory. To develop intuition, we first cover the simpler elliptic analogue, which addresses the loss of compactness due to the action of symmetry groups. We begin with a self-contained discussion of wave and Klein-Gordon equations and carefully cover some basic aspects leading to problems of Fourier analysis, such as Sobolev embeddings and Strichartz estimates. A prior course in PDE is not required, but basic knowledge of ODEs and functional analysis is helpful.