Singularity and Homogenization of PDE 2018

Singularity and Homogenization of PDE 2018


Title: An Introduction to Elliptic Homogenization

Speaker: Fanghua Lin

Abstract: I shall started with some examples and problems that concern with partial differential equations in highly oscillating medium. Then we proceed with various classical results of G- and H-convergences, div-curl, oscillating test function and viscosity methods as well as correctors in the periodic case. Applications and further questions will be addressed at end.
     Further developments including, in particular, various a priori estimates, rates of convergences and behavior of the Green and Poisson kernels would be discussed in Professor Zhongwei Shen's lectures.


Title: Analytic aspects on the heat flow of harmonic maps

Speaker: Changyou Wang

Abstract: In this series of lectures, I plan to discuss some classical theorems and further issues on the heat flow of harmonic maps, which include Eells-Sampson theorem on global smooth solutions into non-positively curvature manifold, Struwe's almost "regular" global solutions in dimension two, and Chen-Struwe, Chen-Lin's existence of partially regular solutions in higher dimensions, and some results on the blow-up analysis by Lin and Wang.


Title: Introduction to Almost-Periodic Homogenization

Speaker:Zhongwei Shen

Abstract: In this series of lectures we will introduce the quantitative homogenization theory of second-order linear elliptic equations and systems in divergence form with almost-periodic coefficients. We will start with the definition and properties of almost-periodic functions, and present the classical qualitative homogenization theory. The main body of the lectures will deal with the key issues in the quantitative theory: estimates of approximate correctors, convergence rates, uniform regularity estimates (H"older, Lipschitz, W^{1, p}). The lectures are designed for advanced graduate students as well as junior researchers in the general areas of analysis and PDEs. We assume that the audience is familiar with the basic material usually covered in the first-year graduate courses on real analysis, functional analysis, and linear elliptic PDEs.


Title: Gluing methods and finite time blow-ups for harmonic map flows into $S^2$

Speaker:Juncheng Wei

 Abstract: I will explain the main ideas in the gluing constructions of Type II blow-ups for harmonic map flows into $S^2$: $$ u_t=\Delta u+ |\nabla u|^2 u, u:\Omega \to S^2$$ where $\Omega$ is a smooth two-dimensional domains.