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Tai-Peng Tsai

2014-04-16
09:30:00 - 11:30:00

101 , Mathematics Research Center Building (ori. New Math. Bldg.)

In the lectures of last semester, we covered the theory for steady states, weak solutions and mild solutions for time dependent solutions, Serrin-type regularity criteria, partial regularity, and self-similar solutions. In this April and May, we plan to cover the following.

1. Uniform L^3 class. For this endpoint Prodi-Serrin class, we first give the uniqueness results of Masuda and Kozono-Sohr. We then study the backward uniqueness of parabolic equations using Carleman-type inequalities, and how it is applied to show the regularity theorem of Escauriaza-Seregin-Sverak.

2. Nonexistence of Type-I singularity of axisymmetric flows. After introducing axisymmetric flows and their applications, we first show their regularity in the no-swirl case. We then show the non-existence of Type-I singularity using DeGiorgi-Nash-Moser approach and Liouville theorem approach. We will also present recent study on elliptic/heat equations with critical drafts.