Partial Differential Equations and Applied Mathematics Seminar
Date/Time: Jan. 17th, Tue.,19th, Thu., 3:30 ~ 5:30 PM
Location: Science Building #254 Yonsei University
Speaker: Prof. Tai-Peng Tsai
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, Canada
Title: On forward self-similar and discretely self-similar solutions of incompressible Navier-Stokes equations
In this minicourse we will consider the construction of forward self-similar and discretely self-similar solutions of 3D incompressible Navier-Stokes equations with arbitrarily large initial data. I will start with a survey on the subject, including the description of all known constructions (Jia-Sverak, Tsai, Korobkov-Tsai, Bradshaw-Tsai, Lemarie-Lieusset and Chae-Wolf), and its relevance to the Navier-Stokes uniqueness problem. After the survey I will focus on the method using the similarity transform and the study of the corresponding Leray equations, used by Korobkov, Bradshaw and myself. It is the only method that works on both the whole space and the half space, and is general enough to allow the construction
of solutions which are self-similar, discretely self-similar, rotated self-similar, or rotated discretely self-similar, for initial data which are very rough, including distributions in Besov spaces of negative differentiability and not necessarily locally L2 integrable. Our machinery includes the wavelet characterization of Besov spaces. We will also comment briefly on the backward case.