Room 515, Cosmology Building, NTU
Speaker(s):
Chi Hin Chan (National Yang Ming Chiao Tung University )
Organizer(s):
Jenn-Nan Wang (National Taiwan University)
一、 課程背景與目的：
The purpose of this short course is to provide scholars and students who are interested in participating in the coming MSRI-NCTS Joint Summer School: Recent Topics on Well-posedness and Stability of Incompressible Fluid and Related Topics, July 19-30, 2021, basic knowledge on the Navier-Stokes equation.
二、課程之大綱：
In this course, we will first build up the existence theory for suitable weak solutions to the time-dependent Navier-Stokes equation. The setting upon which we will build up our suitable weak solutions will be a bounded domain in the 3 Euclidean space with sufficiently smooth boundary. During the proof, we will quote a classical global regularity estimate (as stated in the Textbook by Professor Tsai Tai Peng) for time-dependent Stokes equation written down on a bounded Euclidean domain with sufficiently smooth boundary. One of the key features of the construction is to see the way in which the Lion-Aubin compactness Lemma is applied to pass to the limit on the approximated solutions via strong convergence in some L^p space. This technique of Lion-Aubin runs through almost all the basic researches in the study of existence and regularity of solutions to the incompressible Navier-Stokes equation. This explains why we would like to talk about this topic in this short course. By going through this process of constructing a suitable weak solution arising from a sufficiently regular initial datum, a beginner can get a basic sense of the way in which weak solutions could be constructed. This is the first part of the short course and it will take us at least 5 to 6 hours to complete.
Next, with the foundation of suitable weak solutions available, we will give a detailed exposition of the proof leading to the famous L^{infty}L^3 regularity criteria for the regularity of suitable weak solutions to the incompressible Navier-Stokes equation. Here, we will see how the theory of backward uniqueness of heat equation and the theory of partial regularity will interact with the basics of the subject, thus leading to the final conclusion of the L^{infty}L^3 regularity criteria. This topic will take again 5 to 6 hours to complete.
