2015 CMC Foundations for Mathematical Challenges -1


강연자: 김현석 (서강대)


기간: 2015년 8월 3일 ~ 7일
장소: KAIST 자연과학동 E6-1, 3435

주제: Well-posedness theory for the Navier-Stokes equations

초록: In this series of lectures, we provide a classical theory of the Navier-Stokes equations for incompressible viscous fluids in three-dimensional domains. The first part of the lectures is devoted to giving a rather complete proof of Leray’s fundamental existence result (1934) on global weak solutions without any size condition on the data. The uniqueness and regularity of Leray’s weak solutions has been one of the most outstanding open questions in analysis and are closely related to one of the seven Clay Millennium Problems: the so-called ”Navier-Stokes existence and smoothness problem”. In the second part, as partial answers to the uniqueness question, we establish weak-strong uniqueness results which show that weak solutions are unique if a strong or smooth solution exists. It will be also shown that a strong or smooth solution exists at least for a short time interval and globally in time if the data is sufficiently small. The regularity of weak solutions will be studied in the last part of the lectures. Local regularity theorems, which extends a classical result due to Serrin (1962), are explained in some detail but possibly with outlines of proofs. We finally discuss the partial regularity result of Caffarelli-Kohn-Nirenberg (1982) for suitable weak solutions of the Navier-Stokes equations.