2015 CMC <Daejeon KAIST> Foundations for Mathematical Challenges -1 |
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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.
The lectures are prepared for graduate students in mathematics who have already taken at least one graduate course in the partial differential equations(PDEs). The students are thus expected to have some knowledge (otherwise, please study by yourself) about the basic theory of functional analysis, Sobolev spaces, and weak solutions of elliptic/parabolic PDEs, for instance, given in Chapters 5-7 and Appendix D in the book by Evans. More advanced or specialized materials will be explained in detail during the lectures.
Contents
1. Preliminaries
1.1. The divergence equation
1.2. The Helmholtz-Weyl decomposition
1.3. The stationary Stokes equations
2. Existence of weak solutions
2.1. Denition of weak solutions
2.2. Compactness results
2.3. Global existence of weak solutions
3. Strong solutions and weak-strong uniqueness
3.1. Existence of strong solutions
3.2. Weak-strong uniqueness results
4. Regularity of weak solutions
4.1. The nonstationary Stokes equations
4.2. Regularity properties of weak solutions
4.3. Singular sets of weak solutions
5. Regularity criteria for weak solutions
5.1. The weak Lebesgue spaces L_{w}^{p}
5.2. Estimates in L_{w}^{p} for the heat equation
5.3. Interior regularity criteria in L_{w}^{p}
5.4. Further results
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서울 KIAS Hodge Theory 강연 등록링크:
(http://home.kias.re.kr/MKG/h/FMCS2015/)
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**사전등록 기간: 7월 26일까지
**본 강연은 한국어로 구성되었으며, KAIST CMC와 CMC의 주최로 진행됩니다.
Contact
노희주 (heejunoe@kias.re.kr)
Tel. +82 42 350 8545