報告題目：關于液晶方程的數學分析問題
報 告 人：Changyou Wang，Purdue University
報告地點：騰訊會議 會議 ID：991 4503 5304
校内聯系人：郭斌 bguo@jlu.edu.cn
Abstract: Liquid crystal materials exhibit basic features of both liquid state and solid state, i.e. they enjoy both the crystalline structure and the fluid structure. In particular, it has an optical structure that can be transformed under either temperature changes or the action of external electric or magnetic fields. They have found profound applications in Liquid Crystal Devices (LCD). There have been tremendous studies on liquid crystals by applied physicists, material scientists, and mathematic- ians. In this lecture series, we would like to introduce both the static theory and the dynamic theory of liquid crystals, namely, the Oseen-Frank model and the Ericksen-Leslie model. The former concerns the equiqibriums of the Oseen-Frank energy functional, that can be studied through calculus of variations. The latter concerns the hydrodynamic evolution of liquid crystals, that can be viewed as a dissipative system coupling between the underlying fluid (via the Navier-Stokes equation) and the liquid crystal orientation field (via the heat flow of harmonic maps). The fundamental questions are: (i) the existence of global weak solutions in the finite energy spaces. (ii) the issue of regularity and uniqueness of certain classes of weak solutions . (iii) the existence of finite time singularity of locally smooth solutions . (iV) the local and global well posedness problem for a largest possible function space of initial data
授課日期 Date of Lecture |
課程名稱（講座題目） Name (Title) of Lecture |
授課時間 Duration (Beijing Time) |
參與人數 Number of Participants |
July 5, 2021 |
Static theory and Background I |
8:30-9:30 |
40 |
July 5, 2021 |
Static theory and Background II |
9:30-10:30 |
40 |
July 7, 2021 |
Open problems about LCE I |
8:30-9:30 |
40 |
July 7, 2021 |
Open problems about LCE II |
9:30-10:30 |
40 |
July 9, 2021 |
The method I |
8:30-9:30 |
40 |
July 9, 2021 |
The method II |
9:30-10:30 |
40 |
July 12,2021 |
Hamonic map I |
8:30-9:30 |
40 |
July 12, 2021 |
Hamonic map I |
9:30-10:30 |
40 |
July 14, 2021 |
Application |
8:30-9:30 |
40 |
July 14, 2021 |
Summary |
9:30-10:30 |
40 |
Lecture 1: Static theory and Background I
Lecture 2: Static theory and Background II
Lecture 3: Some problems of the Oseen-Frank model
Existence and regularity of Oseen-Frank energy minimizers
Lecture 4: Some problems of the Ericksen-Leslie mode
compensated compactness property
Lecture 5: The method I,
Direct methods in proving the minimize of the energy functional.
Lecture 6: The method II,
Compasion Principle
Lecture 7: Hamonic map I
Holder continuity, twice differential
Lecture 8: Hamonic map II
Some properties of Harminic Map, Myers estimates
Lecture 9: Application
The latter concerns the hydrodynamic evolution of liquid crystals, that can be viewed as a dissipative system coupling between the underlying fluid (via the Navier-Stokes equation) and the liquid crystal orientation field (via the heat flow of harmonic maps).
Lecture 10: Summary
報告人簡介：
王長友，美國普渡大學教授，從事偏微分方程和幾何分析的研究，在CPAM， AMRA, CPDE, JDE, SIAM J. Math. Anal., JFA, CVPDE等雜志上發表多篇高水平論文。
