Seminar on unbounded operators

This seminar is mainly about the theory of unbounded operators and its application to the PDEs and dynamic systems. We are scheduled to begin at mid-July, 2023, where lecture sources and meeting videos will be uploaded in this post. Our current plan is to meet every Monday from 3 p.m. to 5 p.m.(see meeting id at the last part), but there may be adjustments depending on the progress.

References

[1] 张恭庆、郭懋正：泛函分析讲义（下册），北京大学出版社，1990；
[2] K. Yosida, Functional analysis, second edition, Springer-Verlag, 1968.
[3] B. Helffer, Spectral theory and its applications, Cambrige university press, 2013.

I learned this course given by Prof. Zhang Zhifei in 2021 antumn. A anynomous student gergorously shared his notes, which strictly followed Prof. Zhang's teaching schedule, so it is highly worth referring to.

Main topics

It is necessary to introduce the relevant fundamental topics in advance, so that we have a rough framework in mind for the content of this seminar. Particularly, the Laplace operator $\Delta$ and its variants will always serve as basic examples for applications of the operator theory.

1. Basic notions about unbounded closed operators, and its self-adjoint extension.
2. Spectral analysis, and two characterizations of self-adjoint operators,
3. Functional calculus, and the applications of spectrum theorem.
4. Peturbation theory, and the self-adjointness of Schrodiner operator.
5. Semigroup theory, and the mild solutions of evolutional PDE.

Besides, some extra topics will be given properly:

6. Analytic semigroup, and the well-posedness of mild solutions of incompressible Naiver-Stokes equations.
7. Pseudo-spectrum, and the stability theory of hydrodynamic flows.
8. Spectral gaps, and singualrity of the defocusing nonlinear Schrodinger equation.

Uploaded sources

The sources will be upoladed within two days after the seminar except the typed notes, which may requires additional time to be done. Please download the video recordings ASAP if you need, as the sharing links have a one-month time limit. (though you can ask privately from me by email or wechat)