This seminar is about the singularity of PDEs in hydrodynamics and quantum mechanics, led by Prof. Huang's group. The relevant informations and sources(including recording videos and documents) will be updated in this post. Contents Our seminar proceed as follows: Ni will mainly introduce the recent works by Merle-Raphaël-Rodnianski-Szeftel, which worked out the Millennium problem in compressible version. Some related topics, like shock waves and other types of singularity, will also considered in our seminar. And Meng and Gu will lecture basic theory of elliptic equations. See references at the end. Schedule and sources We schedule on every Friday, 2:00-5:30 pm, located at Siyuan building S615, AMSS, CAS. No. Schedule Topic Lecturer Notes Records Location 1 24.03.08 Singularity & Stability A. Ni written typed offline S615 2 24.03.08 Laplace's equation 1 Y. Gu typed offline S615 3 24.03.15 Smooth implosion 1 A. Ni written offline S615 4 24.03.15 Moser iteration W. Meng typed offline S615 5 24.03.22 Smooth implosion 2 A. Ni written offline S615 6 24.03.22 Spherical blowup 1 Y. Gu written offline S615 7 24.03.29 Smooth implosion 3 A. Ni written offline S615 8 24.03.29 Perron's method W. Meng written offline S615 9 24.04.05 Smooth implosion 4 A. Ni written offline S615 10 24.04.05 Newtonian Potential W. Meng written offline S615 11 24.04.12 Smooth implosion 5 A. Ni written offline S615 12 24.04.12 Iteration method Y. Gu written offline S615 13 24.04.19 Smooth implosion 6 A. Ni written offline S615 13 24.04.19 Laplace equation 4 W. Meng written offline S615 14 24.04.26 Smooth implosion 7 A. Ni written offline…

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.

We will introduce the construction of $k-$solitary waves for focusing NLS in enery sub/super/critical cases respectively, and investigate the blowup phenomeon developed by solitons.

This lecture covers mainly the singularity works On the effect of advection and vortex stretching and Finite time singularity formation for $C^{1,\alpha}$ solutions to the incompressible Euler equations in $\R^3$ by Elgindi. In the former work Elgindi investigates a blowup mechanism for a simplified 1D model for Euler equation, and then he applies this mechanism to 3D Euler case in the latter one. Conclusively, he constructs a self-similar type blowup vorticity formulated as \begin{equation*} \omega(x,t)=\frac{1}{1-(1+\mu)t}F\br{\frac{|x|^\alpha}{(1-(1+\mu)t)^{1+\lambda}} } \end{equation*}for the 3D incompressible Euler flow with initial data $\omega_0\in C^\alpha$($\alpha$ small enough), where $F$ is a bounded profile. The basic idea is to investigate a fundamental model(which preserves the singularity but is easy enough to solve out the explicit solution), and analyze the singularity under a perturbation by the advection term $u\cdot \nabla \omega$ up to some bounded quantities. Some further discussions and open problems will also be discussed in related topics. 1 Introduction We recall that the incompressible Euler flow is governed by the following equations \begin{equation*} \begin{cases} \partial_t u+u\cdot \nabla u +\nabla p=0,\\ \nabla \cdot u=0, \end{cases} \end{equation*}here $u(x,t)$ is the velocity and $p$ is the pressure. The classic open problem is whether a $C^\infty$-initial data $u_0$ can develop a global smooth formulation. A well-known Beale-Kato-Majda criterion tells us that a classic solution loses its regularity as $t\to T$ iff the vorticity $\omega\coloneqq \nabla \times u$ satisfies \begin{equation*} \lim_{t\to T} \norm{\omega}_{L^1(0,t; L^\infty )}=+\infty. \end{equation*}Consequently, it is natural to consider the vorticity equation\begin{equation*} \partial_t \omega+ u\cdot \nabla \omega=\omega \cdot \nabla u, \end{equation*}and investigate the blowup mechanism of $\omega$. The term $u\cdot \nabla…

The C-K-N theory, originated from the paper in 1982, study the singular behavior of weak solutions of the incompressible Navie-Stokes equations. In this lecture, we adapt the compactness method by Lin's paper to prove the whole theory.