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…

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The Hilbert transform arises when we investigate $\nabla u $ in the 1D vorticity equation, which behaves similarly as $H\omega$. We will introduce the basic properties of Hilbert transform and apply it to solve ODE containing $H$.