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 $u\cdot \nabla \omega$ is a transport term and cannot cause any growth of the vorticity since $u$ is divergence-free. The term $\omega \cdot \nabla u$ is called the vortex stretching term and can lead to amplification of the vorticity, which is the main difficulty and causes the singularity. It is conjectured that when the transport term is weaker than the vortex stretching term, finite time singularities will occur. As Biot-Savart law gives out the representation of the velocity $u$ by the vorticity $\omega$:
$$u=-\nabla \times (\Delta^{-1} \omega )=-K* \omega, $$where $\nabla K$ is a Calderon-Zygmund kernel, we can view $\nabla u=-\nabla K* \omega\sim \omega$ with particular singularity. Omitting this singularity and the advection term, it is enlightening is reduce the vorticity equation into the following ODE (with only time variable $t$):
$$\partial_t \omega=\omega^2, $$which has explicit solution $\omega=\frac{1}{1-t}$ if equipped with unit data $w_0=1$. Based on this observation, we will consider a backward self-similar type formulation in our further discussion.
The main content of this lecture is organized as follows: In Section 2, we will consider a simplified 1D model which reveals the significant singularity of vortex stretching, and investigate the self-similar type blowup initiated above under high regular($H^k, k\ge 3$) case and low regular($C^\alpha, 0\sl \alpha \sl1$) case respectively. As the blowup is caused by $\omega \cdot \nabla u$, we take the system without advection as the fundamental model (Section 2.1) and view the advection as a perturbation under some small assumption, whose convergence can be obtained by some Hardy type estimate(Section 2.2). The idea will be applied to 3D Euler flow in Section 3. Particularly, the main extra difficulties is that the selection of fundamental model(Section 3.1) and the analysis of perturbation part will be more complex, for which we will calculate some important elliptic(Section 3.3) and coercive estimate (Section 3.2), and obtain the existence via a propri estimates and the classic compactness argument(Section 3.4-3.5). The appendix will gives some related facts.