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…
We investigate into the local well-posedness of incompressible fluid equations. The method adopted in this lecture is different from mild solution, since the dissipation does not occur in the Euler flow. We will establish the claim by the regular approximation and Picard contraction.
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.
Mild solution is an important object to investigate the strong solution of hydrodynamic flows with dissipation. The investigation will apply semigroup theory and some other functional tools.