Before the derivation from a physical view, we'd like introduce the full compressible Naiver-Stokes(-Fourier) system in $\Omega \subset \R^d$ directly: \begin{equation} \label{NSF} \begin{cases} \partial_t \rho +\nabla \cdot (\rho u )=0, \\ \partial_t (\rho u)+\nabla \cdot (\rho u\otimes u ) = \rho g+ f+ \nabla \cdot \sigma,\\ \partial_t ( E)+\nabla \cdot ( Eu)+\nabla \cdot q=\rho g\cdot u+ f\cdot u+\rho h+\nabla\cdot (\sigma\cdot u ) \end{cases} \end{equation} usually with rheological assumption: \begin{equation} \label{rheological} \sigma =-pI+\tau, \end{equation} and total energy \begin{equation} \label{total energy} E=\frac 12\rho |u|^2+\rho e. \end{equation} The physical meanings of quantities in the system are listed below:

Here we introduce the cylindrical coordinate system for vector fields to describe axisymmetric flows, and derive the expressions of classical differential operators in cylindrical coordinates. Coordinate transform Denote the standard basis as $e_1,e_2,e_3$ and $x_1,x_2,x_3$ the coordinate in $\R^3$. Let $$(r,\theta,h)=\br{\sqrt{x_1^2+x_2^2}, \arctan \br{\frac {x_2}{x_1} },x_3}\Longleftrightarrow (x_1,x_2,x_3)=(r\cos \theta, r\sin \theta,h).$$This induce the Jacobi matrix between $(x_1,x_2,x_3)\sim (r,\theta,z)$ and its inverse \begin{equation*} J\eqqcolon \begin{pmatrix} \dfrac{\partial r }{\partial x_1 } & \dfrac{\partial r }{\partial x_2 } & \dfrac{\partial r }{\partial x_3 }\\ \dfrac{\partial \theta }{\partial x_1 } & \dfrac{\partial \theta }{\partial x_2 } & \dfrac{\partial \theta }{\partial x_3 } \\ \dfrac{\partial h }{\partial x_1 } & \dfrac{\partial h }{\partial x_2 } & \dfrac{\partial h }{\partial x_3 } \\ \end{pmatrix} = \begin{pmatrix} \cos \theta & \sin\theta & 0 \\ -\dfrac{\sin \theta }{r} & \dfrac{\cos \theta }{r} & 0\\ 0 & 0 & 1 \end{pmatrix} \Longleftrightarrow J^{-1}= \begin{pmatrix} \dfrac{\partial x_1 }{\partial r } & \dfrac{\partial x_1 }{\partial \theta } & \dfrac{\partial x_1 }{\partial h } \\ \dfrac{\partial x_2 }{\partial r } & \dfrac{\partial x_2 }{\partial \theta } & \dfrac{\partial x_2 }{\partial h } \\ \dfrac{\partial x_3 }{\partial r } & \dfrac{\partial x_3 }{\partial \theta } & \dfrac{\partial x_3 }{\partial h } \\ \end{pmatrix}= \begin{pmatrix} \cos \theta & -r\sin \theta & 0\\ \sin \theta & r \cos \theta & 0\\ 0 & 0 & 1 \end{pmatrix}. \end{equation*}And we choose the another orthonormal basis as \begin{equation} \label{cylinderical-jacobi} \begin{pmatrix} e_r \\ e_\theta \\ e_h \end{pmatrix} =\begin{pmatrix} \cos \theta & \sin \theta & 0\\ -\sin \theta & \cos \theta & 0\\ 0 & 0 &1\\ \end{pmatrix} \begin{pmatrix} e_1\\ e_2\\…

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 Hilbert transform is formally defined as \begin{equation*} Hf (x) \coloneqq \text{P.V.} \frac 1{\pi x} * f (x)=\frac 1\pi \int_{-\infty}^{+\infty} \frac{f(y)}{x-y}dx. \end{equation*} Some important facts are listed as follows. The Fourier transform of the kernel \begin{equation*} \Fi\br{\text{P.V.} \frac{1}{\pi x} }=-i \sgn . \end{equation*} We have identities for the Hilbert transform: \begin{equation*} \begin{split} H(Hf)=& -f,\\ H(fg)=& fHg+gHf+H(Hg Hf ),\\ H(fHf)=& \frac12 \br{(Hf)^2-f^2 } . \end{split} \end{equation*} We gives out some classic Hilbert transform: $H (\sin x)= -\cos x, H(\cos x)=\sin x.$ $H \br{\frac{1}{1+x^2}}=\frac{x}{1+x^2}, H\br{\frac{x}{1+x^2} }=\frac{-1}{1+x^2}.$ This lecture is unfinished yet, I will add more contents later. And anyone who offer the detailed calculation of the second one will be appreciated.

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