I'm currently working through some lectures notes by Tsuyoshi Yoneda on the "Illposedness of the Incompressible 2D Euler Equations" (the lectures themselves are here but I'm unsure about the lecture notes, as I received them from one of my professors, but they're presumably somewhere on the MSRI website). In these notes, Yoneda provides a simplified proof for the 2D case after Bourgain-Li here
We are in $\mathbb{R}^2$ as the title of the lectures notes implies. $u$ denotes the velocity in the 2D Euler Equation: $$\partial_t u + (u \cdot \nabla)u + \nabla p = 0 \qquad t \geq 0, x \in \mathbb{R}^2$$ $$\nabla \cdot u = 0 \quad u(0) = u_0$$ On the first page of the notes, Yoneda defines the 2D vorticity as follows:
$$\omega = \text{rot } u = -\frac{\partial u_1}{\partial x_2} + \frac{\partial u_2}{\partial x_1}$$ He also states the Biot-Savart law as follows $$u_1(t,x) =\frac{1}{2\pi} \int_{\mathbb{R}^2} -\frac{y_2}{\lvert y \rvert^2} \omega(t, x-y) \text{d}y$$ $$u_2(t, x) = \frac{1}{2\pi} \int_{\mathbb{R}^2} \frac{y_1}{\lvert y \rvert^2} \omega(t, x-y) \text{d}y$$ As an exercise, Yoneda asks the following: By using the above formula for $u$ (the Biot-Savart Law), show that $\omega = -\frac{\partial u_1}{\partial x_2} + \frac{\partial u_2}{\partial x_1}$
I thought I had understood this, but I realized the work I did initially was completely wrong. Now, what I have is the following, which is all from direct calculation $$-\frac{\partial u_1}{\partial x_2} + \frac{\partial u_2}{\partial x_1} =$$ $$ -\frac{\partial}{\partial x_2} (\frac{1}{2\pi} \int_{\mathbb{R}^2} -\frac{y_2}{\lvert y \rvert^2} \omega(t, x-y) \text{d}y) + \frac{\partial}{\partial x_1} (\frac{1}{2\pi} \int_{\mathbb{R}^2} \frac{y_1}{\lvert y \rvert^2} \omega(t, x-y) \text{d}y) = $$ $$\frac{1}{2\pi}[\int_{\mathbb{R^2}} \frac{y_2}{\lvert y \rvert^2} \partial_{x_2}\omega(t, x-y) \text{d}y + \int_{\mathbb{R^2}} \frac{y_1}{\lvert y \rvert^2} \partial_{x_1}\omega(t, x-y) \text{d}y]$$ but from here, I don't know how to further simplify the expression or retrieve $\omega$ from its partial derivatives. I've tried expanding these expressions explicitely into double integrals and combining them into one expression, but neither of these seemed to do much except for confuse me.
For some background, I'm doing this as part of a project in my PDEs course. This is a graduate level course, but is my first course in PDEs, so I am not very well-versed in the subject. I had mostly been focusing on trying to grasp the general contours of Yoneda's argument and had glossed over this exercise until now thinking it was a straightforward direct computation, which I'm sure it is for someone more comfortable manipulating these expressions, but I see now it is a roadblock for me and something I should have addressed sooner. I'll be working more closely with my professor in the coming weeks, but this seems like a calculation I should be able to do before then. In any case, I say all this because
- I'd like to be able to do this as independently as possible, so if it is possible to only solve one of the integrals or only provide a hint as to what direction to go in, I would appreciate that immensely
- if a common technique in PDEs is used that someone new to the subject may not know, some type of source or explanation would be much appreciated. $$$$ Thank you in advance.
Since you asked for hints, here are 3:
$\partial_{x_1} f(x-y) = -\partial_{y_1} f(x-y)$.
Changing $y$ to polar $(r,\theta)$ coordinates, $y_1\partial_{y_1}+y_2\partial_{y_2} = r\partial_r$.