January 2012.  The 2D Euler equation can be written in the following form

$\dot \theta=\nabla \theta\cdot \nabla^\perp \Delta^{-1}\theta \quad\quad (1)$

where $\theta(x,y,t)$ is $2\pi$— periodic in both $x$ and $y$ and it denotes the vorticity of the velocity field. The symbol $\nabla^\perp$ stands for $(\partial_y, -\partial_x)$. To define the Cauchy problem, one needs to specify $\theta(x,y,0)=\theta_0(x,y)$

This is a transport equation with the divergence-free vector field and so all $L^p$ norms of $\theta$ are preserved in time. The global regularity is known for many functional spaces, e.g. $C^\infty$ or $H^s$ with large $s$ will do. The quantitative version is a bound

$\|\theta(t)\|_{H^s}\lesssim (1+\|\theta(0)\|_{H^s})^{e^{Ct}}, s\geq 2$

Is this estimate sharp? I do not know but one can have double exponential growth for arbitrarily large but finite time. The lower bound valid for all time is only superlinear.

Another interesting problem is dynamics of patches. Assume that $\theta_0(x,y)$ is the characteristic function of a domain $\Omega(0)$ and the problem is considered on the whole plane, not on the 2d torus. Then, $\theta(x,y,t)=\chi_{\Omega(t)}$ and the boundary of $\Omega(t)$ is smooth provided that the boundary of $\Omega(0)$ was smooth. However, one can study the growth of curvature and the rate of merging (if there are several patches).

The merging mechanism was studied recently and the INFINITE double exponential rate is possible at least if the regular strain is present. This result is sharp.

If one modifies equation $(1)$ for vorticity by writing $\Delta^{-1/2}$ instead of $\Delta^{-1}$, then the resulting equation is called SQG (surface quasi-geostrophic). The outstanding conjecture in the field is that the singularity forms in finite time. No proof so far.

_____________________________________________________________________

May 2013. I have finished one more revision of the paper on the corner formation and you can check either my webpage or arxiv. The estimates are now sharp and the regularity of the strain is as optimal as the method can give. I think I approach the stage when the paper can finally be read by the students.

_________________________________________________________________________

October 2013. In the recent preprint, Kiselev and Sverak considered the 2d Euler equation on the disc and proved the infinite in time double-exponential growth of the Lipschitz norm for the vorticity. The singularity forms on the boundary and happens as two identical vortices of different sign slide along the boundary towards each other. This interesting result might indicate that singularity formation is generic when the equation is considered on the domain with the boundary. The initial data is smooth but is not zero near the boundary (this is important for the construction). So, if one writes the equation on the Fourier side, the decay of

$\hat \theta(n,t)$

in $n$ is very weak. The stable mechanism for the singularity formation in the bulk is still missing.

________________________________________________________________

October 2013. Another interesting result obtained for 2d Euler. My colleague Andrej Zlatos proved the possibility of infinite exponential growth of the Hessian of vorticity for smooth initial data. This was done for the 2d torus and no boundary effects were used. He also showed that the Lipschitz norm can grow exponentially for $C^{1,\alpha}$ initial data. In this work the hyperbolic scenario is exploited. I think the interesting question is the possibility of the exponential growth of the Lipschitz norm for smooth initial data (e.g., trig polynomial). So far we have only superlinear estimate. Even better problem is to show that the sharp front can form with width decaying exponentially. This would prove that the results of Fefferman and D. Cordoba on the fronts are essentially sharp. There are plausible mechanisms but the justification for Euler equation is hard.