March 3, 2014.

If one considers the 2d Euler equation of incompressible inviscid fluids on the plane in the vorticity form and takes the initial data as the characteristic function of a certain domain, then the Yudovich theory guarantees that the solution will exist globally and will be equal to the characteristic function of a time-dependent domain which is homeomorphic to the original one for all times. The numerical experiments dating back to the works of P. Saffman and Zabusky et al. indicate the existence of the centrally-symmetric V-states, i.e. a symmetric pair of patches that rotates with constant angular velocity around the origin without changing shape. If the distance between the patches in the pair equals to $\lambda$ and $\lambda>0$, then the boundary of the V-state seems to be smooth. However, when $\lambda=0$, the both patches form a 90 degrees angle at the point of contact. The analytical proof for the existence of these V-states has never been obtained and this is an interesting problem. In the recent preprint, I considered the analogous equation with the cut-off. Loosely speaking, this corresponds to looking at the window around the origin where the contact of the patches is supposed to happen. Mathematically, the model with cut-off is important as it possesses the explicit singular solution: $y_0(x)=|x|$. Then, I addressed the problem of existence of the curve of smooth solutions that converge to $y_0$ in the uniform metric when the parameter $\lambda\to 0$.  Technically, this boils down to application of the implicit function theorem and is somewhat tedious. This technique might be important to better understand the mechanism of the merging and the sharp corner formation in the Euler dynamics. Another important problem is to prove that the merging in finite time is possible for the $\alpha$-model when $\alpha<1$ and is close to $1$.