In this talk, we consider a fractional nonlinear wave equation with a general power-type nonlinearity (FNLW) on the two-dimensional torus. Our main goal is to construct invariant global-in-time Gibbs dynamics for FNLW. We first construct the Gibbs measure associated with this equation. By introducing a suitable renormalisation, we then prove almost sure local well-posedness with respect to Gibbsian initial data. Finally, we extend solutions globally in time by applying Bourgain’s invariant measure argument. This talk is based on a joint work with Luigi Forcella (University of Pisa, Italy).

In this talk we are interested in compactly supported solutions of the steady Euler equations. In 3D the existence of such solutions has been an open problem until the recent result of Gavrilov (2019). In 2D, instead, it is easy to construct solutions via radially symmetric stream functions. Low regularity solutions without radial symmetry have been found in the literature, but even the $C^1$ case was open. In this talk we construct such solutions with regularity $C^k$, for any fixed $k$ given. For the proof, we look for stream functions which are solutions to non-autonomous semilinear elliptic equations. In this framework we look for a local bifurcation around a conveniently constructed 1-parameter family of solutions. This is joint work with A. Enciso (ICMAT, Madrid) and Antonio J. Fernández (UAM, Madrid).