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).

The reconstruction in the Calderon inverse conductivity problem is considered. A suitable linearization of the inverse problem allows us to define the so-called Born approximation. This provides a good reconstruction only for the high frequencies of the conductivity. In a recent work, we established a new formula relating this Born approximation with the Dirichlet to Neumann map and the momenta of the conductivity coefficient. Based on this formula, we give a new convergent iterative algorithm that improves the numerical reconstruction given by the Born approximation. We illustrate the process with numerical experiments for radial and non-radial potentials.

 This is part of a collaboration work with J.A. Barceló, F. Macià, D. Mendoza,  and C. Meroño.

 The Hansen and Spies laminated beam model is a model that de-

scribes the interaction between two identical Timoshenko beams that are bonded together by means of an adhesive layer of negligible thickness and mass, allowing interfacial sliding. It is a system of three coupled hy-perbolic equations. In this lecture, we will present several recent results on the stability and lack of stability of the semigroup associated with this model when subjected to the action of various dissipative mechanisms.

In the 1970s, H.O Fattorini and D.L Russell, [4], [5], linked the method of moments to the study of the controllability at time T > 0 of the heat equation. The basic argument is the construction and estimation of bi-orthogonal families in L2(0T) a {exp- landakt, k>= 1} where landak, k>= 1 is the sequence of eigenvalues of Sturm Liouville operators. Muntzs theorem and Weyls formula require us to restrict ourselves to the dimension 1 of space. In recent years works such as, for example, [1], [3], [2] have shown that even in the case of parabolic equations, controllability to 0 at time T could require T >=T0 with T0 > 0. These results, based on high frequency phenomena, are obtained by deepening the work of [5] on the construction and estimation of families bi-orthogonal in L2(0T) a { exp- landakt,landa in F} where F contained in C. Consequently, they are restricted to the dimension 1 of space. The objective of this talk is to take a rst step in the construction of bi-orthogonal families in L2((0T)Xw) (in space-time) a  { exp- landakt,landa in F} . F in R, Phi landa,landa in F contained in L2(Omega ),w contained in Omega,Omega contained in  Rd,d >=1.

 I’ll present the so called Localized Induction Approximation that de- scribes the dynamics of a vortex filament according to the Binormal Curvature Flow (BF). I’ll give a result about the desingularization of the Biot-Savart in- tegral proved with Marco A. Fontelos within the framework of Navier-Stokes equations. Some particular examples regarding BF obtained with Valeria Ban- ica will be also considered. These examples allow to connect BF with the so- called Riemann non-differentiable function and the Frisch-Parisi approach to turbulence.