## Vega, Luis

#### BCAM UPV/EHU Research Professor, Group Leader

Linear and Non-Linear Waves
##### Helbidea

Alameda Mazarredo 14. 48009 Bilbao Basque Country - Spain

##### Harremana

My research is mainly focused in the interplay of Fourier Analysis and Partial Differential Equations of Mathematical Physics. More recently on fluid mechanics and turbulence. Concretely in the so called Localized Induction Approximation, also known as the binormal curvature flow (BF), for the evolution of vortex filaments and the relevance of the presence of corners in the filament. The results concerning regular polygons seem to me quite striking. Motivated by a numerical experiment done by D. Smets, and together with F. De La Hoz, we established in 2014 a numerical connection between the trajectory followed by a corner of, say an equilateral triangle, and a classical analytical problem that goes back at least to Riemann: the existence of continuous functions which are no where differentiable. Very recently (arXiv:2007.07184), and in collaboration with V. Banica, we have proved analytically that this connection is indeed true.

Right now I am also working together with N. Arrizabalaga and A. Mas, on relativistic and non-relativistic equations with singular electromagnetic potentials. The singularities of the potentials are critical from the point of view of the scaling symmetry. In the relativistic setting we consider perturbations of Dirac equation given by singular measures supported on smooth hyper-surfaces. This mathematical problem is closely related to a relevant question in physics, that of the optimal confinement of relativistic quantum particles.

Finally I continuous working on the deep connection between uncertainty principles, that are easily described using the Fourier transform, and lower bounds for solutions of linear and non-linear dispersive equations. This is a topic that I started with L. Escauriaza, Carlos E. Kenig and G. Ponce more than 10 years ago and from which very fruitful branches have emerged. For example, one of the first consequences we obtained using these lower bounds, was that a compact perturbation of a solitary wave or soliton of the Korteweg-De Vries (KdV) equation instantaneously destroys its exponential decay. KdV is a simplified local model about the dynamics of the frontier of a fluid. In particular, it describes with very high accuracy the propagation of a wave along a narrow and sallow channel. However, when the depth is big so that it can be considered close to be infinite the local approximation is too rough and non-local models as the Benjamin-Ono equation has to be considered. It turns out that the answer to the corresponding question requires completely different techniques that are closer to those developed with A. Fernández-Bertolin for the discrete laplacian and with L. Roncal and D. Stan for the fractional laplacian.