Prelectura de tesis de Bruno Vergara

**Prelectura de tesis**

**Doctorando: Bruno Vergara (ICMAT)**

**Director: Alberto Enciso (ICMAT)**

**Tutor: Luis Guijarro (UAM)**

**Título: Weighted inequalities in Fluid Mechanics and General Relativity: Carleman estimates and cusped travelling waves**

**Lunes 25 de marzo, sala 520**

**Módulo 17, 11 horas**

**Abstract: In this seminar we will discuss some results on singular dispersive equations in two different flavors.**

**Firstly, we will see certain aspects of wave equations with coefficients that are very singular at the boundary of a manifold. Carleman inequalities, Boundary Observability, on the one hand, and Dirichlet-to-Neumann maps and fractional wave operators, on the other, are the central objects of study in this part strongly motivated by the study of geometric PDEs in General Relativity.**

**In the second part, we will deal with singular solutions to a model of shallow water waves and prove a recent conjecture in Fluid Mechanics. Here we will develop a new strategy to construct singular solutions to equations with very low regularity that shows the existence of a highest, cusped, periodic travelling wave solution to the Whitham equation that is convex between consecutive crests.**

Location Lunes 25 de marzo, sala 520 Módulo 17, 11 horas