Mathematical and numerical analysis of some partial differential equations and their applications

Objective:

The current project focuses on several mathematical aspects of physical systems that can be modeled by partial differential equations and which applicability ranges from quantum physics to fluid dynamics and economy. An essential part of the current proposal considers theoretical questions related to uniquene continuation and control in different parabolic and elliptic equations. A main tool that we use extensively use are Carleman estimates. The application of these techniques to prove uncertainty principles, which one of the classical questions in Fourier Analysis, has been a breakthrough obtained by our group in recent years. Our impression is that we have just seen the tip of the iceberg and thus we propose to go as deep as possible into the subject. A second part is concerned with the study of evolution problems. In particular, we will study some fundamental equations of Mathematical Physics such as Dirac and Schrödinger equations both in the linear and the non-linear setting. We will look at some spectral problems related to these equations. We will study Oscillatory Integrals and Fourier Integral Operators (FIO) and its connection with some classical maximal functions and its application in the regularity of some PDEs. Also some applications will be considered, as, for example, the evolution of vortex filaments and its connection with turbulence and the possibility of confinement for relativistic particles. Other problems, concern the study of viscous flows, such as the existence of some relevant self-similar solutions of the thin-film equation, in particular those that describe lifting and rupture of the film. Analysis of PDEs with fractional diffusion of porous medium type and the role of non local diffusion terms in combination with dispersive terms will also be subject of study. In many of the problems already mentioned, numerical computations represent a complementary technique that is useful in the analysis of the underlying properties of solutions. However, we are also interested in some theoretical questions of numerical analysis such as spectral and pseudo-spectral methods for PDEs posed in unbounded domains.