Theoretical and Numerical Analysis of Evolution Equations

Objective:

The current project focuses on several mathematical aspects of the evolution of physical systems that can be modeled by partial differential equations and whose applicability ranges from quantum physics to fluid dynamics. An essential part of the current proposal considers theoretical questions related to uniqueness and inverse problems. A main tool that we use extensively are the Carleman estimates. The application of these techniques to prove uncertainty principles, which is 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 scratched the surface 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 the Dirac and Schrödinger equations, both in the linear and in the non-linear settings. We will look at some spectral problems related to these equations. Also some applications will be considered such 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 and the study of toy models of porous media flows that shall mimic some hysteresis phenomena characteristic of these flows. Analysis of partial differential equations with fractional diffusion and its role in combination with dispersive terms, as well as the study of some special solutions of nonlinear dispersive equations will also be subject to 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. Some are as classical as the successful treatment of absorption of energy at the boundary of a truncated domain for the wave equation, issue that can be addressed from the point of view of inverse problems. Other questions concern the adequacy of spectral and pseudo-spectral methods for the simulation of hyperbolic systems of nonlinear partial differential equations, being the shallow water equations over a sphere a practical example.