Broadly speaking, the area of my project is partial differential equations (PDEs). This is the branch of Mathematics which uses the tools of calculus to model phenomena in nature. Indeed, many laws in Physics, Biology, Economics, Social Studies, can be formulated as PDEs.
The study of PDEs is a very broad field within Mathematics and can encompass both theoretical and more applied perspectives. For instance, Euler equations are a set of PDEs that describe how the velocity, pressure and density of a moving fluid are related. These equations neglect the effects of the viscosity which are included in the Navier-Stokes equations. A solution of the Euler equations is therefore only an approximation to a real fluids model. For some problems, like the lift of a thin airfoil at low angle of attack, a solution of the Euler equations provides a good model of reality. For other problems, like the growth of the boundary layer on a flat plate, the Euler equations do not properly model the problem.
My main interest is for the purely mathematical aspects of the study of PDEs. The typical questions that arise in the study of PDEs include: Do solutions of a given equation (theoretically) exist? (If not, our model is not capturing something essential.) Are they stable under perturbations of the initial data? (If not, they may be difficult or impossible to observe in nature.) Do they have some inherent symmetry that reflects the underlying physical or biological phenomena being modeled? (Nature is intrinsically economical, and often the 'simplest' solutions have the most symmetry.) Do the solutions vary smoothly over time and space, or are abrupt changes possible (what mathematicians refer to as formation of singularities in PDEs)?
In this proposal I will address all these questions for specific non-linear PDEs, with main emphasis on the last one: the mathematical analysis of formation of singularities. In many models, static or dynamic in nature, governed by non-linear PDEs, one observes the formation of singularities or some form of concentration of their solutions, as the time-variable or some parameter approaches a limit value. This happens when solutions become concentrated on lower-dimensional sets, or some expressions dependent on the solution become arbitrarily large. From a PDEs' point of view, this phenomenon reflects lack of compactness in the variational formulation of the problem or loss of regularity in the solution set, which is usually related with relevant episodes of the modeled event. Think of the explosion of some substance triggered by a chemical reaction or the appearance of fractures in planes or bridges.
We propose the construction of solutions with singularities for some significant non-linear PDEs, such as for Euler equations for incompressible inviscid fluids, for Ginzburg-Landau model in superconductivity, for sine-Gordon equations, for Keller-Segel model in chemotaxis and for the prescribed mean curvature problem. My aim is to elaborate new refined gluing techniques to carry out these constructions and to derive precise descriptions on why, where and how formation of singularities takes place. My results will be of interest not only in Mathematical Analysis, but also in Geometric Flows, Geometric Partial Differential Equations and Boundary Value Problems for Nonlinear PDE's.