Concentration phenomena in nonlinear partial differential equations.

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.

Partial Differential Equations (PDEs) are key equations for anyone wanting to use mathematics to solve real life problems. Mathematical descriptions of continuous systems are typically phrased in terms of rates of change, or derivatives. Many laws in Physics, Biology, Ecomonics, Social Studies, can be formulated as PDEs. To mention some examples: Newton's laws of motion, non-linear diffusion equations that describe density fluctuations in a material undergoing diffusion, parabolic equation of dissipation type to describe movement of prices in economics, the Fokker-Planck equation for the evolution of individuals in a population, among others. Another interesting feature of PDEs is their universal applicability and their flexibility in allowing us to model and understand changes in different systems.
My project focuses on the theoretical analysis of important classes of PDEs, and on the formation of possible singularities in their solutions. A singularity is a location in space and time where some relevant quantity related to the equation becomes infinity, in a way that does not depend on the chosen coordinate system. Detecting singularities has an important impact in the applications the PDEs are modeling.
As an example, in aerodynamics, a mayor issue is the optimal design of a wing or an airfoil. The designer seeks to optimize the geometric shape of a configuration taking into account the trade-offs between aerodynamic performance, structure weight, and the requirement for internal volume to contain fuel and payload. The subtlety and complexity of fluid flow is such that it is unlikely that repeated trials in an interactive analysis and design procedure can lead to a truly optimum design. Progress toward automatic design has been restricted by the extreme computing costs that might be incurred from brute force numerical optimization. However, useful design methods have been devised for various simplified models, such as two-dimensional wings in inviscid flows, governed by Euler equations, which are part of my research proposal. Theoretical studies of formation of singularity in the vorticity of the fluid's velocity, and of its dynamics, are fundamental to optimize the shape of a wing or an airfoil. Indeed, the aerodynamic properties of the airfoil are linked to the vortices shed at the trailing edge, and understanding these vortices is the key to quantifying lift.
Another example, in mathematical biology, is the movement of cells by chemotaxis in response to a chemical stimulus, which is widely observed in various biological situations, as morphogenesis, bacterial self-organization and inflammatory processes. The first and most studied mathematical model of chemotaxis is the Keller-Segel system of PDEs, one of the topics of my project, whose analysis has helped to understand certain characteristics of bacterial chemotaxis. Even after 40 years of research in this problem, very little is known on formation of singularities, here represented by the cells density becoming arbitrarily large.
My research will also have a strong human resource impact component. I will employ and train a highly skilled post-doctoral research associate (PDRA) who will gain new training and practice in cutting edge energy critical heat equations and in the analysis of blow-up solutions for such problems. Furthermore, the research project will benefit two doctoral students and train them similarly. These doctoral students are funded respectively by a grant from the Royal Society and by EPSRC. The PDRA and students will be part of a much larger community of mathematicians and benefit, through the attendance to seminars and other activities (such as the Integrated Think Tanks organized by the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics SAMBa based at the University of Bath), of a broad exposure to applied, computational and industrial mathematics.