Grupo de Ecuaciones Diferenciales y Aplicaciones (GEDA)

Many phenomena of real life, experimental sciences and technology are modelled using partial differential equations (PDEs): as examples all those involving atoms and molecules, fluids, weather, financial markets, populations, …

In GEDA we deal with the mathematical study of some priority scientific questions related to different aspects of physics and ecology. Under this premise, we present different proposals with differentiated starting points, but with common threads. On the one hand, its applicability, and its clearly innovative character, putting mathematics together with other sciences of special relevance in the present day. On the other hand, its theoretical development, combines the study of non-local or higher order operators, reaction-diffusion equations, different boundary conditions and the development of numerical models that validate the results.

Biological and Ecological models

We propose to analyse biological and ecological models both theoretically and numerically, to find an answer to some crucial questions such as: how do populations interact? What happens if they live in separate regions and interact through a boundary? Is it possible to find safe areas?

Theoretical analysis of these and related questions should provide us with a series of protocols to increase species productivity or, perhaps, to avoid unpleasant situations in which productivity is overestimated, ultimately causing species extinction. These issues
seem to be crucial nowadays given the situation we live in because of the industrial revolution, the development of medicine, transports or communications, which among other things have shaped the welfare state and the exponential growth of the population, with the ecological consequences that this entails.

From the mathematical point of view, we would like to point out that it is a novel proposal to combine in the same problem the study of the behaviour of populations
with heterogeneities (spatial or temporal heterogeneities) with nonlocal or higher-order diffusions. To date, there is no literature on this subject; however, we believe that this type of approach will allow us to design more realistic models.

Contact: Pablo Álvarez-Caudevilla (pacaudev@math.uc3m.es), Cristina Brändle (cristina.brandle@uc3m.es)

Quantum Mechanics through the Calculus of Variations

Many physical models, such as in nonlinear optics or in multispecies Bose-Einstein condensates (BEC), consist of systems of Schrödinger-type equations coupled through nonlinear terms, or coupled nonlinear Schrödinger-Korteweg de Vries equations related with the study of Tsunamis. If one considers a particular type of solutions, the solitary wave solutions, then these systems give rise to elliptic systems with nonlinear coupling.

The story of “BEC” goes back to the 1920's when Satyendra Nath Bose (1894-1974) sent a paper to be published in a scientific journal and it was rejected. Bose revised his paper and sent it to Albert Einstein (1879-1955) in 1924. Einstein saw the potential of the investigation by Bose and extended its results with new applications. This led them to meet in Berlin 1925 and to finish the definition of the known “Bose-Einstein Condensates” model. The BEC were shown experimentally in 1990’s by the German Wolfgang Ketterle and the Americans Eric A. Cornell and Carl E. Wieman. In 2001, Ketterle-Cornell-Wieman won the Nobel Prize in Physics by the conquest of the “BEC”, the fifth state of matter which holds at ultra-cold temperature in which atoms behave like a unique super-atom following the quantum-mechanic laws. That is, in the BEC state, all atoms condensate into a so-called quantum-mechanical ground state.

Inspired by previous works, we will deal with systems of two nonlinear Schrödinger equations or systems with one of the components being nonlinear Schrödinger equation while the other is a Korteweg de Vries equation, defined in bounded domains under mixed boundary conditions.

Contact: Pablo Álvarez-Caudevilla (pacaudev@math.uc3m.es), Eduardo Colorado (ecolorad@math.uc3m.es)

Optimization Problems

The calculus of variations is a field that deals with the optimization of certain types of functions called functionals. These functionals, related to the energy of a certain system, appear when considering, for example, minimal surfaces, transport optimization or some optical phenomena and, therefore, one type of interesting solutions are the so-called minimum energy solutions or minimizers. Two questions naturally arise: Do solutions exist in general? And if so, are the solutions of regular problems in the calculus of variations always necessarily regular?

Contact: Pablo Álvarez-Caudevilla (pacaudev@math.uc3m.es), Eduardo Colorado (ecolorad@math.uc3m.es)

Problems assuming membrane regions

We consider systems of reaction-diffusion equations defined in different domains and coupled only through a common boundary. The examples we are dealing with model, for example, ecological or migratory issues, in which different populations live separately, in different regions of a territory, and interact only at a common border; proteins crossing a biological membrane in a cell; or problems of quantum mechanics where we find different quantum systems on different sides of the domain, while connected by the interaction through a barrier acting as insulation region. As a canonical model, we first consider the Kedem-Katchalsky conditions which were introduced in a thermodynamic context. They are represented by the equality of the flow going through a membrane and are proportional to the difference of densities on both sides of the domain. The main aim will be to to provide answers concerning the existence, multiplicity and regularity of solutions of systems involving those conditions on the interface region.

Contact: Pablo Álvarez-Caudevilla (pacaudev@math.uc3m.es), Cristina Brändle (cristina.brandle@uc3m.es)

Optimal transportation: theory and applications

We propose the study of optimal transportation and gradient flow methods for partial differential equations (PDE), differential geometry, functional inequalities, and/or applications. The term “optimal transportation” comes from the classical problem of how best to transport mass from one configuration to another, and in mathematical language is posed in terms of minimizing a cost functional. This is a modern and multidisciplinary branch of mathematical analysis that has recently achieved great recognition, as evidenced by the Fields Medal awarded to Alessio Figalli in 2018. The theory of optimal transportation has also found applications in topics such as image processing and machine learning.

This thesis aims at first obtaining a solid understanding of the theory of optimal transportation, where metric measure spaces, functional analysis, and some differential geometry naturally appear, before studying one or more applications, preferably the proofs of existence of solutions to evolution PDE, the derivation of sharp functional inequalities, or image analysis.

Contact: Troy W. Petitt (tpetitt@math.uc3m.es)

Geometric analysis using Ricci curvature

We propose the study of elliptic partial differential equations on Riemannian manifolds, and the ubiquitous role that Ricci curvature lower bounds and the Bochner formula plays. Such topics constitute an extremely active field of mathematics, bridging the gap between analysis and geometry.

The first goal is to obtain a working familiarity of Riemannian geometry concepts and definitions before quickly passing to study the key tools that allow one to study PDEs in such spaces, i.e. the Bochner formula, model manifolds, and various comparison theorems. Finally, using these tools we propose to study some specialized topics, such as splitting theorems, rigidity theorems, or Ricci flow.