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.
01
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.
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Contact: Pablo Álvarez (pacaudev@math.uc3m.es), Cristina Brändle (cristina.brandle@uc3m.es)
02
Elliptic equations: from Differential Geometry to Quantum Mechanics through the Calculus of Variations:
1) Differential Systems of subcritical or critical elliptic equations derived from the famous Schrödinger equation
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 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 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.
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Contact: Alejandro Ortega (alortega@math.uc3m.es), Eduardo Colorado (ecolorad@math.uc3m.es)
03
Elliptic equations: from Differential Geometry to Quantum Mechanics through the Calculus of Variations:
2) Analityticity or regularity of solutions
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?
These are the questions that Hilbert, in his 20th and 19th problems, raised. Answered in the affirmative in the 50's (independently by E. De Giorgi and J. Nash - "A beautiful mind"), this study developed techniques that have been successfully applied to important problems.
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Contact: Alejandro Ortega (alortega@math.uc3m.es), Eduardo Colorado (ecolorad@math.uc3m.es)
04
Elliptic equations: from Differential Geometry to Quantum Mechanics through the Calculus of Variations:
3) Elliptic critical problems whose origin goes back to important questions of
differential geometry raised in the 60's
The well-known Yamabe problem is: "to determine whether given a Riemann manifold there exists a conformal metric such that the scalar curvature in the new metric is constant”. This problem gives rise to problems typically known as Brezis-Nirenberg type problems.
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Contact: Alejandro Ortega (alortega@math.uc3m.es), Eduardo Colorado (ecolorad@math.uc3m.es)
05
Optimal windows
The problem we have in mind was stated by Denzler (1996): “Suppose you are in a thermically well-insulated room, except for small holes in the insulation, which we will call windows, although in reality, cracks near the frame of a window may be more responsible for the loss of heat. What can be said about the loss of heat through a noninsulated window? How does it depend on the area of the window e.g., asymptotically for small area, and how does it depend on its geometry? What kind of geometry of the window would be preferable to minimize the heat-leak rate fixing the area?”
Having this problem as our starting point, we consider different versions of it. For example: what happens inside a room in which a window is periodically opened and closed? What can we say about windows in more dimensions? Is there an optimal shape for the window?
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Contact: Pablo Álvarez (pacaudev@math.uc3m.es), Cristina Brändle (cristina.brandle@uc3m.es), Alejandro Ortega (alortega@math.uc3m.es), Eduardo Colorado (ecolorad@math.uc3m.es)
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