Description
Brief description:
Most of the models the group is dealing with are given by nonlinear partial differential equations PDEs (purely deterministic or, sometimes, involving some stochastic terms).
A first set of aims of this Program deals with mathematical intrinsic techniques like the modelisation of complex systems through many different techniques: nonlinear analysis, homogenization of nonlinear PDEs, averaged process in nonlinear PDEs, decomposition methods for Numerical Analysis, stochastic equations, integrodifferential equations, fractional equations, etc. These techniques are very powerful and generic tools that may find unexpected future applications.
Among the second block of aims consists of the application of the previous techniques to some of the above mentioned Prioritary Research Lines in the Regional, National and International calls such as, for example, Modelisation and Simulation of the Earth Climate System, Modelisation and Simulation of Elementary Particles and High Energy Physics. Applications to many other fields as, for instance, Chemical Engineering, Genetics or Economics are also regarded as future goals.
The last block of searches is the application of our techniques to problems Modelling and Simulation for Food Technologies, Medical Images and their processing.
Thus, the wide range of application goes from advance topics of the basic science (like Elementary Particles or Earth Sciences) to new technological fields with high sociological consequences (like Nuclear Fusion, Food Technologies, Animal Health or Medical Images).
Aims of the program:

Mathematical models for Lithiumion batteries in multiple portable electronic device:A problem of the models is that they need several electrochemical parameters whose values are not easy to obtain. An aim of program is to carry out a mathematical study of these models and to develop a methodology in order to identify some of these parameters.

Ecological problems related to polluted water: Mathematical modeling, analysis and numerical simulation of systems arising in various application fields such as population dynamics, ecology, phase transitions and flame propagation, as well as the transport of contaminants in aquifers.

Patterns in Complex Reaction Diffusion Systems: Many problems are characterized by the presence of moving interfaces. Spacetime phenomena giving rise to fronts and interfaces are central issues in many contexts of Science and Technology. As a matter of fact, the analysis and control of interfaces is one of the key issues of the European ITN project Fronts and Interfaces in Sciences and Technology linking 8 teams of universities of 8 countries and two industrial partners (Guigues in France and Siemens in Germany), coordinated by Prof. J. I. Diaz from UCM.

Highpressure processes and Food Engineering: Significant issues are modeling and simulation of protein behaviour under high pressure processes; optimization of high pressure  moderate temperature processes in Food Technology (for instance for enzyme inactivation and reduction of microbial population); modeling and simulation of freezing processes at high pressure; identification of parameters involved in the above models by solving mathematical inverse problems with the help of experimental data.

Animal epidemics: Topics to be addressed include modeling and simulation of Epidemics of Classical Swine Fever Virus by means of a spatialstochastic model to evaluate transmission, both within herds and between them; modeling the spread of animal diseases transmitted by vectors (i.e., organisms, often invertebrate arthropods, that transmit a pathogen from reservoir to host) as the blue tongue disease; and development of a professional software so simulate the different scenarios and the risk of animal epidemics related with the problems considered.

Controllability of nonlinearitzables problems: To show local controllability near trajectories for different PDEs which are not linearizatbles in these trajectories.

Cessation of the initial and random influences in finite time to stochastic and deterministic fast parabolic equations: study of the cessation of the initial and random influences after a stopping time in the solutions of a stochastic nonlinear problem with a random forcing term involving a sequence of independent real standard Brownian motions on a filtered probability space.

Schrödinger and GinzburgLandau nonlinear Systems: An aim is to show that if the singularity of the nonlinear potential occur for null values of the unknown then a phenomenon of localization of the support appears. The effects of the presence of memory terms in equations with delay will be also studied.

Models of systems of equations of nonlinear PDEs with different special dimension.

Very weak solutions of PDEs and singular degenerate problems.

GeodynamicsModels.

Mathematical analysis of the total variation based denoising problem: total variation flows.

Stochastic and deterministic fast parabolic partial differential equations.
Website:
http://www.ams.org/mathscinet/search/publications.html?pg1=IID&s1=57710
http://www.ams.org/mathscinet/search/publications.html?pg1=IID&s1=614471