This project is devoted both to the study of theoretical and numerical aspects of Partial Differential Equations (PDE), and to the numerical simulation, control and applications to optimal design, with applications to aeronautics, nanotechnology, and oil prospecting in view.

We shall also address some closely related models like Stochastic Partial Differential Equations (SPDE) and hybrid discrete-continuous kinetic equations.

In particular, we shall consider the problem of the numerical simulation and design of nonlinear waves (solitons, travelling waves, etc), of optimal shapes in aeronautics and of refrigeration systems for electronic nanodevices. We shall also investigate the difficult problem of the propagation of discrete waves on irregular meshes and heterogeneous media, the propagation and dispersion properties of discontinuous Galerkin methods and discrete Carleman inequalities.

From a more analytical and theoretical viewpoint we shall also analyze quantitative versions of isoperimetric inequalities, Carleman inequalities for parabolic equations with irregular coefficients, and the homogenization of heterogeneous media, for which particularly interesting acoustic and electromagnetic wave propagation properties have been predicted in the Physics literature. We shall also continue working in the development of a systematic counterpart of the already existing theory por PDE, where the team has been intensively active. Finally we shall address some kinetic models for coagulation-fragmentation of gases and particles, and discuss solutions and its asymptotic stability.