This project is devoted on the one hand to the study of some analytic and numerical aspects of Partial Differential Equations (PDEs), and on the other to the numerical simulation, control and development of software for optimal industrial design, with particular emphasis on aeronautics, turbomachinery and electrical networks, all subjects in which the team collaborates with leading Spanish and Basque companies in the sector, such as AIRBUS-E, Baltogar S. A. or the Arteche Group. We also study models involving Stochastic Partial Differential Equations (SPDEs) and kinetic-type hybrid discrete-continuous PDEs employed in Material Sciences.

In the context of the analytic theory of PDEs, we will analyze various but interconnected issues such as Hardy and Carleman inequalities for parabolic equations with singular or discontinuous coefficients, hypoellipticity and the asymptotic behaviour of conservation laws. We will likewise develop the stochastic analogues of the existing control theory and stability for deterministic PDEs. Furthermore, we will analyze some kinetic models relevant in the study of coagulation and fragmentation phenomena of gases and particles, paying special attention to the study of global existence issues and explosion of solutions, the existence of self-similar solutions and their asymptotic stability as well as the control of these types of problems.

We will also develop a numerical analysis theory combining the most advanced and sophisticated Mathematical and Numerical Analysis tools, which enables us to generate numerical methods capable of reproducing the fine qualitative properties of the approximate PDEs, such as the velocity of propagation, dispersivity, shock location, etc.. Thus, for example, we will address the important problem of the propagation of discrete waves on irregular meshes and heterogeneous media; we will continue the study of the propagation and dispersion properties of some classes of Galerkin discontinuous methods; we will investigate the effectiveness of adaptive finite element methods for the approximation of problems with singular coefficients as well as the discrete versions of the relevant notions of hypoellipticity and hypocoercivity.

We will also tackle the control and shape optimization of fluids in the presence of singularities for the optimal design of shapes in aeronautics and turbomachinery. Analytically speaking, this is a particularly complex problem and of great importance in applications because of the extreme sensitivity that the most vital functionals such as drag and lift present in the design parameters with regard to solutions with shock discontinuities.

We will likewise analyze other problems concerning nonlinear waves (for example, solitons and travelling waves), their design, and in particular their propagation in one-dimensional networks and their numerical approximation, all topics of great importance in various fields such as irrigation, neurosciences and oceanic flow. Finally, and taking advantage of the BCAM and UPM infrastructures and the Basque computational network, we will develop a computational environment for the control and design of fluids and vibrations that will be useful in various industrial applications, and in particular in the fields of aeronautics and electrical networks, by maintaining and strengthening the existing collaboration with the R+D teams of Baltogar S.A. in the design and computation of turbomachinery, with those of the Arteche Group in the design of the most efficient electrical transport networks and AIRBUS-E in aeronautics.