Nonlinear partial differential equations and applications

The general theme of research study of this Group is the quantitative and qualitative properties of solutions of equations and systems of nonlinear partial differential equations, with special emphasis on the evolution equations of mathematical physics. The study of the numerical approximation of the solutions of the equations is also developed, especially in the case of fluid mechanics and in problems of calculus of variations and optimization. In particular, the numerical simulation of some physical models is performed.

We emphasize the study of the existence and possible uniqueness of weak solutions of certain nonlinear systems and the systematic approach to the analysis of the possible blow-up of smooth solutions.

As examples of equations and systems we study the Euler and Navier-Stokes equations, the boundary layer problem, variants of non-Newtonian fluids, the porous media equations, the nonlinear Schrödinger equation, parabolic and hyperbolic equations with non-standard growth conditions, quasilinear hyperbolic equations and the Oberbeck-Boussinesq systems.

Other important aspects are the study of localization and extinction properties of solutions, and the analysis of the existence and smoothness of free boundaries.

Applications of homogenization theory to elasticity problems in order to obtain the characterization of solutions to nonlinear optimization problems in terms of microstructures.

In the context of distributional solutions, problems of existence of solitary waves for certain linear and quasilinear equations are also considered.

Another research topic is about the analytical and geometrical properties of elastic bodies with defects such as fractures or dislocations.

Most of this work is performed with a strong collaboration with reference research centers in Paris, Prague, Oxford, Rio de Janeiro and St.Petersburg, among others.