Weak convergence methods and applications

Project outline

Summary

Weak convergences are the foundation of most successful methods in the investigation of partial differential equations today. The approach known as Tartar's programme proved particularly convenient for partial differential equations modelling physical laws; the basic assumption being that weak convergences of sequences of solutions model the transition from microscale to macroscale. Of importance are also mathematical objects being, in a generalised sense, accumulation points of weakly convergent sequences, such as defect measures or H-measures.

The research will be directed towards improving the existing weak convergence methods, designing new mathematical objects for that purpose, as well as developing new methods, with particular attention paid to their applications to practical problems.

Problems known as homogenisation are tied to properties of mixtures of materials, with important applications in optimal design. Problems related to evolutionary equations and their admissibility conditions mainly concern transport problems in free and porous media, where it is sometimes natural to include continuous time random walk variables in the model. Such problems sometimes also lead to fractional derivative equations.

Some problems are related to hyperbolic conservation laws and the propagation and interaction of singularities. High-frequency waves corresponding to solutions of hyperbolic systems are governed by ODEs, allowing for final interpretation as particles.

Particularly interesting are situations where the equation changes its type (like transonic fluid flow), motivating our interest in Friedrichs systems. Further results for semilinear systems could open a new approach to the Maxwell-Dirac system, with potential applications in quantum field theory. Practical computation of coefficients in specific applications is also important; as the unknown functions have the domain in multidimensional space, quite often more efficient numerical methods have to be developed.

Introductory lecture was held on December 17th, 2014 (in Croatian): Weak convergence methods and applications
In June 2016 we organised a workshop in Zagreb. More details can be found at the workshop homepage (click here).
In September 2016 we took part in the organisation of The International Conference on Generalized Functions GF2016 in Dubrovnik. More details can be found at the conference webpage (click here).
In January 2017 we organised a mini-workshop in Zagreb. For more information please visit the workshop homepage (click here).
The second project workshop was held in June 2018 in Zagreb. More details can be found at the workshop homepage (click here).

Project entry in the CROSBI database (Croatian scientific bibliography).