This project is concerned with the mathematical treatment of generalized nonlinear Fokker-Planck equations (well-posedness, asymptotic behavior, approximation and controllability) and their applications to statistical physics, mathematical biology and technique of image processing. Nonlinear Fokker-Planck equations are widely used in statistical physics to describing the anomalous diffusion and open systems far from equilibrium. Several equations arising in Bose-Einstein diffusion, Fermi-Dirac statistics and thermostatistics are of this form. The solutions to these equations represent probability densities of stochastic equations which describe the random dynamics of particles and there exist an equivalent representation of nonlinear Fokker-Planck equations in terms of stochastic differential equations (McKean-Vlasov equations). However, a rigorous mathematical treatment of the existence and longtime behavior of solutions to these equations is missing and this is the main objective of this project which continues a research developed by the principal investigator (V. Barbu) at University of Bielefeld (Germany) in the framework of a scientific project funded by DFG in the period 2017-2021. A large part of this project will be devoted to applications in biology, fluid dynamics and new image processing techniques based on nonlinear Fokker-Planck equations.
The main objective of this project is the rigorous mathematical treatment of the existence and asymptotic behavior of the Fokker-Planck type equations, as well as their applications in biology, fluid dynamics and image processing techniques. We expect to produce important new scientific theoretical results with impact in the scientific community by the novelty and importance of the problem for applied sciences and physics. This project will contribute to the foundation of a new direction of research in the host institution as well as in the institutes of research associated to this project. We expect to publish a number of original works with top mathematical journals in Europe and USA and develop new models in mathematical biology, image processing techniques based on nonlinear Fokker-Planck equations. The scientific results will be also communicated to important conferences and workshops. Also, these will be communicated in public conferences for specialists and students. If one take into account the interest that the mathematical community for our previous results on nonlinear Fokker-Planck equations, we expect a similar interest and echo of this research.