Non-local dynamics in incompressible fluids
Principal investigator: Diego Córdoba Gazolaz
Project reference: 788250
Start Date: September 2018
End Date: August 2023
Abstract: The search for singularities in incompressible flows has become a major challenge in the area of non-linear partial differential equations and is relevant in applied mathematics, physics and engineering. The understanding of these singularities could lead to important insights on the structure of turbulence, one of the major open scientific problems of classical physics. Recently we have made a breakthrough in the formation of singularities and the creation of a mixing zone for the evolution of an interface generated between two immiscible incompressible fluids. These contour dynamics problems are given by basic fluid mechanics systems, such as Euler´s equation, Darcy´s law and the Quasi-geostrophic equation. These give rise to problems such as internal wave, Muskat, two-phase Hele-Shaw and evolution of sharp fronts. The fundamental questions to address are local-existence, uniqueness, global-existence of solutions or on the other hand, formation of singularities on the free boundary or ill-posedness.
The goal of this project is to pursue new methods in the mathematical analysis of non-local and non-linear partial differential equations. For this purpose, several physical scenarios of interest in the context of incompressible fluids are presented from a mathematical point of view as well as for its applications: both from the standpoint of global well-posedness, existence and uniqueness of weak solutions and as candidates for blow-up.
The equations we consider are the incompressible Euler equations, incompressible porous media equation and the generalized Quasi-geostrophic equation. This research will lead to a deeper understanding of the nature of the set of initial data that develops finite time singularities as well as those solutions that exist for all time for incompressible flows.
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme grant agreement No 788250.