Partial differential equations (PDEs) are the conceptual framework to describe most natural phenomena. While writing down a PDE that models a particular physical system is often easy, when that PDE is non-linear, it is hard to do anything else with it.
The equations describing incompressible fluids are a paradigmatic example: despite more than two centuries of intense study, a good mathematical understanding of their implications is still lacking.
The GDSFLOWS project's aim was to sharpen the mathematical tools we use to analyse the equations of incompressible fluids, by bringing in new techniques from differential topology, harmonic analysis, and dynamical systems. We had two sets of problems in mind:
(1) regular vs singular behaviour (that is: after a smooth start, do some physical magnitudes of fluid become irregular in finite time)?
(2) qualitative dynamics (existence of invariant sets, attractors, equilibrium solutions) of the
trajectories in the phase space (that is, in the space of velocity or vorticity).
During the project, we focused on (2). We were particularly interested in finding finite dimensional invariant manifolds of the Euler equations (the equations describing the motion of incompressible inviscid fluids), exhibiting interesting dynamics.
An invariant manifold is a parametric family of velocity fields, with the property that the solutions of the Euler equation with initial condition in the family exist and remain in the family there for all time, defining a finite-dimensional ODE on the space of parameters. For example if each vector field in the familly is identified by 10 angular parameters, the invariant manifold is a 10-dimensional torus, and if the ODE on the torus is a linear flow, we get families of periodic or quasiperiodic solutions of the Euler equation.
