Periodic Reporting for period 1 - GDSFLOWS (Geometry, Dynamics and Singularities in Fluid Flows)
Reporting period: 2022-07-01 to 2024-06-30
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.
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.
In collaboration with A. Enciso and D. Peralta-Salas, we built upon a construction of N. Crouseilles and E. Faou to show the existence of invariant tori for the 3D Euler equation. The solutions we constructed are the first and only known quasiperiodic solutions of the Euler equation in 3 dimensions. We also showed that Crouseilles and Faou invariant tori for the 2D Euler equation are in fact dense in the strong L^p norm on the space of stream functions.
During that time I also started a collaboration with Robert Cardona, who visited the University of Seville financed by the GDSFLOWS grant. This resulted in our recent work on the non-mixing of the 3D Euler equation on lower regularity. The non-mixing problem consists in showing that there are pairs of smooth velocity fields, sharing some properties (energy, helicity, homotopy class of vorticity), such that no field close to one can ever evolve into a field close to the other (by velocity fields being close we mean that their values, and the values of their derivatives up to certain order k, are similar). This means that, no matter how much time passes, the velocity field of a fluid with initial velocity similar to v1 will never become similar to v2. Khesin, Kuksin and Peralta-Salas proved this in 2014 for k>=4, their proof was based on the KAM theorem. Proving the same result for smaller values of k remained an open problem, and we proved it for k=1. For this we introduced a new framework that assigns contact/symplectic geometry invariants to large sets of time-dependent solutions to the Euler equations on any 3-manifold.
During that time I also started a collaboration with Robert Cardona, who visited the University of Seville financed by the GDSFLOWS grant. This resulted in our recent work on the non-mixing of the 3D Euler equation on lower regularity. The non-mixing problem consists in showing that there are pairs of smooth velocity fields, sharing some properties (energy, helicity, homotopy class of vorticity), such that no field close to one can ever evolve into a field close to the other (by velocity fields being close we mean that their values, and the values of their derivatives up to certain order k, are similar). This means that, no matter how much time passes, the velocity field of a fluid with initial velocity similar to v1 will never become similar to v2. Khesin, Kuksin and Peralta-Salas proved this in 2014 for k>=4, their proof was based on the KAM theorem. Proving the same result for smaller values of k remained an open problem, and we proved it for k=1. For this we introduced a new framework that assigns contact/symplectic geometry invariants to large sets of time-dependent solutions to the Euler equations on any 3-manifold.
-Construction of invariant tori of any finite dimension for the 3D Euler equations. This is the first and only known construction of quasiperiodic solutions for the 3D Euler equation.
-Proof of non-mixing of the 3D Euler equations in the C^1 topology.
-Proof of non-mixing of the 3D Euler equations in the C^1 topology.