GDSFLOWS · Geometry, Dynamics and Singularities in Fluid Flows
Horizon Europe — Marie Skłodowska-Curie Actions
- Duration
- 2022-07-01 → 2024-06-30
- EU contribution
- €165,313
- Participants
- 1
- Scheme
- HORIZON-TMA-MSCA-PF-EF
Lines connect the coordinator with its partners.
Results in brief
Geometry, Dynamics and Singularities in Fluid Flows
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.
Data: CORDIS, © European Union
Project objective
The GDSFLOWS project aims to re-shape the mathematics we use to understand fluid flows. More precisely, the goal is to develop completely new tools, at the crossroads of differential topology, harmonic analysis, and dynamical systems, to address two of the most pressing problems on the PDEs of incompressible fluids: (1) if, and how, do solutions blow-up (that is: after a smooth start, do the physical magnitudes of the problem become irregular in finite time)? and (2) when solutions do not blow-up, what are the qualitative dynamics (attractors, equilibrium solutions) of the trajectories in the phase space (that is, in the space of velocity or vorticity fields?The project proposes 3 horizons: 1) Extending the recently obtained universality results for the Euler equation on certain Riemannian manifolds to the case of PDEs modelling the evolution of fluid interfaces, where the existence of solutions blowing-up in finite time is rigurously known. 2) Proving that the Euler equations on high-dimensional Euclidean spaces are universal, and using this to study whether solutions in very high dimensions that blow-up in finite time exist. 3) Proving the existence of chaotic invariant sets in the infinite dimensional phase space of the 2D Euler equation. The GDSFLOWS project will be carried out by the researcher, an expert in the study of geometric properties of PDEs coming from mathematical physics and hydrodynamics. He recently developed a method for embedding any finite-dimensional dynamical system into the Euler equation on certain high-dimensional Riemannian manifolds, building on T. Tao's recent program to prove blow-up of solutions to the high-dimensional Euler equation. The researcher will collaborate with the Supervisor, a prominent expert in the formation of singularities in the PDEs of fluid dynamics, and one of the authors of the first rigurous proof of blow-up in well-posed PDEs modelling incompressible fluids.
Original text from CORDIS.
Participants
- UNIVERSIDAD DE SEVILLA · SevillaCoordinatorSpain
Links
- View on CORDIS
- DOI: 10.3030/101063565
- https://ec.europa.eu/research/participants/documents/downloadPublic?documentIds=080166e5feb3a65f&appId=PPGMS
- https://ec.europa.eu/research/participants/documents/downloadPublic?documentIds=080166e5feb4a975&appId=PPGMS
Data: CORDIS, © European Union