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.
Fields of science
- natural sciencesmathematicsapplied mathematicsmathematical physics
- natural sciencesmathematicspure mathematicstopology
- natural sciencesphysical sciencesclassical mechanicsfluid mechanicsfluid dynamics
- natural sciencesmathematicsapplied mathematicsdynamical systems
- natural sciencesmathematicspure mathematicsmathematical analysisdifferential equationspartial differential equations
- HORIZON.1.2 - Marie Skłodowska-Curie Actions (MSCA) Main Programme