In this talk we consider the following non-mixing problem for the 3D Euler equations: show that there are pairs of smooth velocity fields, with same energy and helicity, such that no field that is C^k close to one can evolve into a field that is C^k close to the other.
Khesin, Kuksin and Peralta-Salas proved this in 2014 for k>4 and non-integer, on any closed Riemannian 3-manifold, for all values of helicity and large enough energies (https://arxiv.org/abs/1401.5516). They did so by defining new functionals on the space of velocity fields that are integrals of motion of the Euler equation and have good enough continuity properties in the C^{4, s} topology and above. The 4 comes from KAM theory, as these functionals measure volume occupied by knotted invariant tori of the vorticity.
Proving the same result for smaller values of k remained an open problem (https://arxiv.org/abs/2205.01143). We will present joint work with Robert Cardona where we do this for all k>=1 (https://arxiv.org/abs/2312.03514). We will also give some further applications, for which we will introduce new integrals of motion of the 3D Euler equations that are defined in certain open regions of the space of velocities and are continuous in the C^{1,s}-topology. These new functionals equal very special linear combinations of circulations, those that are precisely embedded contact homology spectral invariants.
The talk will be via Zoom at:
https://utoronto.zoom.us/j/99576627828
Passcode: 448487