Bursting Dynamics of the 3D Euler Equations in Cylindrical Domains

A class of three-dimensional initial data characterized by uniformly large vorticity is considered for the 3D incompressible Euler equations in bounded cylindrical domains. The fast singular oscillating limits of the 3D Euler equations are investigated for parametrically resonant cylinders. Resonances of fast oscillating swirling Beltrami waves deplete the Euler nonlinearity. These waves are exact solutions of the 3D Euler equations. We construct the 3D resonant Euler systems; the latter are countable uncoupled and coupled SO(3;C) and SO(3;R) rigid body systems. They conserve both energy and helicity. The 3D resonant Euler systems are vested with bursting dynamics, where the ratio of the enstrophy at time t = t to the enstrophy at t = 0 of some remarkable orbits becomes very large for very small times t; similarly for higher norms Hs, s ≥ 2. These orbits are topologically close to homoclinic cycles. For the time intervals where Hs norms, s ≥ 7/2 of the limit resonant orbits do not blow up, we prove that the full 3D Euler equations possess smooth solutions close to the resonant orbits uniformly in strong norms. Key-Words: Incompressible Euler Equations, Rotating Fluids, Rigid Body Dynamics, Enstrophy Bursts MSC: 35Q35, 76B03, 76U05 [email protected] and Laboratoire J.-L. Lions, Université Paris Diderot-Paris 7 [email protected] [email protected]