Analysis Seminar

This talk concerns regularity and boundedness properties of solutions of the incompressible Navier-Stokes equations. We highlight two recent constructions of solutions that exhibit an energy cascade from high to low frequencies. The first shows that, even from critically bounded data in BMO^{-1}, classical solutions can become arbitrarily large. The second construction (joint with A. Cheskidov and M. Dai) demonstrates formation of an "anomalous" blow-up from smooth data, arising from an infinite energy cascade. At the blow-up time, the classical solution splits into a continuum of non-unique solutions with interesting features such as period breaking and various near-optimal bounds.