New a priori estimates for navier-stokes equations in dimension 3

Euler and Navier-Stokes equations

“…Vortex reconnection is a manifestation of a regularizing mechanism. This result ( [17]) follows using the ideas of [37]. The sufficient conditions for regularity involving gradients are obtained easily.…”

Book Review: Initial-boundary value problems and the Navier-Stokes equations

“…The regularity properties of these solutions are not sufficiently understood. It is known that the spatial BV norm is bounded uniformly in time [C2,CLMS] and various temporal averages of higher derivatives are also a priori bounded [C2,FGT]. Also, the Hausdorff dimension of the set of possible space-time singularities of any suitable weak solution is at most one [CKN].…”

Weak and Strong Solutions of the Navier-Stokes Initial Value Problem

“…For n =3 up to now no zero-dimensional estimates for Leray-Hopf solutions are known. Foias, Guillope and Temam [8] gives new one-dimensional estimates for weak solutions for n -3, when the boundary condition is periodic. They prove P \A r ' 2 u\* 2 >-dt Jo is finite for r = l, 2,..., a r = 2/(2r -1).…”