On a new scale of regularity spaces with applications to Euler's equations

We introduce a new ladder of function spaces which is shown to fill in the gap between the weak L-p infinity spaces and the larger Morrey spaces, M-p. Our motivation for introducing these new spaces, denoted by V-pq, is to ga in more accurate information on (compact) embeddings of Morrey spaces in ap propriate Sobolev spaces. It is here that the secondary parameter q (and a further logarithmic refinement parameter alpha, denoted by V-pq(log V)(alph a)) gives a finer scaling, which allows us to make the subtle distinctions necessary for embedding in spaces with a fixed order of smoothness. We utilize an H-1-stability criterion which we have recently introduced (Lo pes Filho M C, Nussenzveig Lopes H J and Tadmor E 2001 Approximate solution of the incompressible Euler equations with no concentrations Ann. Insitut H Poincare C 17 371-412), in order to study the strong convergence of appro ximate Euler solutions. We show how the new refined scale of spaces, V-pq ( log V)(alpha), enables us to approach the borderline cases which separate b etween H-1-compactness and the phenomena of concentration-cancellation. Exp ressed in terms of their V-pq(logV)(alpha) bounds, these borderline cases a re shown to be intimately related to uniform bounds of the total (Coulomb) energy and the related vorticity configuration.