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

Abstract

We introduce a new ladder of function spaces which is shown to fill in thegap between the weak Lp∞ spaces and the larger Morrey spaces,Mp.Our motivation for introducing these new spaces, denotedby ∨pq, is to gain more accurate information on(compact) embeddings of Morrey spaces in appropriateSobolev spaces. It is here that the secondary parameterq (and a further logarithmic refinement parameterα, denoted by ∨pq(log ∨)α)gives a finer scaling, which allows us to make the subtledistinctions necessary for embedding in spaces with afixed order of smoothness.We utilize an H-1-stability criterion which we have recentlyintroduced (Lopes Filho M C, Nussenzveig Lopes H J and Tadmor E 2001Approximate solution of the incompressible Eulerequations with no concentrationsAnn. Insitut H Poincaré C 17 371-412),in order to study the strong convergence ofapproximate Euler solutions. We show how the new refined scale ofspaces, ∨pq(log ∨)α, enables us toapproach the borderline cases which separate betweenH-1-compactness and the phenomena ofconcentration-cancellation. Expressed in terms of their∨pq(log ∨)α bounds, these borderline cases are shown to be intimatelyrelated to uniform bounds of the total (Coulomb) energy and the relatedvorticity configuration.