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

Abstract

We introduce a new ladder of function spaces which is shown to fill in the gap between the weak $L^{p\infty}$ spaces and the larger Morrey spaces, $M^p$. Our motivation for introducing these new spaces, denoted $\V^{pq}$, is to gain a more accurate information on (compact) embeddings of Morrey spaces in appropriate Sobolev spaces. It is here that the secondary parameter q (-- and a further logarithmic refinement parameter $\alpha$, denoted $\V^{pq}(\log \V)^{\alpha}$) 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 in {Lopes Filho M C, Nussenzveig Lopes H J and Tadmor E 2001 Approximate solution of the incompressible Euler equations with no concentrations Ann. Institut H Poincare C 17 371-412}, in order to study the strong convergence of approximate Euler solutions. We show how the new refined scale of spaces, $\V^{pq}(\log \V)^{\alpha}$, enables us approach the borderline cases which separate between $H^{-1}$-compactness and the phenomena of concentration-cancelation. Expressed in terms of their $\V^{pq}(\log \V)^{\alpha}$ bounds, these borderline cases are shown to be intimately related to uniform bounds of the total (Coulomb) energy and the related vorticity configuration.