On a new scale of regularity spaces with applications to Euler’s equations
نویسندگان
چکیده
We introduce a new ladder of function spaces which is shown to fill in the gap between the weak Lp∞ spaces and the larger Morrey spaces, Mp. Our motivation for introducing these new spaces, denoted ∨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 α, denoted ∨pq(log∨)α) 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 [21], in order to study the strong convergence of approximate Euler solutions. We show how the new refined scale of spaces, ∨pq(log∨)α, enables us approach the borderline cases which separate between H−1-compactness and the phenomena of concentration-cancelation. Expressed in terms of their ∨pq(log∨)α bounds, these borderline cases are shown to be intimately related to uniform bounds of the total (Coulomb) energy and the related vorticity configuration.
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