A sharp stability criterion for Euler equations via sparseness

We introduce sparse versions of function spaces that are relevant to characterize the solutions of Euler equations without concentration.The standard Sobolev space H 1 is given a sparse structure that allows measuring the degree of compactness of embeddings into H 1 and provides new quantitative general criteria for H 1 -stability.Indices of sparseness are defined, and function spaces whose indices have prescribed decay are constructed, resulting in an improvement of the classical H 1 -stability results: sparse stability.The analysis relies on the introduction of sparse Riesz-Morrey-Tadmor spaces, that are characterized via maximal operators and new sparse domination theorems, together with extrapolation techniques.Our methods also yield improvements on recent results on the conservation of energy of physically realizable solutions of 2D-Euler.1. Preamble 659 2. Background 669 3. Characterization of sparse RMT spaces via maximal operators 670 4. A sparse approach to H 1 -stability 680 5. Computability of sparse indices 687 6. Proof of Theorem 2 690 7. Sharpening Morrey regularity of DiPerna-Majda via V ‰ 691 8. Sharpening Tadmor's regularity via T ‰ 701 9. Comparison between V ‰ and T ‰ 711 10.A sparse approach to energy conservation 713 Added in proof 718