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We introduce sparse versions of function spaces that are relevant tocharacterize the solutions of Euler equations without concentration. Thestandard Sobolev space $H^{-1}$ is given a sparse structure that allows tomeasure the degree of compactness of embeddings into $H^{-1}$ and provides newquantitative general criteria for $H^{-1}$-stability. Indices of sparseness aredefined, and function spaces whose indices have prescribed decay areconstructed, resulting in an improvement of the classical $H^{-1}$-stabilityresults: sparse stability. The analysis relies on the introduction of sparseRiesz-Morrey-Tadmor spaces, that are characterized via maximal operators andnew sparse domination theorems, together with extrapolation techniques. Ourmethods also yield improvements on recent results on the conservation of energyof physically realizable solutions of $2$D-Euler.

A sharp stability criterion for Euler equations via sparseness