A sharp stability criterion for Euler equations via sparseness

Domínguez, Óscar; Milman, Mario

doi:10.2140/apde.2026.19.659

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A sharp stability criterion for Euler equations via sparseness

Óscar Domínguez and Mario Milman

Vol. 19 (2026), No. 4, 659–720

DOI: 10.2140/apde.2026.19.659

Abstract

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 $2$ D-Euler.

Keywords

$H^{-1}$-stability, Euler equations, energy conservation, sparseness, Morrey spaces, approximate solutions, physical solutions, extrapolation spaces

Mathematical Subject Classification

Primary: 42B35, 42B37, 46B70, 76B03

Milestones

Received: 15 December 2023

Revised: 23 February 2025

Accepted: 8 June 2025

Published: 17 May 2026

Authors

Óscar Domínguez
Department of Mathematics CUNEF Universidad Madrid Spain

Mario Milman
Instituto Argentino de Matemática Buenos Aires Argentina

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Vol. 19, No. 4, 2026