A wavelet-inspired L³-based convex integration framework for the Euler equations

2023/05/29 by Vikram Giri, Giri, Vikram, Hyunju Kwon +3 · 2 citations
Earth and Planetary Sciences · Medicine · #35D30 #35Q31 #Analysis of PDEs (math.AP) #FOS: Mathematics #Pelvic floor disorders treatments #Seismic Imaging and Inversion Techniques

paper · pdf · doi:10.48550/arxiv.2305.18142

openalex publication_date 2023/05/29 · openalex created_date 2023/05/31 · openalex updated_date 2026/07/28

Abstract

In this work, we develop a wavelet-inspired, L³-based convex integration framework for constructing weak solutions to the three-dimensional incompressible Euler equations. The main innovations include a new multi-scale building block, which we call an intermittent Mikado bundle; a wavelet-inspired inductive set-up which includes assumptions on spatial and temporal support, in addition to L^p and pointwise estimates for Eulerian and Lagrangian derivatives; and sharp decoupling lemmas, inverse divergence estimates, and space-frequency localization technology which is well-adapted to functions satisfying L^p estimates for p other than 1, 2, or ∞. We develop these tools in the context of the Euler-Reynolds system, enabling us to give both a new proof of the intermittent Onsager theorem (An Intermittent Onsager Theorem, Inventiones Mathematicae, (2023), 233) in this paper, and a proof of the L³-based strong Onsager conjecture in a companion paper (arXiv:2305.18509).

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• Fine dissipative properties of Euler solutions with measure first derivatives
• Sharp Non-uniqueness in Law for Stochastic Differential Equations on the Whole Space

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