A Wavelet-Inspired \(L^3\)-Based Convex Integration Framework for the Euler Equations

Abstract

In this work, we develop a wavelet-inspired, \(L^3\)-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 \(\infty \). 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 from Novack and Vicol (Invent Math 233(1):223–323, 2023) in this paper, and a proof of the \(L^3\)-based strong Onsager conjecture in the companion paper Giri et al. (The \(L^3\)-based strong Onsager theorem, arxiv).