A Wavelet-Inspired L3-Based Convex Integration Framework for the Euler Equations

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 infinity. 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).