A wavelet-inspired $L^3$-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 $\infty$. 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 (An Intermittent Onsager Theorem, Inventiones Mathematicae, (2023), 233) in this paper, and a proof of the $L^3$-based strong Onsager conjecture in a companion paper (arXiv:2305.18509).