Non-uniqueness and h-principle for Hölder-continuous weak solutions of the Euler equations

TL;DR

This work analyzes the Cauchy problem for the incompressible Euler equations on $\mathbb{T}^3$ in the Hölder setting and shows that for any $\theta<1/5$, wild Hölder-$\theta$ initial data are dense in $L^2$, i.e., there exist infinitely many admissible Hölder-$\theta$ weak solutions emanating from such data. Central to the approach is a convex-integration framework using Mikado flows to realize arbitrary positive-definite Reynolds stresses, establishing a Nash–Kuiper–type h-principle: smooth subsolutions with a positive Reynolds tensor can be approximated by Hölder-$\theta$ weak solutions with prescribed energy distribution $\bar v\otimes\bar v+\bar R$. The paper also develops adapted subsolutions that permit vanishing Reynolds stress at $t=0$ with a controlled blow-up of $C^1$ norms, enabling energy-controlled approximations of arbitrary background flows and refined energy accounting across the iteration. Together, these results illuminate the flexible, non-unique nature of Hölder Euler solutions under admissibility constraints and provide a robust mechanism to approximate a wide class of background flows within the convex-integration paradigm.

Abstract

In this paper we address the Cauchy problem for the incompressible Euler equations in the periodic setting. Based on estimates developed in [Buckmaster-De Lellis-Isett-Székelyhidi], we prove that the set of Hölder $1\slash 5-\eps$ wild initial data is dense in $L^2$, where we call an initial datum wild if it admits infinitely many admissible Hölder $1\slash 5-\eps$ weak solutions. We also introduce a new set of stationary flows which we use as a perturbation profile instead of Beltrami flows to recover arbitrary Reynolds stresses.

Theorems & Definitions (32)

• Definition 1.1
• Theorem 1.1: Density of wild initial data
• Theorem 1.2: h-principle
• Definition 2.1
• Lemma 2.1: $\mathcal{R}=\mathrm{div\,}^{-1}$
• Lemma 2.2: Stationary phase lemma
• proof
• Lemma 2.3
• Lemma 2.4
• proof : Proof of Lemma \ref{['l:Mikado']}