Wild solutions of the Navier-Stokes equations whose singular sets in time have Hausdorff dimension strictly less than 1

TL;DR

The paper advances the theory of weak solutions for the 3D Navier–Stokes equations by constructing non-unique solutions with bounded energy whose singular set in time has Hausdorff (box-counting) dimension strictly less than 1. It extends the convex integration framework with a gluing technique and introduces intermittent jets as time-dependent building blocks to achieve fractal-time singularities while preserving control of the Reynolds stress. The method yields a weak solution that matches two given strong solutions on complementary time intervals, thereby proving non-uniqueness for NSE (and the energy-supercritical hyperdissipative variant with α in [1,5/4)) within a precisely characterized regularity class. This work tightens the understanding of well-posedness thresholds for weak solutions and highlights the nuanced interplay between energy bounds, regularity, and temporal fractal singularities in fluid dynamics.

Abstract

We prove non-uniqueness for a class of weak solutions to the Navier-Stokes equations which have bounded kinetic energy, integrable vorticity, and are smooth outside a fractal set of singular times with Hausdorff dimension strictly less than 1.

Theorems & Definitions (27)

• Definition 1.1: Weak solution
• Theorem 1.1: Main result
• Remark 1.2: Non-uniqueness of weak solutions for strong initial datum
• Remark 1.3: Weak solutions with partial regularity in time
• Remark 1.4: Non-uniqueness of very weak solutions for any $L^2$ initial datum
• Theorem 1.5: The hyperdissipative problem
• Proposition 2.1: Main Iteration Proposition
• Proposition 2.2
• Proposition 3.1
• proof : Proof of Proposition \ref{['prop:local:existence']}