Pathological solutions of Navier-Stokes equations on \mathbbT² with gradients in Hardy spaces

2025/09/09 by Jan Burczak, Burczak, Jan, Antonio Hidalgo-Torné +1
Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2509.08168

openalex publication_date 2025/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For an arbitrary smooth initial datum, we construct multiple nonzero solutions to the 2d Navier-Stokes equations, with their gradients in the Hardy space H^p with any p ∈ (0,1). Thus, in terms of the path space C(H^p) for vorticity, p=1 is the threshold value distinguishing between non-uniqueness and uniqueness regimes. In order to obtain our result, we develop the needed theory of Hardy spaces on periodic domains.

Citations

• Non-Uniqueness of Smooth Solutions of the Navier-Stokes Equations from Critical Data
• Flexibility of Two-Dimensional Euler Flows with Integrable Vorticity
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• Nonuniqueness of solutions to the Euler equations with vorticity in a Lorentz space
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• A Proof of Onsager's Conjecture
• Are the incompressible 3d Navier-Stokes equations locally ill-posed in the natural energy space?
• A local version of real Hardy spaces

Related

• Non-uniqueness and energy dissipation for 2D Euler equations with vorticity in Hardy spaces
• Trapping regions and an ODE-type proof of the existence and uniqueness theorem for Navier-Stokes equations with periodic boundary conditions on the plane
• Epochs of regularity for wild Hölder-continuous solutions of the Hypodissipative Navier-Stokes System
• Wild solutions of the Navier-Stokes equations whose singular sets in time have Hausdorff dimension strictly less than 1
• Non-Uniqueness of Smooth Solutions of the Navier-Stokes Equations from Critical Data