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Non-uniqueness of smooth solutions to the Navier-Stokes equations on torus $\TTT^2$

Changxing Miao, Yao Nie, Weikui Ye

TL;DR

The paper resolves the two-dimensional local well-posedness problem for incompressible Navier–Stokes on $\mathbb{T}^2$ in the largest known critical space ${\rm BMO}^{-1}$ by constructing two distinct global smooth solutions with the same initial data. It develops a heat-dominated Fourier mode flow based on steady 2D Euler flows and executes a careful inductive scheme that decomposes each iterates into a primary oscillatory part, a secondary inverse-cascade part, and a small nonlinear perturbation, with sharp Besov-type control. Through a detailed analysis of the interaction terms and error estimates, the authors prove convergence of alternating subsequences to two different limits, $u^{\mathrm{even}}$ and $u^{\mathrm{odd}}$, thereby demonstrating non-uniqueness in ${\rm BMO}^{-1}$. The results build on and extend the framework used in the 3D setting, adapting it to 2D with novel geometric and frequency-localized techniques, and show that non-uniqueness persists in the plane as a fundamental feature of the critical-scale Navier–Stokes dynamics.

Abstract

The local well-posedness theory for the incompressible Navier-Stokes equations in $\BMO^{-1}$ has attracted considerable attention over the past two decades. In a recent breakthrough, Coiculescu and Palasek (Invent. Math., 2025) settled the three-dimensional case by demonstrating the existence of two distinct global solutions, both smooth for $t>0$, evolving from a common initial datum in ${\rm BMO}^{-1}(\mathbb{T}^3)$. However, the two-dimensional case remains open. In this paper, we solve the two-dimensional problem. Unlike its three-dimensional counterpart, the two-dimensional setting presents additional difficulties stemming from the geometric intersections of two-dimensional Mikado flows. To overcome these difficulties, we develop a heat-dominated Fourier mode flow built upon steady two-dimensional Euler flows, and present the proof using a new iterative scheme.

Non-uniqueness of smooth solutions to the Navier-Stokes equations on torus $\TTT^2$

TL;DR

The paper resolves the two-dimensional local well-posedness problem for incompressible Navier–Stokes on in the largest known critical space by constructing two distinct global smooth solutions with the same initial data. It develops a heat-dominated Fourier mode flow based on steady 2D Euler flows and executes a careful inductive scheme that decomposes each iterates into a primary oscillatory part, a secondary inverse-cascade part, and a small nonlinear perturbation, with sharp Besov-type control. Through a detailed analysis of the interaction terms and error estimates, the authors prove convergence of alternating subsequences to two different limits, and , thereby demonstrating non-uniqueness in . The results build on and extend the framework used in the 3D setting, adapting it to 2D with novel geometric and frequency-localized techniques, and show that non-uniqueness persists in the plane as a fundamental feature of the critical-scale Navier–Stokes dynamics.

Abstract

The local well-posedness theory for the incompressible Navier-Stokes equations in has attracted considerable attention over the past two decades. In a recent breakthrough, Coiculescu and Palasek (Invent. Math., 2025) settled the three-dimensional case by demonstrating the existence of two distinct global solutions, both smooth for , evolving from a common initial datum in . However, the two-dimensional case remains open. In this paper, we solve the two-dimensional problem. Unlike its three-dimensional counterpart, the two-dimensional setting presents additional difficulties stemming from the geometric intersections of two-dimensional Mikado flows. To overcome these difficulties, we develop a heat-dominated Fourier mode flow built upon steady two-dimensional Euler flows, and present the proof using a new iterative scheme.
Paper Structure (10 sections, 13 theorems, 120 equations)

This paper contains 10 sections, 13 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Induction scheme
  3. Parameters
  4. Iterative procedure
  5. Proof of Proposition \ref{['iteration']}
  6. Construction of ${w}^{\text{(p)}} _{2q}$ and ${w}^{\text{(s)}} _{2q}$.
  7. Key estimates for ${w}^{\text{(p)}} _{2q}$ and ${w}^{\text{(s)}} _{2q}$.
  8. Estimates of the error terms
  9. Construction of $w^{\textup{(ns)}}_{2q}$ and completion of iteration
  10. Proof of Theorem \ref{['t:main']}

Key Result

Theorem 1.1

The system NS admits two distinct global solutions $u, \widetilde{u} \in C^\infty_{t,x}(\mathbb R^{+}\times\mathbb{T}^2)$ with the same initial data $u^{\textup{in}}$ and for all $1\le r<\infty$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1: Inductive step
  • Proposition 3.1
  • Proposition 3.2: Estimates for ${w}^{\text{(p)}} _{2q}$ and ${w}^{\text{(s)}} _{2q}$
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • Proposition 3.5
  • ...and 13 more