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Non-uniqueness of Leray solutions of the forced Navier-Stokes equations

Dallas Albritton, Elia Brué, Maria Colombo

TL;DR

The paper proves non-uniqueness of Leray–Hopf solutions to the forced 3D Navier–Stokes equations with zero initial data by constructing two distinct, smooth-after-positive-time Leray solutions sharing the same forcing. Central to the approach is a self-similar background, built from a compactly supported axisymmetric vortex ring, which is linearly unstable and serves as a forcing-tolerant unstable state; a second solution is obtained as a trajectory on the unstable manifold of this background. The authors connect a forced 3D Euler steady state to Navier–Stokes in similarity variables, proving the persistence of instability in the NS setting and then implement a nonlinear fixed-point construction to produce two distinct solutions, thereby demonstrating non-uniqueness at the critical regularity borderline. The results illuminate the delicate boundary between well-posedness and ill-posedness for NS and offer a concrete mechanism—via instability on an unstable manifold—to realize non-uniqueness in a Leray framework.

Abstract

In the seminal work [39], Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. We exhibit two distinct Leray solutions with zero initial velocity and identical body force. Our approach is to construct a `background' solution which is unstable for the Navier-Stokes dynamics in similarity variables; its similarity profile is a smooth, compactly supported vortex ring whose cross-section is a modification of the unstable two-dimensional vortex constructed by Vishik in [43,44]. The second solution is a trajectory on the unstable manifold associated to the background solution, in accordance with the predictions of Jia and Šverák in [32,33]. Our solutions live precisely on the borderline of the known well-posedness theory.

Non-uniqueness of Leray solutions of the forced Navier-Stokes equations

TL;DR

The paper proves non-uniqueness of Leray–Hopf solutions to the forced 3D Navier–Stokes equations with zero initial data by constructing two distinct, smooth-after-positive-time Leray solutions sharing the same forcing. Central to the approach is a self-similar background, built from a compactly supported axisymmetric vortex ring, which is linearly unstable and serves as a forcing-tolerant unstable state; a second solution is obtained as a trajectory on the unstable manifold of this background. The authors connect a forced 3D Euler steady state to Navier–Stokes in similarity variables, proving the persistence of instability in the NS setting and then implement a nonlinear fixed-point construction to produce two distinct solutions, thereby demonstrating non-uniqueness at the critical regularity borderline. The results illuminate the delicate boundary between well-posedness and ill-posedness for NS and offer a concrete mechanism—via instability on an unstable manifold—to realize non-uniqueness in a Leray framework.

Abstract

In the seminal work [39], Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. We exhibit two distinct Leray solutions with zero initial velocity and identical body force. Our approach is to construct a `background' solution which is unstable for the Navier-Stokes dynamics in similarity variables; its similarity profile is a smooth, compactly supported vortex ring whose cross-section is a modification of the unstable two-dimensional vortex constructed by Vishik in [43,44]. The second solution is a trajectory on the unstable manifold associated to the background solution, in accordance with the predictions of Jia and Šverák in [32,33]. Our solutions live precisely on the borderline of the known well-posedness theory.
Paper Structure (20 sections, 19 theorems, 195 equations)

This paper contains 20 sections, 19 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. Strategy
  3. Comparison with existing literature
  4. Linear instability: from two dimensions to axisymmetry
  5. Spectral preliminaries
  6. Two-dimensional instability
  7. Axisymmetric instability without swirl
  8. Functional analytic set-up
  9. Main result
  10. Linear instability: from Euler to Navier-Stokes equations
  11. Nonlinear instability
  12. The linear operator
  13. Nonlinear construction
  14. Functional setting
  15. Abstract claim
  16. ...and 5 more sections

Key Result

Theorem 1.2

There exist $T>0$, $f \in L^1_t L^2_x(\mathbb{R}^3_+ \times (0,T))$, and two distinct suitable Leray--Hopf solutions $u$, $\bar{u}$ to the Navier--Stokes equations on $\mathbb{R}^3 \times (0,T)$ with body force $f$ and initial condition $u_0 \equiv 0$.

Theorems & Definitions (34)

  • Definition 1.1
  • Theorem 1.2: Non-uniqueness
  • Theorem 1.3: Non-uniqueness, refined
  • Theorem 2.1
  • Proposition 2.2: Truncated unstable vortex
  • proof
  • Corollary 2.3: Instability in weighted space
  • proof
  • Lemma 2.4: Correcting the divergence
  • proof
  • ...and 24 more