Nonuniqueness of Leray-Hopf solutions to the unforced incompressible 3D Navier-Stokes Equation

1. Problem Statement

The paper addresses one of the most fundamental open problems in mathematical fluid dynamics: the uniqueness of Leray–Hopf weak solutions to the three-dimensional incompressible Navier–Stokes equations (NSE) in the absence of external forcing.

• The Equations: The study focuses on the unforced NSE on $\mathbb{R}^3$:
  $$\partial_t u + u \cdot \nabla u - \Delta u + \nabla p = 0, \quad \text{div } u = 0$$
• The Context: While Leray (1934) and Hopf (1951) proved the global existence of weak solutions satisfying energy inequalities, their uniqueness remains unproven. Uniqueness is guaranteed only if solutions satisfy the Prodi–Serrin condition ($u \in L^s_t L^q_x$ with $3/q + 2/s \le 1$). Solutions that fail this condition (often associated with singular initial data or blow-up scenarios) are the primary candidates for non-uniqueness.
• The Goal: To provide the first rigorous computer-assisted proof demonstrating that there exist infinitely many distinct Leray–Hopf solutions starting from the same divergence-free initial data with compact support.

2. Methodology

The authors employ a multi-stage strategy combining self-similar analysis, spectral theory, and rigorous numerical verification.

A. Self-Similar Reduction

Inspired by Jia and Sverak, the authors seek non-uniqueness via forward self-similar solutions. They transform the time-dependent NSE into a stationary equation for a profile $\tilde{U}(\xi)$ where $\xi = x/\sqrt{t}$:
$$-\frac{1}{2}\tilde{U} - \frac{1}{2}\xi \cdot \nabla \tilde{U} + \Pi(\tilde{U} \cdot \nabla \tilde{U}) - \Delta \tilde{U} = 0$$
The initial data is singular ($O(|x|^{-1})$), but the authors localize this to create a valid Leray–Hopf solution with compact support.

B. Spectral Stability Analysis

The core of the non-uniqueness argument relies on the linearized operator $\mathcal{L}_{\tilde{U}}$ around the self-similar profile:
$$\mathcal{L}_{\tilde{U}} v = \frac{1}{2}v + \frac{1}{2}\xi \cdot \nabla v - \Pi(\tilde{U} \cdot \nabla v + v \cdot \nabla \tilde{U}) + \Delta v$$

• Hypothesis: If $\mathcal{L}_{\tilde{U}}$ possesses an unstable eigenvalue (real part $\lambda > 0$), then the self-similar solution is unstable.
• Consequence: This instability allows for the construction of a second solution by perturbing the profile along the unstable manifold, leading to infinitely many distinct solutions for the same initial data.

C. Rigorous Computer-Assisted Proof Framework

To prove the existence of such a profile and eigenvalue rigorously, the authors develop a novel numerical framework:

1. High-Precision Approximation: They use a gradient descent method (projected onto the solution manifold) to find a candidate profile $\tilde{U}$ and an eigenpair $(\lambda, v)$ that minimizes the residual of the equations.
2. Decomposition of the Linear Operator: The linearized operator is decomposed into a coercive part ($L_1$) and a compact perturbation ($L_2$).
   • Coercive Part: The viscous term $-\Delta$ provides strong coercivity (unlike the Euler equations), allowing for standard $L^2/H^1$ estimates without singular weights.
   • Compact Part: Approximated by a finite-rank operator ($L_{ap}$) constructed via a spectral basis (eigenfunctions of a related Laplacian problem).
3. Fixed-Point Argument:
   • The error equation is split into a linear part (solved exactly on the finite-rank subspace) and a remainder.
   • The remainder is shown to be small enough to be absorbed by the coercivity of $L_1$.
   • The Schauder Fixed-Point Theorem is used to prove the existence of an exact solution $(\tilde{U} + \text{error})$ and an exact eigenpair in the neighborhood of the numerical candidates.
4. Rigorous Bounds: All numerical errors (residuals, quadrature errors, interpolation errors) are bounded using interval arithmetic and analytic estimates (e.g., Bernstein inequalities for trigonometric polynomials, Bessel function bounds).

3. Key Contributions

1. First Rigorous Proof of Nonuniqueness: This is the first mathematically rigorous proof (verified by computer) that Leray–Hopf solutions to the unforced 3D NSE are non-unique. Previous works (e.g., Albritton–Brue–Colombo) required singular forcing terms; this work removes that requirement.
2. Novel Numerical Framework: The authors introduce a method to rigorously verify the existence of unstable eigenmodes for the linearized NSE operator. This involves:
   • Decomposing the operator into coercive and compact parts.
   • Using a finite-rank approximation with rigorous error control.
   • Leveraging the "genuine coercivity" of viscosity to simplify the analysis compared to the Euler singularity program.
3. Construction of Specific Profiles: They explicitly construct a smooth, axisymmetric, divergence-free self-similar profile $\tilde{U}$ and an associated unstable eigenfunction $v$ (with odd parity in $z$) that satisfy the necessary spectral conditions.
4. Localization Strategy: They demonstrate how to localize the singular self-similar solution to a compactly supported initial condition $u_{in}$ while preserving the instability mechanism, thereby generating infinitely many Leray–Hopf solutions.

4. Key Results

• Theorem 1 (Nonuniqueness): There exist infinitely many distinct suitable Leray–Hopf solutions to the 3D NSE on $\mathbb{R}^3 \times [0, 1]$ with the same divergence-free initial condition $u_{in} \in C^\infty$ with compact support. These solutions are smooth for $t > 0$ but fail the Prodi–Serrin condition.
• Theorem 2 (Existence of Unstable Profile): There exists a weak solution $(\tilde{U}, \tilde{v}, \tilde{\lambda})$ to the self-similar system where $\tilde{\lambda} < 0$ (in the authors' sign convention, corresponding to an unstable eigenvalue $\lambda > 0$ in the time-evolution sense).
• Numerical Verification:
  • The residual of the approximate profile is bounded by $\|E_U\|_{L^2} \le 6.5 \times 10^{-7}$.
  • The eigenvalue is verified to be $\lambda \approx -0.113$ (unstable).
  • The coercivity constants are verified to be positive (e.g., $\approx 0.051$ for the profile equation and $\approx 0.164$ for the eigenfunction equation), ensuring the fixed-point argument closes.

5. Significance

• Resolution of a Major Open Problem: This work provides a definitive answer to the non-uniqueness question for Leray–Hopf solutions, a central problem in fluid dynamics for nearly a century.
• Methodological Breakthrough: The paper establishes a robust template for computer-assisted proofs in nonlinear PDEs. By combining high-order numerical methods with rigorous interval arithmetic and functional analysis, it bridges the gap between numerical evidence and mathematical proof.
• Implications for Regularity: The existence of non-unique solutions implies that the Prodi–Serrin condition is sharp for uniqueness. It suggests that the "regularity" of solutions is not guaranteed solely by the energy inequality, and that singularities (or near-singularities) can lead to branching solution paths.
• Future Directions: The authors suggest that their localization techniques could be adapted to study non-uniqueness on bounded domains (e.g., with no-slip boundary conditions), potentially resolving the uniqueness question in physical settings.

In summary, this paper represents a landmark achievement in mathematical fluid dynamics, utilizing advanced computational techniques to rigorously prove that the 3D Navier–Stokes equations admit multiple distinct solutions for the same initial data, fundamentally challenging the deterministic nature of the equations in the weak solution regime.