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

Abstract: The nonuniqueness of Leray-Hopf solutions to the unforced incompressible 3D Navier-Stokes equations is one of the central open problems in mathematical fluid dynamics. In this paper, we provide, to our knowledge, the first rigorous computer-assisted proof demonstrating such nonuniqueness. Inspired by earlier works in this area, we construct a Leray-Hopf solution in the self-similar setting and then establish the existence of a second solution by analyzing the stability of the linearized operator around this profile, showing that it corresponds to an unstable perturbation. To achieve this, we develop an innovative numerical method that computes candidate solutions with high precision and propose a framework for rigorously establishing exact solutions in a neighborhood of these candidates. A key step is to decompose the linearized operator into a coercive part plus a compact perturbation, which is further approximated by a finite-rank operator up to a small error. The invertibility of the linearized operator restricted to the image of this finite-rank approximation is then rigorously verified using computer-assisted proofs. This certifies the existence of an unstable eigenpair and, consequently, yields a second solution - indeed, infinitely many Leray-Hopf solutions.

• The paper establishes nonuniqueness of Leray-Hopf solutions for unforced 3D Navier-Stokes using a rigorous computer-assisted proof.
• It employs a self-similar framework and finite-rank spectral approximation to verify instability in the linearized operator.
• The proof combines high-precision numerical computation with a posteriori error analysis, providing a template for future nonuniqueness studies in nonlinear PDEs.

Nonuniqueness of Leray-Hopf Solutions to the Unforced Incompressible 3D Navier-Stokes Equation

Introduction and Context

This paper provides a rigorous computer-assisted proof of the nonuniqueness of Leray-Hopf solutions to the unforced incompressible 3D Navier-Stokes equations on $\mathbb{R}^3 \times [0,1]$ with compactly supported, divergence-free initial data. The result directly addresses one of the central open problems in mathematical fluid dynamics: whether the classical weak solutions constructed by Leray and Hopf are unique under the energy inequality and local energy inequality constraints, in the absence of external forcing. The authors establish the existence of infinitely many distinct suitable Leray-Hopf solutions for the same initial data, just missing the Prodi-Serrin regularity threshold for uniqueness.

Mathematical Framework and Main Theorem

The analysis is conducted in the self-similar setting, focusing on scale-invariant initial data of the form $u_{in}(x) = |x|^{-1}A(x/|x|)$, with $A$ satisfying the divergence-free constraint. The self-similar ansatz $u(t,x) = t^{-1/2}\widetilde{U}(x/\sqrt{t})$ leads to a stationary nonlinear elliptic system for $\widetilde{U}$, which is then linearized to study the spectral properties of the associated operator $\mathcal{L}_{\widetilde{U}}$.

The main theorem asserts the existence of infinitely many distinct suitable Leray-Hopf solutions in $L^s([0,1];L^q(\mathbb{R}^3))$ for any $q\geq 2$, $\frac{3}{q}+\frac{2}{s}>1$, with the same compactly supported initial data $u_{loc}\in L^q$ for $q<3$. These solutions are constructed via a bifurcation mechanism associated with an unstable eigenvalue of the linearized operator around a self-similar profile.

Computer-Assisted Proof Strategy

The proof combines high-precision numerical computation with rigorous a posteriori analysis. The approach consists of:

1. Numerical Construction of Candidate Profiles: The authors develop a gradient descent method to optimize the self-similar profile $\overline{U}$ and its associated unstable eigenmode $\overline{v}$, enforcing axisymmetry and parity constraints. The profiles are computed in spherical coordinates using finite element and spectral methods, and then interpolated onto an exactly divergence-free basis.
2. Finite-Rank Spectral Approximation: The linearized operator is decomposed into a coercive part and a compact perturbation. The compact part is approximated by a finite-rank operator using the leading generalized eigenfunctions of the compact matrix $Q$ (derived from the symmetric part of the gradient of $\overline{U}$). The invertibility of the linearized operator on the image of this finite-rank approximation is verified numerically, with rigorous error bounds established via interval arithmetic.
3. A Posteriori Error Analysis: The authors derive sharp estimates for the residuals, nonlinear terms, and finite-rank approximation errors, ensuring that the fixed-point map induced by the perturbed operator is a contraction in a suitable function space. The Schauder fixed-point theorem is then applied to obtain an exact solution in a neighborhood of the numerical candidate.
4. Localization and Bifurcation: The self-similar solution is localized via a Bogovskii decomposition and heat smoothing, yielding compactly supported initial data. The bifurcation analysis, using a Riesz projection onto the unstable eigenspace, produces infinitely many solutions parameterized by the unstable coordinates at $t=1$.

Technical Highlights and Numerical Results

• Coercivity and Spectral Gap: The viscous term provides strong coercivity in $L^2$ and $H^1$, with numerically verified lower bounds (e.g., $0.054$ for the profile equation and $0.16$ for the eigenmode equation). This allows the analysis to be conducted in standard Sobolev spaces without singular weights.
• Low-Rank Verification: The finite-rank spectral approximation is highly efficient, requiring only $m=19$ modes for the profile and $m=24$ for the eigenmode, with matrix sizes on the order of $1200\times 1200$ and spectral checks reduced to $25\times 25$ generalized eigenproblems.
• Analytic Trigonometric Calculus: The use of trigonometric tensor-product representations in $(\beta,\theta)$ enables analytic evaluation of $L^2$ residuals and pointwise envelopes, with rigorous $L^\infty$ bounds and quadrature errors controlled via interval arithmetic.
• Symmetry Breaking: The bifurcating branch of Leray-Hopf solutions is shown to break the base even symmetry of the self-similar profile, consistent with previous numerical observations.
• Strong Numerical Bounds: The residuals for the profile and eigenmode are certified to be $\leq 7\times 10^{-7}$ and $\leq 1.8\times 10^{-5}$, respectively, with all error budgets closed under the fixed-point argument.

Implications and Theoretical Significance

The paper establishes, for the first time, the nonuniqueness of suitable Leray-Hopf solutions to the unforced 3D Navier-Stokes equations via a rigorous computer-assisted proof. This result demonstrates that the energy inequality and local energy inequality are insufficient to guarantee uniqueness in the absence of Prodi-Serrin regularity, even for compactly supported initial data. The construction is robust, relying on spectral instability of self-similar profiles and localization techniques, and is not reliant on convex integration or nonphysical forcing.

The framework developed for finite-rank spectral approximation and a posteriori error analysis is broadly applicable to other nonlinear PDEs with compact perturbations, and provides a template for future computer-assisted proofs in fluid dynamics and related fields.

Future Directions

• Extension to Other Initial Data: The methodology could be adapted to study nonuniqueness for a wider class of initial data, including those with less symmetry or different decay properties.
• Higher Regularity and Singularities: The approach may inform investigations into the regularity and potential singularity formation in Navier-Stokes and related systems, especially in the context of self-similar and backward self-similar solutions.
• Algorithmic Improvements: The analytic trigonometric calculus and interval arithmetic techniques may be further optimized for large-scale spectral problems in computational PDE analysis.
• Connections to Convex Integration: While the solutions constructed here are much more regular than those obtained via convex integration, understanding the interplay between these two mechanisms for nonuniqueness remains an open problem.

Conclusion

This work rigorously demonstrates the nonuniqueness of Leray-Hopf solutions to the unforced incompressible 3D Navier-Stokes equations, using a combination of high-precision numerical computation, finite-rank spectral approximation, and a posteriori error analysis. The result provides a definitive answer to a longstanding open problem in mathematical fluid dynamics, and establishes a robust framework for future computer-assisted proofs in nonlinear PDEs. The implications for the theory of weak solutions, regularity, and uniqueness in Navier-Stokes are substantial, and the techniques developed herein are likely to have broad impact in computational and theoretical analysis of fluid equations.