[Paper Review] Non-uniqueness of Weak Solutions to Hyperviscous Navier-Stokes Equations -- On Sharpness of J.-L. Lions Exponent

Tianwen Luo, Edriss S. Titi|arXiv (Cornell University)|Aug 23, 2018

Navier-Stokes equation solutions16 references8 citations

TL;DR

This paper establishes the non-uniqueness of weak solutions to the 3D hyperviscous Navier-Stokes equations with fractional Laplacian viscosity (−Δ)θ for exponents θ < 5/4, proving that J.-L. Lions' exponent 5/4 is sharp. Using a convex integration scheme with intermittent Beltrami flows, the authors construct infinitely many compactly supported weak solutions, including those with zero initial data, by controlling viscosity-induced error through concentration and oscillation mechanisms in the building blocks.

ABSTRACT

Abstract: Using the convex integration technique for the three-dimensional Navier–Stokes equations introduced by Buckmaster and Vicol, it is shown the existence of non-unique weak solutions for the 3D Navier–Stokes equations with fractional hyperviscosity (-Δ)θ, whenever the exponent θ is less than Lions’ exponent 5/4, i.e., when θ&lt;5/4.

Motivation & Objective

• Address the sharpness of J.-L. Lions' exponent 5/4 in the context of weak solution uniqueness for the 3D hyperviscous Navier-Stokes equations.
• Extend the non-uniqueness result of Buckmaster and Vicol from the standard Navier-Stokes equations to the fractional hyperviscous case.
• Establish the existence of infinitely many weak solutions with compactly supported time and zero initial data for θ < 5/4.
• Analyze the role of the fractional viscosity (−Δ)θ in controlling the error terms within the convex integration framework.
• Show that the critical threshold θ = 5/4 is sharp by demonstrating that the error control fails for θ ≥ 5/4.

Proposed method

• Adapt the convex integration framework of Buckmaster and Vicol using intermittent Beltrami flows as building blocks for the hyperviscous system.
• Construct a sequence of approximate solutions (vq, Rq) satisfying a perturbed system with stress tensor Rq, iteratively improving the error control.
• Employ a frequency-localized, intermittent Beltrami wave construction with parameters λ (frequency), σ (spacing), r (number of modes), and µ (temporal oscillation) tuned via scaling laws.
• Control the viscous error term ν(−Δ)θwq+1 by balancing the concentration of intermittent flows with high-frequency oscillations, ensuring decay for θ < 5/4.
• Use Fourier projection and Littlewood-Paley decomposition to estimate the Lp norms of error terms, particularly focusing on the critical term ∥|∇|2θ−1wq+1∥L∞tLp x.
• Apply a symmetric anti-divergence operator to reconstruct the stress tensor and ensure divergence-free structure throughout the iteration.

Experimental results

Research questions

• RQ1 Is J.-L. Lions' exponent 5/4 sharp for the uniqueness of weak solutions in the 3D hyperviscous Navier-Stokes equations?
• RQ2 Can non-uniqueness of weak solutions be established for θ < 5/4 using convex integration techniques?
• RQ3 What is the role of intermittent Beltrami flows in controlling the viscous error term ν(−Δ)θv in the convex integration scheme?
• RQ4 How does the scaling of the frequency parameter λ and the viscosity exponent θ interact to determine the controllability of the error terms?
• RQ5 Can the construction yield weak solutions that are compactly supported in time and have zero initial data for θ < 5/4?

Key findings

• For any θ ∈ [1, 5/4), there exist infinitely many weak solutions to the 3D hyperviscous Navier-Stokes equations that are compactly supported in time.
• Non-uniqueness is established via a convex integration scheme that constructs weak solutions arbitrarily close in W^{2θ−1,1}_x norm to a given smooth, compactly supported divergence-free vector field.
• The critical threshold θ = 5/4 is sharp: the viscous error term ∥|∇|2θ−1wq+1∥L∞tLp x becomes uncontrollable for θ ≥ 5/4, preventing convergence of the iteration.
• Infinitely many weak solutions with zero initial data exist, as demonstrated by constructing solutions close to a nontrivial compactly supported initial field.
• By choosing parameters λq+1 sufficiently large and setting r = λαq+1, σ = λ−(α+1)/2q+1, µ = λ(5α+1)/4q+1 with α ∈ (max{0, 2(2θ−1)/3}, 1), the error estimates (4), (6), and (7) in the iteration lemma are satisfied.
• The construction yields weak solutions in C0_t L2_x ∩ L∞_t W^{2θ−1,1}_x, confirming the regularity threshold for non-uniqueness.