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Nonlocal Hyperdissipative Perturbations of the Three Dimensional Navier-Stokes System

Veli Shahmurov, Rishad Shahmurov

Abstract

We study the three-dimensional incompressible Navier-Stokes system on $\mathbb{R}^3$ with an additional dissipative nonlocal term \[ \partial_t u + (u\cdot\nabla)u + \nabla p = νΔu + Lu, \qquad {\rm div}\, u = 0, \] where $L$ is a self-adjoint Fourier multiplier whose symbol is comparable to $-|ξ|^{2α}$ for some $α>1$. We first identify a sharp Fourier-symbol criterion distinguishing lower-order convolution perturbations from genuinely regularizing nonlocal corrections. In the resulting hyperdissipative class we prove the exact $L^2$ energy identity, global weak solvability for every $α>1$, and local strong well-posedness in $H^s(\mathbb{R}^3)$ for $s>\frac52$. We then show that the Lions exponent $α=\frac54$ remains the critical energy-growth threshold in this nonlocal setting: if $α\ge \frac54$, every $H^s$ solution is global, while for every $α>1$ one has global strong solvability for sufficiently small $H^s$ data. Finally, for the vanishing-hyperdissipation approximation of the classical three-dimensional Navier-Stokes equations, we prove a near-singular divergence principle: if the classical flow blows up at a first singular time $T_*$ in a continuation norm $X$, then the corresponding regularized family cannot remain uniformly bounded in $X$ on any interval approaching $T_*$. This identifies the precise point at which the fixed-parameter global theory degenerates in the Navier-Stokes limit.

Nonlocal Hyperdissipative Perturbations of the Three Dimensional Navier-Stokes System

Abstract

We study the three-dimensional incompressible Navier-Stokes system on with an additional dissipative nonlocal term where is a self-adjoint Fourier multiplier whose symbol is comparable to for some . We first identify a sharp Fourier-symbol criterion distinguishing lower-order convolution perturbations from genuinely regularizing nonlocal corrections. In the resulting hyperdissipative class we prove the exact energy identity, global weak solvability for every , and local strong well-posedness in for . We then show that the Lions exponent remains the critical energy-growth threshold in this nonlocal setting: if , every solution is global, while for every one has global strong solvability for sufficiently small data. Finally, for the vanishing-hyperdissipation approximation of the classical three-dimensional Navier-Stokes equations, we prove a near-singular divergence principle: if the classical flow blows up at a first singular time in a continuation norm , then the corresponding regularized family cannot remain uniformly bounded in on any interval approaching . This identifies the precise point at which the fixed-parameter global theory degenerates in the Navier-Stokes limit.

Paper Structure

This paper contains 27 sections, 18 theorems, 137 equations, 2 figures, 1 table.

Table of Contents

  1. Keywords.
  2. MSC 2020.
  3. Introduction
  4. Model, assumptions, and main results
  5. Fourier-symbol criterion, scaling, and energy identity
  6. A symbol criterion for convolution modifications
  7. Scaling and the Lions threshold
  8. The exact energy identity
  9. Global weak solutions
  10. Local strong well-posedness
  11. Global regularity at and above the Lions exponent, and global small-data theory
  12. Large-data global regularity for $\alpha\ge \frac{5}{4}$
  13. Global strong solutions for sufficiently small data
  14. Vanishing hyperdissipation, Galerkin truncation, and the Navier--Stokes limit
  15. The vanishing-hyperdissipation family
  16. ...and 12 more sections

Key Result

Theorem 2.6

Assume that $m\in\mathfrak M_\alpha$ for some $\alpha>1$. $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Linear damping rate $\lambda_\alpha(k)=\nu k^2+\mu k^{2\alpha}$ for $\alpha=1$, $\alpha=\frac{5}{4}$, and $\alpha=\frac{3}{2}$. The separation between the curves increases rapidly at high wavenumbers, reflecting the growing strength of the nonlocal dissipation.
  • Figure 2: Decay of a representative Fourier mode with $k_0=8$ under the linearized evolution. Larger $\alpha$ produces noticeably faster high-frequency damping.

Theorems & Definitions (48)

  • Definition 2.1: Hyperdissipative symbol of order $2\alpha$
  • Remark 2.2: Hilbert-space operator variant
  • Remark 2.3
  • Definition 2.4: Weak solution
  • Definition 2.5: Strong solution
  • Theorem 2.6: Main results
  • Proposition 3.1: Harmless versus genuinely regularizing kernels
  • proof
  • Remark 3.2: Time convolution
  • Proposition 3.3: Scaling and critical growth
  • ...and 38 more