Shape Differentiation and Anisotropy Onset of Quadratic Invariant-Manifold Jets in Surface Navier–Stokes Flow

Abstract

For a C1 family of smooth closed genus-zero surfaces, we study the quadratic normal jet of unforced surface Navier–Stokes flow near the zero equilibrium. Under an isolated rank-three Riesz slow block and a three-weight slow–fast gap, local C2 invariant graphs share this quotient-valued second derivative after tangent-input identification, without requiring uniqueness of the full graph. A fixed-complement representative is C1 in shape in a bilinear Sobolev-operator norm with infinite-dimensional output. An exact covariance law handles admissible comparisons, including spatial transports with derivative loss. At the unit sphere, the jet vanishes. For symmetric affine deformations, its first variation is a degree-three spherical-harmonic tensor with coefficient −1/(45ν), vanishing exactly for scalar deformation matrices. For each fixed nonscalar D, the jet is therefore nonzero for all sufficiently small 0<|δ|<δ0(D). The onset is local and non-quantitative; no explicit value of δ0(D) is obtained.

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BibTeX RIS

BibTeX

@article{abdelaty2026shape,
  title = {{Shape Differentiation and Anisotropy Onset of Quadratic Invariant-Manifold Jets in Surface Navier–Stokes Flow}},
  author = {Amr M. Y. Abdelaty and Hesham A. Abdelghafar and Ibrahim S. Elshazly},
  journal = {Mathematics},
  year = {2026},
  month = nov,
  doi = {10.3390/math14214044},
  url = {https://www.mdpi.com/2227-7390/14/21/4044},
  note = {CC BY 4.0},
}

RIS

TY  - JOUR
TI  - Shape Differentiation and Anisotropy Onset of Quadratic Invariant-Manifold Jets in Surface Navier–Stokes Flow
AU  - Amr M. Y. Abdelaty
AU  - Hesham A. Abdelghafar
AU  - Ibrahim S. Elshazly
PY  - 2026
DA  - 2026/11/04
JO  - Mathematics
DO  - 10.3390/math14214044
UR  - https://www.mdpi.com/2227-7390/14/21/4044
AB  - For a C1 family of smooth closed genus-zero surfaces, we study the quadratic normal jet of unforced surface Navier–Stokes flow near the zero equilibrium. Under an isolated rank-three Riesz slow block and a three-weight slow–fast gap, local C2 invariant graphs share this quotient-valued second derivative after tangent-input identification, without requiring uniqueness of the full graph. A fixed-complement representative is C1 in shape in a bilinear Sobolev-operator norm with infinite-dimensional output. An exact covariance law handles admissible comparisons, including spatial transports with derivative loss. At the unit sphere, the jet vanishes. For symmetric affine deformations, its first variation is a degree-three spherical-harmonic tensor with coefficient −1/(45ν), vanishing exactly for scalar deformation matrices. For each fixed nonscalar D, the jet is therefore nonzero for all sufficiently small 0<|δ|<δ0(D). The onset is local and non-quantitative; no explicit value of δ0(D) is obtained.
SN  - 2227-7390
ER  -

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