The representative-to-Cartesian L² transfer for eq:aniso:closure:mixed:bound #
The Cauchy–Schwarz step of the mixed-term estimate needs L² bounds, in terms of the tested
enstrophy Y = cutoffEnstrophy χ u and its dissipation M = cutoffEnstrophyDissipation χ u,
for the four cylindrical factors of eq:aniso:closure:mixed — ∂_r u_z, ∂_z∂_r u_z,
ω_z, ∂_z ω_z — read at the meridional representative meridional (polarR x) (x 2) of
a general point x, since that is how mixedTermFactorDz is built. The elliptic and
div–curl bounds (CIV.Closure.EllipticBoundsAnnulusBound) instead bound these quantities read
at x itself.
The transfer is rotation invariance: ∂_r u_z and ∂_z∂_r u_z are each one term of a
Frobenius-norm sum (gradientSq_eq_polarR, dzGradientSq_eq_polarR) that agrees at x and at
its representative, and likewise ω_z and ∂_z ω_z are each one term of a curl-energy sum
(curlSq_eq_polarR, and gradientSq_eq_polarR applied to the vorticity field itself, which is
axisymmetric by isAxisymmetricOn_vorticityField). Each pointwise domination integrates
directly against χ² and composes with the x-level bound.
The polar representative map, on the open unit ball #
Global continuity and L² membership from a cutoff #
A function continuous on the unit ball and vanishing off the support of a compactly supported cutoff inside it is continuous everywhere.
A function continuous on the unit ball and vanishing off the support of a compactly
supported cutoff inside it is in L².
The square of a function continuous on the unit ball and vanishing off the support of a compactly supported cutoff inside it is integrable.
ω_z, read at the representative, transfers to Y exactly #
The χ-weighted L² norm of ω_z, read at the meridional representative, is bounded by
the tested enstrophy Y: ω_z is one term of the rotation-invariant curl-energy sum.
∂_z ω_z, read at the representative, transfers to M exactly #
The χ-weighted L² norm of ∂_z ω_z, read at the meridional representative, is bounded
by the enstrophy dissipation M: the vorticity field is itself axisymmetric
(isAxisymmetricOn_vorticityField), so ∂_z ω_z is one term of the rotation-invariant
Frobenius-gradient sum of ω, which M already is
(cutoffEnstrophyDissipation_eq_integral_sq_fderiv_curlVec).
∂_r u_z, read at the representative, transfers to Y up to a constant #
The χ-weighted L² norm of ∂_r u_z, read at the meridional representative, is bounded
by C (Y + 1) for a constant C from the order-≤ 1 annulus data: ∂_r u_z is one term of
the rotation-invariant Frobenius-gradient sum of u (gradientSq_eq_polarR), and the full sum
is bounded at x itself by integral_cutoff_sq_spatialPartial_le_of_multiPartial_le.
∂_z∂_r u_z, read at the representative, transfers to M up to a constant #
The χ-weighted L² norm of ∂_z∂_r u_z, read at the meridional representative, is
bounded by C (M + 1) for a constant C from the order-≤ 2 annulus data: ∂_z∂_r u_z
commutes to ∂_r∂_z u_z (spatialSecondPartial_comm), which is one term of the
rotation-invariant Frobenius-gradient sum of ∂_z u (dzGradientSq_eq_polarR), and the full
sum is bounded at x itself by integral_cutoff_sq_spatialSecondPartial_le_of_multiPartial_le.
The two representative-level bounds, multiplied by the swirl factor #
The χ-weighted L² norm of u_θ · ∂_z∂_r u_z, both read at the meridional
representative, is bounded by W² C (M + 1): |u_θ| ≤ W on tsupport χ (hW), and the
∂_z∂_r u_z factor alone is bounded by C (M + 1)
(exists_integral_sq_cutoff_mul_secondPartial_rep_le).
The χ-weighted L² norm of u_θ · ∂_r u_z, both read at the meridional representative,
is bounded by W² C (Y + 1): |u_θ| ≤ W on tsupport χ (hW), and the ∂_r u_z factor
alone is bounded by C (Y + 1) (exists_integral_sq_cutoff_mul_firstPartial_rep_le).