Doubled source cylinders for clipped origin cells #
The pressure slice estimate on a cylinder of radius 2 * r controls its
half-ball of radius r. Weak derivative uniqueness on the intersection with
the origin carrier transfers that bound to a field defined only on the
carrier ball. The resulting time integral uses the clipped cell window.
For centres in the closed origin cylinder, this construction is admissible
through radius (1 - R₁) / 2, twice the origin margin scale.
Carrier cell bounds from slicewise bounds on the pressure #
The one-sided estimate of prop:bootstrap sees the selected pressure gradient
only through the indicator of the backward carrier
parabolicCylinder 0 0 R₁. A cell bound for that indicator is a bound for the
integral of the gradient over the intersection of the cell with the carrier,
and by the product decomposition of that intersection the integral splits into
a time integral of spatial slice integrals whose times all lie in
(-R₁ ^ 2, 0].
This module turns a multiscale slicewise L^{6/5} majorant for the weak
gradients of the pressure slices into exactly those carrier cell bounds. The
majorant is measured on the intersection vec3Ball x r ∩ vec3Ball 0 R₁ of the
cell ball with the carrier ball, and its two time integrals are taken over the
clipped windows. Nothing here refers to the gradient at a time outside the
time factor of the unit cylinder.
The integral of a gradient component over the part of a cell that meets
the backward carrier is bounded by the clipped time integral of a slicewise
L^{6/5} majorant. Only the times in (-R₁ ^ 2, 0], which the domain
hypothesis of thm:A places inside the solution interval, are used.
The two carrier integral constants of the one-sided Morrey transfer, from a
multiscale slicewise majorant with clipped time windows. The majorant
M i x r s bounds the L^{6/5} norm of the slice weak gradient on the
intersection of the cell ball with the carrier ball; hgrowth and hglobal
record the two clipped time integrals it has to satisfy. Every cell scale is
used, which a single-scale slicewise bound cannot replace.
The doubled-radius slice bound transfers to a weak gradient on the carrier ball by uniqueness on the intersection of the two spatial balls.
The carrier-cell integral consumer applied to the actual norm of the fixed field, followed by its doubled-radius bound on the clipped time window.
Centres in the closed origin cylinder admit doubled source cylinders through twice the margin scale, including the endpoint radius.
Suitability supplies the doubled-radius slice estimate for a clipped origin cell, while the fixed derivative is required only on the carrier ball.
One measurable carrier gradient, obtained from suitability, satisfies all clipped cell estimates through twice the origin margin scale.