The spatial gradient of a pressure slice on a whole ball #
Display (3.5) produces, on a backward cylinder of radius ρ, a weak spatial
gradient of the pressure slice only on the concentric ball of radius ρ / 2,
and the radius ρ is limited by the time window that the cylinder must fit
into. A single application therefore never reaches a prescribed ball.
Applying it instead on every sufficiently small cylinder of a fixed countable
family and gluing the outcomes reaches every ball whose closure stays inside
the spatial domain, at almost every time of the whole time set. The gluing is
exists_weakPartialDerivOn_of_local; the countable family comes from a
countable dense set of centres together with rational radii and rational top
times, so that the almost-everywhere conditions can be intersected.
Because a locally integrable weak partial derivative is unique almost everywhere on an open set, every bound proved for a slice gradient on a sub-ball is inherited by the glued field there. That is the mechanism that makes a bound available at every cell scale rather than at one scale only.
Small backward cylinders around an interior space-time point #
Around a point of an open Euclidean ball and a time interior to the time set there is a backward parabolic cylinder with rational radius and rational top time, centred at a point of any prescribed dense set, whose closure lies in the product of the ball and the time set, whose half-radius ball still contains the point and lies in the ball, and whose time window contains the given time. This is the geometric step that lets one apply a slice estimate on arbitrarily small cylinders drawn from a fixed countable family.
Around a point x of an open Euclidean ball and a time s interior to an open time
set there is a backward parabolic cylinder with rational radius and rational top time,
centred at a point of any prescribed dense set S, whose half-radius ball contains x
and lies in the original ball, whose closure lies in the product of the ball and the time
set, and whose time window contains s.
Gluing weak partial derivatives over an open cover #
Weak partial derivatives produced separately on the members of an open cover agree almost everywhere on the overlaps, because a locally integrable weak partial derivative is unique almost everywhere on an open set. They therefore glue: over a countable open cover the pieces are represented by one function, and by locality that function is the weak partial derivative on the union.
The second statement is the form used in practice. It says that the existence
of a weak partial derivative is a purely local matter: if every point of an
open set has a neighbourhood on which u has a locally integrable ith weak
partial derivative, then u has one on the whole set. Second countability of
Fin d → ℝ reduces the given family to a countable subfamily.
Weak partial derivatives given on the members of a countable open cover of
U glue to a single locally integrable weak partial derivative on U, which
agrees almost everywhere with each given piece.
Existence of a weak partial derivative is local: a function with a locally
integrable ith weak partial derivative near every point of an open set has
one on the whole set.
Any bound proved for one weak partial derivative on an open subset is inherited by every weak partial derivative of the same function on a larger set. Both are locally integrable there, so they agree almost everywhere.
From the slice estimate on every admissible small backward cylinder to a weak spatial derivative on a whole ball, at almost every time.
The hypothesis is exactly the shape display (3.5) delivers: for each centre, top time and radius whose closed cylinder lies in the space-time domain, a weak spatial derivative on the concentric ball of half the radius, for almost every time of that cylinder's window.
The same statement with the slice estimate's own carrier notation.
The unit-cylinder domain hypothesis of the origin carrier puts every ball of radius below one inside the spatial domain.
The carrier that the origin-cell slice clause asks for, produced from the small-cylinder form of display (3.5). The conclusion is the first clause of the origin-pressure-gradient slice data with the majorant clause removed: the majorant has to come from the quantitative slice bound, not from this qualitative statement.