Slice Selected Gradient Force Slices #
Part of the Caffarelli–Kohn–Nirenberg partial regularity proof.
The slice data of the two force potentials of eq:pk #
The local pressure decomposition eq:pk carries the force through the two potentials
pressureP7 η f s and pressureP8 η f s of CKN.Pressure.DecompositionPotentials,
whose densities on a time slice are η y * f (y, s) j and
spatialDeriv η j y * f (y, s) j. Display (3.5) of the pressure-gradient section
differentiates these potentials on the inner ball of η, and therefore needs, on almost
every time slice of a suitable weak solution in the sense of def:sws: the L^{6/5}
membership and compact support of the first density, the integrability of the second
density, and an L¹ bound for the second density in terms of the force on the ball.
The first fact is an immediate consequence of the compact support of the cutoff. The
second and third transfer the slice membership of the force, supplied by
sws_force_memLp_slice_ae, to the two densities: the cutoff is bounded by one, so the
first density inherits every L^p bound of the force, while the derivative of the cutoff
is bounded by cutoffGradientConstant / ρ, which gives both the integrability of the
second density and its L¹ estimate.
The first density η y * f (y, s) j of the force potentials pressureP7 η f s and
pressureP8 η f s of eq:pk has compact support, so that it is a legitimate source for
the Newtonian potentials of display (3.5) of the pressure-gradient section.
The slice data of the two force densities of eq:pk on a ball, given only that the
force is L^q on that ball for some q ≥ 6/5: the force itself is L¹ there, the first
density is L^{6/5} on the whole space, the second density is integrable, and its L¹
size is controlled by cutoffGradientConstant / ρ times the L¹ size of the force on
the ball. These are the hypotheses that display (3.5) of the pressure-gradient section
uses when differentiating the potentials pressureP7 η f s and pressureP8 η f s.
The slice data of the two force densities of eq:pk holds on almost every time slice
of the lower half of a parabolic cylinder whose closure lies in the carrier of a suitable
weak solution: the force is L^q there with q > 5/2, hence L^1, the first density is
L^{6/5} on the whole space, the second density is integrable, and its L¹ size is
controlled by cutoffGradientConstant / ρ times the L¹ size of the force on the ball.
This is the almost-everywhere statement that display (3.5) of the pressure-gradient
section presupposes when it differentiates pressureP7 η f s and pressureP8 η f s.