Display (3.5) on a slice, with no condition on the divergence of the force #
Display (3.5) of the pressure-gradient section selects, for almost every time
of the one-sided interval J_ρ, a weak spatial gradient of the pressure slice
on the half ball B_{ρ/2}(x₀) together with its L^{6/5} bound. The general
form of that selection carries three analytic hypotheses: the interior
regularity of the harmonic part of the local pressure decomposition, the
first-order identification of the first pressure potential, and the weak
gradient of the two force potentials.
The class def:sws imposes no condition on the spatial divergence of the
force, so the two force potentials of eq:pk do not cancel and the display
must carry them. This file discharges the first hypothesis outright, reduces
the second to the divergence-form characterization of the slice source V, and
discharges the third from the force data of the solution alone: the bound
gains the ρ^{-1/2}-weighted force term sliceForceGradientBound.
Conditional display (3.5) on a slice of a suitable weak solution. The
weak-gradient construction hP1 remains an explicit input; the interior
regularity of the harmonic part, first-order identification, and force slot
are discharged from the solution data, with no condition on the divergence
of the force.
Almost every slice of the pressure has a Vec3-valued weak spatial gradient on
B_{ρ/2}(x₀), in L^{6/5} there, bounded by the Calderón–Zygmund norm of the
divergence-form source, the ρ^{-1/2}-weighted L^{3/2} norm of the pressure,
and the force term of the display.