From interior smallness below the top slice to boundedness at the origin #
Three closing facts for the Gustafson–Kang–Tsai criterion used at the end of the proof of
lem:aniso:closure: the finite halving recursion used to iterate the pressure decay; the passage
from bounds on the interior cylinders Q_r(0, s), s ∈ (-σ, 0), to the Theorem A integrand on
the backward cylinder B_r × (-r², 0) (monotone convergence in the top time, with the force
term controlled by a pointwise bound); and the conclusion BoundedNearOrigin from smallness of
that integrand, by the top-boundary form of CKN's Theorem A and continuity of u inside Q.
If the velocity and pressure integrals over Q_r(0, s) are at most τ r² for every s ∈ (-σ, 0), σ ≤ r², then the Theorem A integrand over the backward cylinder B_r × (-r², 0) is
at most (2τ + (4π/3) M_f^q r^{3q}) r², where M_f bounds the force on Q.
The top-boundary Theorem A in the form used here: there is ε₀ > 0, depending only on q,
such that smallness ≤ ε₀ r² of the Theorem A integrand on one backward cylinder B_r × (-r², 0), r ≤ 1/2, makes (0, 0) a regular point, for a velocity continuous inside Q.