The Gustafson–Kang–Tsai criterion at the endpoint p* = 6, q = 4 #
If ∫_{-r²}^0 ‖u(t)‖_{L⁶(B_r)}^4 dt → 0 as r → 0 for a suitable weak solution on
Q = B(1) × (-1, 0) in the energy class, smooth inside Q, with a C²-bounded force, then
(0, 0) is a regular point. This is Theorem 1.1(i) of Gustafson, Kang and Tsai 2007 at the
endpoint 3/p* + 2/q = 1 of its range, where their Lemmas 3.1 and 3.2 are not needed: Hölder
gives the scale-invariant L³ smallness of u directly on every small cylinder, the integrated
Lin estimate (their Lemma 3.4) iterated finitely many times makes the pressure quantity small on
interior cylinders Q_r(0, s) uniformly in s close to 0, and CKN's Theorem A at the top
boundary concludes. This is the criterion applied at the end of the proof of lem:aniso:closure.
The pressure recursion along the scales θ^n ρ₀, n < N, at every base time s ∈ (-3 (θ^N ρ₀)², 0): each step halves D up to an additive term controlled by the velocity level η and
the force constant C_f.
The uniform descent: for every τ > 0 and ρ_max > 0 there are a radius r ≤ ρ_max, r ≤ 1/2, and a window σ ∈ (0, r²] such that the L³ integral of u and the L^{3/2} integral
of π over Q_r(0, s) are at most τ r² for every s ∈ (-σ, 0).
The Gustafson–Kang–Tsai criterion with p* = 6, q = 4 at the top boundary point (0, 0)
(Theorem 1.1(i) of Gustafson, Kang and Tsai 2007), in the form consumed at the end of the proof
of lem:aniso:closure.