Gkt Criterion #
Supporting results for the Constantin–Ignatova–Vicol axisymmetric regularity theorem.
The interior regularity criterion of Gustafson, Kang and Tsai (their Theorem 1.1(i) with
p_* = 6, q = 4), in the form in which the proof of lem:aniso:closure consumes it: if the
L⁴_t L⁶_x norm of the velocity on the backward cylinders
B(r) × (−r², 0) tends to zero as r → 0, then (0,0) is a regular point, that is, u is
bounded on a backward cylinder at the origin (design note R3).
The hypothesis is the fourth power of ‖u‖_{L⁴(−r²,0;L⁶(B(r)))}, which tends to zero exactly
when the norm does. It is stronger than the smallness that Gustafson, Kang and Tsai 2007
requires, and it is what the
proof of lem:aniso:closure produces.
This statement bundles three things:
the criterion itself; its application at the top boundary point (0,0), which the footnote in the
proof of lem:aniso:annulus justifies from the structure of the proof in Gustafson, Kang and
Tsai 2007 (backward cylinders inside Q,
and a local energy inequality which, for a pair smooth on compact subsets of Q, is an identity
at every s < 0 — this is why hu, hp, hf are premises here and not mere conveniences);
and the reduction of foot:aniso:pressure, since the manuscript applies the criterion to the
pair (u, π − π_f) with the divergence-free force f − ∇π_f, whereas this statement takes the
manuscript's own bounded force (design note R1). The pressure in L^{3/2}(Q) and the energy
class up to t = 0, which the backward cylinders reach, come from henergy (design note R6).