One step of the pressure decay at interior base points below the top slice #
The pressure quantity D(z, r) = r⁻² ∫_{Q_r(z)} |π|^{3/2} obeys the integrated Lin estimate of
CKN (general force clause): for r = θρ with θ ≤ 1/2,
D(θρ) ≤ A θ D(ρ) + A (8 θ⁻² γ(ρ)³ + θ^{3/2} λ(ρ)^{3/2}), where the mean-free velocity quantity
has been bounded by 8 γ³. This file states that inequality at base points (0, s), s < 0,
together with the auxiliary facts used to iterate it uniformly in s: the containment of the
closed cylinders in Q, the comparison of an interior cylinder with a backward cylinder of twice
the radius at the origin, the conversion between the L³ integral and γ³, between the
pressure integral and D, and the crude bound on D by the global L^{3/2} norm of π.
This is Lemma 3.4 of Gustafson, Kang and Tsai 2007 in the form used for the criterion of
lem:aniso:closure.
The closed parabolic cylinder of radius r < 1 based at (0, s), s < 0, lies in the unit
cylinder when its bottom time s - r² exceeds -1.
An interior cylinder Q_ρ(0, s) with -3ρ² ≤ s < 0 lies in the backward cylinder of radius
2ρ at the origin, so an L³ bound η (2ρ)² there gives the bound 4ηρ² on Q_ρ(0, s).
An L³ bound c ρ² on Q_ρ(z) is the bound γ(z, ρ)³ ≤ c.
For a suitable weak solution on a cylinder with closure in the domain, the pressure integral
∫_{Q_r(z)} |π|^{3/2} equals r² D(z, r).
The pressure quantity D is nonnegative.
One step of the integrated Lin pressure estimate with a general force at ratio θ ≤ 1/2,
with the mean-free velocity quantity bounded by 8 γ³.
The crude bound D((0, s), ρ) ≤ ρ⁻² ‖π‖_{L^{3/2}(Q)}^{3/2}, uniform in the base time s.