Lin34 Slice Mean Free #
Part of the Caffarelli–Kohn–Nirenberg partial regularity proof.
Integrability of the mean-free velocity cube on a parabolic cylinder #
This file supplies the integrability of the integrand of the local quantity
C_hat(z,ρ) on a contained parabolic cylinder, the estimate used in
prop:lin34 of paper/ckn.tex (equation eq:Chat). Concretely, if the
velocity u and its cube |u|^3 are integrable on the one-sided parabolic
cylinder parabolicCylinder x t r, then so is the cube of the mean-free
velocity meanFreeVec u x r w.2 w.1, the spatial mean-free part of u at
time w.2 over the ball vec3Ball x r.
The proof converts the cylinder integral to the product measure on
vec3Ball x r ×ˢ Ioc (t - r^2) t, uses the pointwise mean-oscillation bound
on almost every spatial slice, and applies Tonelli's theorem. The two helper
lemmas below are private.
On a spatial slice at time s on which u(·,s) and |u(·,s)|^3 are
integrable, the L³ mean oscillation of u(·,s) over the ball vec3Ball x r
is bounded by 8 times the L³ norm of u(·,s). This is the slicewise
form of the mean-oscillation estimate behind eq:Chat in paper/ckn.tex.
The cube of the mean-free velocity is integrable on a parabolic cylinder
parabolicCylinder x t r whenever the velocity and its cube are. This is the
integrability of the integrand of C_hat(z,ρ) used in prop:lin34 of
paper/ckn.tex (equation eq:Chat).