The commutator-term slice integration by parts #
Before the time s, ∫ Γ (w (∂ₜχ − Δχ) − 2 ∇χ · ∇w) = ∫ w K₀, where χ is the backward
parabolic cutoff and K₀ = Γ (∂ₜχ + Δχ) + 2 ∇_zΓ · ∇χ is the commutator kernel. The proof
integrates by parts in each spatial direction i against F_i := Γ ∂ᵢχ, which (unlike Γχ
itself) is compactly supported inside the window before the time s, since ∂ᵢχ already
vanishes off the window there; this gives ∫ F_i ∂ᵢw = -∫ (∂ᵢΓ · ∂ᵢχ + Γ ∂ᵢ²χ) w, and summing
over i and rearranging produces exactly K₀. This is the third and last of the three slice
integrations by parts in the cut-off heat representation of the interior estimates in the proof
of lem:aniso:annulus.
Γ ∂ᵢχ, the summand of the divergence term of K₀.
Equations
- CIV.serrinHeatWeightDeriv x s ρ i z = CKN.Foundation.Heat.heatKernel (x - z.1) (s - z.2) * CKN.spatialPartial (CIV.serrinCutoff x s ρ) i z
Instances For
The directional derivative of the ball cutoff vanishes strictly outside its radius.
Before the time s and off the window, serrinHeatWeightDeriv vanishes; at or after s
it vanishes because the heat kernel itself does.
Before the time s, serrinHeatWeightDeriv is smooth on the whole slice, compactly
supported in the window's spatial ball, and its own spatial partial in the same direction
recovers the two pieces of the divergence-form expansion of K₀.
The spatial partial of serrinHeatWeightDeriv in the same direction, before the time s,
by the product rule.
serrinHeatWeightDeriv vanishes on a whole neighbourhood of any point before the time s
that lies off the window.
Before the time s and off the window, the spatial partial of serrinHeatWeightDeriv
vanishes.
serrinHeatWeightDeriv is smooth before the time s.
The commutator-term integration by parts in a single spatial direction: before s,
∫ (Γ ∂ᵢχ) ∂ᵢw = -∫ ∂ᵢ(Γ ∂ᵢχ) w, together with the integrability of the left integrand.
The commutator-term slice integration by parts: before s,
∫ Γ (w (∂ₜχ − Δχ) − 2 ∇χ · ∇w) = ∫ w K₀.