Uniform collar coefficient #
The cutoff derivative constants and the lower order coefficient give a radius-independent bound on the collar error.
theorem
ESS.uc_collar_coefficients_uniform
{ρ c₁ scale : ℝ}
(hρ : 4 ≤ ρ)
(hc₁ : 0 ≤ c₁)
(hscale0 : 0 ≤ scale)
(hscale1 : scale ≤ 1)
:
have A := 32 + 3 * (CKN.cutoffSecondDerivativeConstant / ρ ^ 2) + 3 * c₁ * scale * (CKN.cutoffGradientConstant / ρ);
have B := 18 * (CKN.cutoffGradientConstant / ρ);
have M :=
32 + 3 * CKN.cutoffSecondDerivativeConstant + 3 * c₁ * CKN.cutoffGradientConstant + 18 * CKN.cutoffGradientConstant;
0 ≤ A ∧ 0 ≤ B ∧ A ≤ M ∧ B ≤ M
The quadratic collar coefficient is bounded independently of the normalized radius and of scales at most one.
theorem
ESS.uc_gaussian_collar_error_uniform_integral_le
{ρ c₁ scale : ℝ}
(hρ : 4 ≤ ρ)
(hc₁ : 0 ≤ c₁)
(hscale0 : 0 ≤ scale)
(hscale1 : scale ≤ 1)
{v : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3}
{Dv : CKN.Foundation.Parabolic.ParabolicPoint → Fin 3 → CKN.Foundation.Parabolic.Vec3}
{D2v : CKN.Foundation.Parabolic.ParabolicPoint → Fin 3 → Fin 3 → CKN.Foundation.Parabolic.Vec3}
{Dtv : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3}
(hweak : CKN.HasSpaceTimeWeakDerivs (CKN.Foundation.Parabolic.vec3Ball 0 ρ) (Set.Ioo 0 2) v Dv D2v Dtv)
(hL2 :
∫⁻ (z : CKN.Foundation.Parabolic.ParabolicPoint) in ucCylinder ρ, ‖v z‖ₑ ^ 2 + ‖Dv z‖ₑ ^ 2 + ‖D2v z‖ₑ ^ 2 + ‖Dtv z‖ₑ ^ 2 < ⊤)
:
have a := 1 / 100 * ρ ^ 2 / (2 * Real.log (CKN.gaussCarlemanTimeWeight (3 / 2)));
have A := 32 + 3 * (CKN.cutoffSecondDerivativeConstant / ρ ^ 2) + 3 * c₁ * scale * (CKN.cutoffGradientConstant / ρ);
have B := 18 * (CKN.cutoffGradientConstant / ρ);
have M :=
32 + 3 * CKN.cutoffSecondDerivativeConstant + 3 * c₁ * CKN.cutoffGradientConstant + 18 * CKN.cutoffGradientConstant;
∫ (z : CKN.Foundation.Parabolic.ParabolicPoint) in ucCylinder ρ, ucGaussianWeight a z * (3 * (if z ∈ ucCutoffRegion ρ then
A * CKN.Foundation.Parabolic.vec3EuclideanNorm (v z) + B * √(CKN.spatialGradientSq v Dv z)
else 0) ^ 2) ≤ 6 * Real.exp (-ρ ^ 2 / 100) * M ^ 2 * ∫ (z : CKN.Foundation.Parabolic.ParabolicPoint) in ucCylinder ρ, CKN.Foundation.Parabolic.vec3EuclideanNorm (v z) ^ 2 + CKN.spatialGradientSq v Dv z
The integrated collar error has a radius-independent coefficient depending only on the lower order constant.