Local data under the Gaussian rescaling #
The hypotheses of lem:bu-gaussian restrict to the physical cylinder used by
its parabolic change of variables, and then transfer to the normalized ball.
theorem
ESS.buGaussian_rescaled_data
(c₁ A : ℝ)
(hc₁ : 0 < c₁)
(w : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3)
(Dw : CKN.Foundation.Parabolic.ParabolicPoint → Fin 3 → CKN.Foundation.Parabolic.Vec3)
(D2w : CKN.Foundation.Parabolic.ParabolicPoint → Fin 3 → Fin 3 → CKN.Foundation.Parabolic.Vec3)
(Dtw : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3)
(hcont : ContinuousOn w (buHalfSpace ×ˢ Set.Ico 0 1))
(hzero : ∀ (x : CKN.Foundation.Parabolic.Vec3), 0 < x 2 → w (x, 0) = 0)
(hderiv : CKN.HasSpaceTimeWeakDerivs buHalfSpace (Set.Ioo 0 1) w Dw D2w Dtw)
(hL2 :
∀ S ⊆ buHalfCylinder,
Bornology.IsBounded S →
∫⁻ (z : CKN.Foundation.Parabolic.ParabolicPoint) in S, ‖Dw z‖ₑ ^ 2 + ‖D2w z‖ₑ ^ 2 + ‖Dtw z‖ₑ ^ 2 < ⊤)
(hineq :
∀ᵐ (z : CKN.Foundation.Parabolic.ParabolicPoint) ∂MeasureTheory.volume.restrict buHalfCylinder, (CKN.Foundation.Parabolic.vec3EuclideanNorm fun (i : Fin 3) => Dtw z i + ∑ j : Fin 3, D2w z i j j) ≤ c₁ * (√(CKN.spatialGradientSq w Dw z) + CKN.Foundation.Parabolic.vec3EuclideanNorm (w z)))
(hgrowth :
∀ z ∈ buHalfCylinder,
CKN.Foundation.Parabolic.vec3EuclideanNorm (w z) ≤ Real.exp (A * CKN.Foundation.Parabolic.vec3EuclideanNorm z.1 ^ 2))
(x : CKN.Foundation.Parabolic.Vec3)
(t γ : ℝ)
(hx₃ : 2 < x 2)
(ht : 0 < t)
(hγ : γ ≤ 1 / 12)
(htγ : t < γ)
:
have scale := √(3 * t);
have ρ := (x 2 - 1) / scale;
CKN.HasSpaceTimeWeakDerivs (CKN.Foundation.Parabolic.vec3Ball 0 ρ) (Set.Ioo 0 2) (ucScaledField x scale w)
(ucScaledDw x scale Dw) (ucScaledD2w x scale D2w) (ucScaledDtw x scale Dtw) ∧ ∫⁻ (z : CKN.Foundation.Parabolic.ParabolicPoint) in ucCylinder ρ, ‖ucScaledField x scale w z‖ₑ ^ 2 + ‖ucScaledDw x scale Dw z‖ₑ ^ 2 + ‖ucScaledD2w x scale D2w z‖ₑ ^ 2 + ‖ucScaledDtw x scale Dtw z‖ₑ ^ 2 < ⊤ ∧ ContinuousOn (ucScaledField x scale w) (CKN.Foundation.Parabolic.vec3Ball 0 ρ ×ˢ Set.Ico 0 2) ∧ (∀ y ∈ CKN.Foundation.Parabolic.vec3Ball 0 ρ, ucScaledField x scale w (y, 0) = 0) ∧ ∀ᵐ (z : CKN.Foundation.Parabolic.ParabolicPoint) ∂MeasureTheory.volume.restrict (ucCylinder ρ), CKN.Foundation.Parabolic.vec3EuclideanNorm
(ucWeakHeatVector (ucScaledD2w x scale D2w) (ucScaledDtw x scale Dtw) z) ≤ c₁ * scale * (CKN.Foundation.Parabolic.vec3EuclideanNorm (ucScaledField x scale w z) + √(CKN.spatialGradientSq (ucScaledField x scale w) (ucScaledDw x scale Dw) z))
The bounded source data in lem:bu-gaussian give the weak derivatives,
finite quadratic data, and heat inequality on the rescaled cylinder.