Short-time vanishing from Gaussian integrability

The compact-set phase estimate and spatial unique continuation give vanishing on the initial physical time interval.

theorem ESS.bu_short_time_from_integrable_densities : ∃ (a₀ : ℝ) (c : ℝ), 0 < a₀ ∧ 0 < c ∧ ∀ (M scale c₁ C₁ C₂ : ℝ), 0 < M → 0 < scale → scale ≤ 1 / 2 → 0 < c₁ → 0 ≤ C₁ → (∀ (scale y : ℝ), |deriv (deriv (buShortNormalCutoff scale)) y| ≤ C₁) → 0 ≤ C₂ → (∀ (x : ℝ), |deriv (deriv CKN.smoothTransitionProfile) x| ≤ C₂) → c * 36 * (c₁ * scale) ^ 2 ≤ 1 / 2 → ∀ (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), ContinuousOn w ({x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} ×ˢ Set.Ico 0 1) → (∀ (x : CKN.Foundation.Parabolic.Vec3), 0 < x 2 → w (x, 0) = 0) → CKN.HasSpaceTimeWeakDerivs {x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} (Set.Ioo 0 1) w Dw D2w Dtw → (∀ S ⊆ CKN.spaceTimeSet {x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} (Set.Ioo 0 1), Bornology.IsBounded S → ∫⁻ (z : CKN.Foundation.Parabolic.ParabolicPoint) in S, ‖Dw z‖ₑ ^ 2 + ‖D2w z‖ₑ ^ 2 + ‖Dtw z‖ₑ ^ 2 < ⊤) → (∀ᵐ (z : CKN.Foundation.Parabolic.ParabolicPoint) ∂MeasureTheory.volume.restrict (CKN.spaceTimeSet {x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} (Set.Ioo 0 1)), CKN.Foundation.Parabolic.vec3EuclideanNorm (ucWeakHeatVector D2w Dtw z) ≤ c₁ * (√(CKN.spatialGradientSq w Dw z) + CKN.Foundation.Parabolic.vec3EuclideanNorm (w z))) → (∀ z ∈ CKN.spaceTimeSet {x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} (Set.Ioo 0 1), CKN.Foundation.Parabolic.vec3EuclideanNorm (w z) ≤ Real.exp (M * CKN.Foundation.Parabolic.vec3EuclideanNorm z.1 ^ 2)) → MeasureTheory.IntegrableOn (buShortTangentialDensity scale w Dw) (buShortWideGapRegion scale) MeasureTheory.volume → (∀ (a : ℝ), max a₀ 1 < a → MeasureTheory.IntegrableOn (buShortWeightedPolynomialDensity scale a w Dw) (buShortHighStrip scale) MeasureTheory.volume) → (∀ (a : ℝ), max a₀ 1 < a → MeasureTheory.IntegrableOn (buShortWeightedDensity scale a w Dw) (buShortHighStrip scale) MeasureTheory.volume) → ∀ (x : CKN.Foundation.Parabolic.Vec3), 0 < x 2 → ∀ (t : ℝ), 0 < t → t < scale ^ 2 / 2 → w (x, t) = 0

The short-time conclusion follows from the three Gaussian energy integrability bounds used to remove the spatial cutoff (lem:bu-small-time).