Backward uniqueness from Gaussian averages #
The Gaussian average estimate supplies uniform short-time vanishing, which then propagates through the half-space time slab.
theorem
ESS.bu_backwardUniqueness_from_gaussian
(c₁ M : ℝ)
(hc₁ : 0 < c₁)
(hM : 0 < M)
(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 ({x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} ×ˢ Set.Ico 0 1))
(hinit : ∀ (x : CKN.Foundation.Parabolic.Vec3), 0 < x 2 → w (x, 0) = 0)
(hderiv : CKN.HasSpaceTimeWeakDerivs {x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} (Set.Ioo 0 1) w Dw D2w Dtw)
(hL2 :
∀ 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 < ⊤)
(hineq :
∀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 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 ∈ 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))
(hGaussian :
∃ (γ : ℝ),
0 < γ ∧ γ < 1 / 12 ∧ ∀ (A : ℝ),
0 ≤ A →
A ≤ 1 / 10 ^ 12 →
∃ (C : ℝ),
0 < C ∧ ∀ (u : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3)
(Du : CKN.Foundation.Parabolic.ParabolicPoint → Fin 3 → CKN.Foundation.Parabolic.Vec3)
(D2u : CKN.Foundation.Parabolic.ParabolicPoint → Fin 3 → Fin 3 → CKN.Foundation.Parabolic.Vec3)
(Dtu : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3),
ContinuousOn u ({x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} ×ˢ Set.Ico 0 1) →
(∀ (x : CKN.Foundation.Parabolic.Vec3), 0 < x 2 → u (x, 0) = 0) →
CKN.HasSpaceTimeWeakDerivs {x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} (Set.Ioo 0 1) u Du D2u
Dtu →
(∀ 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, ‖Du z‖ₑ ^ 2 + ‖D2u z‖ₑ ^ 2 + ‖Dtu 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 D2u Dtu z) ≤ c₁ * (√(CKN.spatialGradientSq u Du z) + CKN.Foundation.Parabolic.vec3EuclideanNorm (u z))) →
(∀ z ∈ CKN.spaceTimeSet {x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} (Set.Ioo 0 1),
CKN.Foundation.Parabolic.vec3EuclideanNorm (u z) ≤ Real.exp (A * CKN.Foundation.Parabolic.vec3EuclideanNorm z.1 ^ 2)) →
∀ (X : CKN.Foundation.Parabolic.Vec3),
2 < X 2 →
∀ (t : ℝ),
0 < t →
t < γ →
t.rpow (-(5 / 2)) * ∫ (z : CKN.Foundation.Parabolic.ParabolicPoint) in CKN.spaceTimeSet (Metric.ball X √(3 * t)) (Set.Ioo t (5 * t / 2)), CKN.Foundation.Parabolic.vec3EuclideanNorm (u z) ^ 2 ≤ C * Real.exp
(8 * max A (1 / 10 ^ 12 / 2) * CKN.Foundation.Parabolic.vec3EuclideanNorm X ^ 2) * Real.exp (-(1 / 10 ^ 6 * X 2 ^ 2 / (12 * t))))
(z : CKN.Foundation.Parabolic.ParabolicPoint)
:
z ∈ CKN.spaceTimeSet {x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} (Set.Ioo 0 1) → w z = 0
Backward uniqueness follows from the Gaussian average estimate
and the half-space Carleman argument (thm:bu).