Integrated weak heat inequality on the cutoff support #
The squared differential inequality is integrated with the half-space Carleman weight over the compact cutoff support.
theorem
ESS.bu_short_cutoff_heat_integral_le
(scale R ε c₁ a : ℝ)
(hscale : 0 < scale)
(hscale1 : scale ≤ 1)
(hR : 0 < R)
(hε : 0 < ε)
(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 ({x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} ×ˢ Set.Ico 0 1))
(hweak : 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 (ucWeakHeatVector D2w Dtw z) ≤ c₁ * (√(CKN.spatialGradientSq w Dw z) + CKN.Foundation.Parabolic.vec3EuclideanNorm (w z)))
:
have K := buCutSupportSet (buShortFullCutoff scale R hR ε);
have κ := buShortFullCutoff scale R hR ε;
have v := buAffineField (-scale ^ 2 / 2) scale w;
have Dv := buAffineDw (-scale ^ 2 / 2) scale Dw;
have D2v := buAffineD2w (-scale ^ 2 / 2) scale D2w;
have Dtv := buAffineDtw (-scale ^ 2 / 2) scale Dtw;
have W := buCutField κ v;
have DW := buCutDw κ v Dv;
have D2W := buCutD2 κ v Dv D2v;
have DtW := buCutDt κ v Dtv;
∫ (z : CKN.Foundation.Parabolic.ParabolicPoint) in K, buShortCarlemanWeight a z * CKN.Foundation.Parabolic.vec3EuclideanNorm (ucWeakHeatVector D2W DtW z) ^ 2 ≤ (36 * (c₁ * scale) ^ 2 * ∫ (z : CKN.Foundation.Parabolic.ParabolicPoint) in K, buShortCarlemanWeight a z * (CKN.Foundation.Parabolic.vec3EuclideanNorm (W z) ^ 2 + CKN.spatialGradientSq W DW z)) + 2 * ∫ (z : CKN.Foundation.Parabolic.ParabolicPoint) in K, buShortCarlemanWeight a z * buShortCutoffHeatErrorSize scale R hR ε c₁ v Dv z ^ 2
The weighted weak heat energy is bounded by cutoff energy and cutoff-derivative error on the compact support.