Quadratic heat inequality for the short-time cutoff #
The weak heat inequality separates a quadratic term in the cutoff field from an error containing derivatives of the scalar cutoff.
noncomputable def
ESS.buShortCutoffHeatErrorSize
(scale R : ℝ)
(hR : 0 < R)
(ε c₁ : ℝ)
(v : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3)
(Dv : CKN.Foundation.Parabolic.ParabolicPoint → Fin 3 → CKN.Foundation.Parabolic.Vec3)
(z : CKN.Foundation.Parabolic.ParabolicPoint)
:
The scalar cutoff error appearing in the squared heat bound.
Equations
- One or more equations did not get rendered due to their size.
Instances For
theorem
ESS.bu_short_cutoff_heat_sq_ae_bound
(scale R ε c₁ : ℝ)
(hscale : 0 < scale)
(hscale1 : scale ≤ 1)
(hR : 0 < 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)
(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 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;
∀ᵐ (z :
CKN.Foundation.Parabolic.ParabolicPoint) ∂MeasureTheory.volume.restrict
(CKN.spaceTimeSet {x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} (Set.Ioo (1 / 2) 1)), CKN.Foundation.Parabolic.vec3EuclideanNorm (ucWeakHeatVector D2W DtW z) ^ 2 ≤ 36 * (c₁ * scale) ^ 2 * (CKN.Foundation.Parabolic.vec3EuclideanNorm (W z) ^ 2 + CKN.spatialGradientSq W DW z) + 2 * buShortCutoffHeatErrorSize scale R hR ε c₁ v Dv z ^ 2
The square of the compact field's heat vector is bounded by a term that can be absorbed in the Carleman left side and a cutoff error.