Local energy bound on a short-time dyadic cell #
The extended rescaled weak heat equation and the Caccioppoli estimate control each shifted cell by one Gaussian velocity average.
theorem
ESS.bu_short_shifted_cell_energy_bound
(k : ℤ)
(m : Fin 3 → ℤ)
(ell : Fin 2048)
(c : ℝ)
(hc : 0 ≤ c)
(hδhalf : (CKN.Foundation.buSmallTimeDyadicCellCenter k m ell).2 < 1 / 2)
(hball :
CKN.Foundation.Parabolic.vec3Ball (CKN.Foundation.buSmallTimeDyadicCellCenter k m ell).1
(2 * (√(CKN.Foundation.buSmallTimeDyadicCellCenter k m ell).2 / 8)) ⊆
{x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2})
(havgBall :
CKN.Foundation.Parabolic.vec3Ball (CKN.Foundation.buSmallTimeDyadicCellCenter k m ell).1
√(3 * (CKN.Foundation.buSmallTimeDyadicCellCenter k m ell).2 / 2) ⊆
{x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2})
{v : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3}
{Dv : CKN.Foundation.Parabolic.ParabolicPoint → Fin 3 → CKN.Foundation.Parabolic.Vec3}
{D2v : CKN.Foundation.Parabolic.ParabolicPoint → Fin 3 → Fin 3 → CKN.Foundation.Parabolic.Vec3}
{Dtv : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3}
(hcont : ContinuousOn v ({x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} ×ˢ Set.Ico (1 / 2) (3 / 2)))
(hweak :
CKN.HasSpaceTimeWeakDerivs {x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} (Set.Ioo (1 / 2) (3 / 2)) v Dv D2v Dtv)
(hlocal :
∀ S ⊆ CKN.spaceTimeSet {x : CKN.Foundation.Parabolic.Vec3 | 0 < x 2} (Set.Ioo (1 / 2) (3 / 2)),
Bornology.IsBounded S →
∫⁻ (z : CKN.Foundation.Parabolic.ParabolicPoint) in S, ‖v z‖ₑ ^ 2 + ‖Dv z‖ₑ ^ 2 + ‖D2v z‖ₑ ^ 2 + ‖Dtv 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 (1 / 2) (3 / 2))), CKN.Foundation.Parabolic.vec3EuclideanNorm (ucWeakHeatVector D2v Dtv z) ≤ c * (√(CKN.spatialGradientSq v Dv z) + CKN.Foundation.Parabolic.vec3EuclideanNorm (v z)))
:
have δ := (CKN.Foundation.buSmallTimeDyadicCellCenter k m ell).2;
have Y := (CKN.Foundation.buSmallTimeDyadicCellCenter k m ell).1;
have r := √δ / 8;
have Avg :=
CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball Y √(3 * δ / 2)) (Set.Ioo (1 / 2 + δ / 2) (1 / 2 + 5 * δ / 4));
∫ (z : CKN.Foundation.Parabolic.ParabolicPoint) in buShortShiftedDyadicCell k m ell, CKN.Foundation.Parabolic.vec3EuclideanNorm (v z) ^ 2 + CKN.spatialGradientSq v Dv z ≤ (1 + 256 * (1 + c ^ 2 + 1 / r ^ 2)) * ∫ (z : CKN.Foundation.Parabolic.ParabolicPoint) in Avg, CKN.Foundation.Parabolic.vec3EuclideanNorm (v z) ^ 2
A short-time dyadic cell's energy is bounded by its Gaussian
velocity average (lem:bu-small-time).