Linear scale bound for the fixed pressure remainder #
The fixed harmonic remainder has a time-integrated spatial supremum bound; parabolic rescaling converts it to the critical linear factor in the radius.
theorem
ESS.pressureSplit_remainder_mixed_scale_bound
{u : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3}
{Du : CKN.Foundation.Parabolic.ParabolicPoint → Fin 3 → CKN.Foundation.Parabolic.Vec3}
{p : CKN.Foundation.Parabolic.ParabolicPoint → ℝ}
(hu : MeasureTheory.AEStronglyMeasurable u (MeasureTheory.volume.restrict pressureSplitDomain))
(hDu : MeasureTheory.AEStronglyMeasurable Du (MeasureTheory.volume.restrict pressureSplitDomain))
(hpmeas : MeasureTheory.AEStronglyMeasurable p (MeasureTheory.volume.restrict pressureSplitDomain))
(hL2 :
essSup (fun (t : ℝ) => ∫⁻ (x : CKN.Foundation.Parabolic.Vec3) in pressureSplitBall, ‖u (x, t)‖ₑ ^ 2)
(MeasureTheory.volume.restrict pressureSplitTime) < ⊤)
(henergy : ∫⁻ (z : CKN.Foundation.Parabolic.ParabolicPoint) in pressureSplitDomain, ‖u z‖ₑ ^ 2 + ‖Du z‖ₑ ^ 2 < ⊤)
(hp : MeasureTheory.MemLp p (ENNReal.ofReal (3 / 2)) (MeasureTheory.volume.restrict pressureSplitDomain))
(hL3 :
essSup
(fun (t : ℝ) =>
∫⁻ (x : CKN.Foundation.Parabolic.Vec3) in pressureSplitBall, ENNReal.ofReal (CKN.Foundation.Parabolic.vec3EuclideanNorm (u (x, t))) ^ 3)
(MeasureTheory.volume.restrict pressureSplitTime) < ⊤)
(hgrad :
∀ᵐ (t : ℝ) ∂MeasureTheory.volume.restrict pressureSplitTime, ∀ (i : Fin 3),
CKN.HasWeakGradientOn pressureSplitBall (fun (x : CKN.Vec 3) => u (x, t) i) fun (x : CKN.Vec 3) => Du (x, t) i)
(hdiv :
∀ χ ∈ CKN.spaceTimeTestFunction pressureSplitBall pressureSplitTime,
∫ (z : CKN.Foundation.Parabolic.ParabolicPoint) in pressureSplitDomain, ∑ i : Fin 3, u z i * CKN.spatialPartial χ i z = 0)
(hMomentum :
∀ φ ∈ CKN.spaceTimeTestFunction pressureSplitBall pressureSplitTime,
∫ (z : CKN.Foundation.Parabolic.ParabolicPoint) in pressureSplitDomain, -∑ i : Fin 3, u z i * CKN.timePartial (fun (y : CKN.Foundation.Parabolic.ParabolicPoint) => φ y i) z - ∑ i : Fin 3,
∑ j : Fin 3,
u z i * u z j * CKN.spatialPartial (fun (y : CKN.Foundation.Parabolic.ParabolicPoint) => φ y i) j z + ∑ i : Fin 3,
∑ j : Fin 3,
Du z i j * CKN.spatialPartial (fun (y : CKN.Foundation.Parabolic.ParabolicPoint) => φ y i) j z - p z * ∑ i : Fin 3, CKN.spatialPartial (fun (y : CKN.Foundation.Parabolic.ParabolicPoint) => φ y i) i z - ∑ i : Fin 3, 0 z i * φ z i = 0)
(x₀ : CKN.Foundation.Parabolic.Vec3)
(hx₀ : x₀ ∈ closure (CKN.Foundation.Parabolic.vec3Ball 0 (1 / 2)))
(t₀ : ℝ)
{Ω : Set CKN.Foundation.Parabolic.Vec3}
(hΩmeas : MeasurableSet Ω)
(hΩbounded : Bornology.IsBounded Ω)
:
let F := CKN.pressureSplitTensor u;
have hF := ⋯;
have p₁ := pressureSplitRieszPressure F hF;
have p₂ := pressureSplitRemainder p p₁;
∃ C < ⊤,
∃ (δ : ℝ),
0 < δ ∧ ∀ (r : ℝ),
0 < r →
r < δ →
∫⁻ (t : ℝ), MeasureTheory.eLpNorm
(fun (x : CKN.Foundation.Parabolic.Vec3) =>
parabolicRescalePressure x₀ t₀ r (goodPointDomain.indicator p₂) (x, t))
⊤ (MeasureTheory.volume.restrict Ω) ^ (3 / 2) ≤ C * (ENNReal.ofReal r * (MeasureTheory.eLpNorm (fun (z : CKN.Foundation.Parabolic.Vec3 × ℝ) => p (z.1, z.2)) (3 / 2)
((MeasureTheory.volume.restrict (CKN.euclideanBall 0 1)).prod
(MeasureTheory.volume.restrict (Set.Ioo (-1) 0))) ^ (3 / 2) + ENNReal.ofReal (pressureSplitVelocityLpBound u ^ 3)))
The zero-extended fixed pressure remainder has the critical linear scale bound on every bounded measurable spatial region.