Source suitability and the first blow-up bounds #
The local hypotheses supply suitability on the interior source cylinder and uniform pressure and gradient bounds on each fixed rescaled past cylinder.
theorem
ESS.blowup_limit_local_energy_pressure_of_source_data
{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 (CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0))))
(hDu :
MeasureTheory.AEStronglyMeasurable Du
(MeasureTheory.volume.restrict (CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0))))
(hpmeas :
MeasureTheory.AEStronglyMeasurable p
(MeasureTheory.volume.restrict (CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0))))
(hL2 :
essSup
(fun (t : ℝ) => ∫⁻ (x : CKN.Foundation.Parabolic.Vec3) in CKN.Foundation.Parabolic.vec3Ball 0 1, ‖u (x, t)‖ₑ ^ 2)
(MeasureTheory.volume.restrict (Set.Ioo (-1) 0)) < ⊤)
(henergy :
∫⁻ (z : CKN.Foundation.Parabolic.ParabolicPoint) in CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0), ‖u z‖ₑ ^ 2 + ‖Du z‖ₑ ^ 2 < ⊤)
(hpLp :
MeasureTheory.MemLp p (ENNReal.ofReal (3 / 2))
(MeasureTheory.volume.restrict (CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0))))
(hL3 :
essSup
(fun (t : ℝ) =>
∫⁻ (x : CKN.Foundation.Parabolic.Vec3) in CKN.Foundation.Parabolic.vec3Ball 0 1, ENNReal.ofReal (CKN.Foundation.Parabolic.vec3EuclideanNorm (u (x, t))) ^ 3)
(MeasureTheory.volume.restrict (Set.Ioo (-1) 0)) < ⊤)
(hgrad :
∀ᵐ (t : ℝ) ∂MeasureTheory.volume.restrict (Set.Ioo (-1) 0), ∀ (i : Fin 3),
CKN.HasWeakGradientOn (CKN.Foundation.Parabolic.vec3Ball 0 1) (fun (x : CKN.Vec 3) => u (x, t) i)
fun (x : CKN.Vec 3) => Du (x, t) i)
(hS2 :
∀ ψ ∈ CKN.spaceTimeTestFunction (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0),
∫ (z : CKN.Foundation.Parabolic.ParabolicPoint) in CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0), ∑ i : Fin 3, u z i * CKN.spatialPartial ψ i z = 0)
(hS3 :
∀ φ ∈ CKN.spaceTimeTestFunction (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0),
∫ (z : CKN.Foundation.Parabolic.ParabolicPoint) in CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0), -∑ 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)
(t₀ : ℝ)
(r : ℕ → ℝ)
(hx₀ : x₀ ∈ closure (CKN.Foundation.Parabolic.vec3Ball 0 (1 / 2)))
(ht₀ : t₀ ∈ Set.Icc (-(1 / 4)) 0)
(hr : ∀ (k : ℕ), 0 < r k)
(hr0 : Filter.Tendsto r Filter.atTop (nhds 0))
(R a : ℝ)
(hR : 0 < R)
(ha : a < 0)
:
CKN.IsSuitableWeakSolution (CKN.Foundation.Parabolic.vec3Ball 0 (3 / 4)) (Set.Ioo (-1) 0) 3 u Du p 0 ∧ ∃ (Cg : ℝ) (Cp : ℝ),
0 ≤ Cg ∧ 0 ≤ Cp ∧ ∀ᶠ (k : ℕ) in Filter.atTop, MeasureTheory.IntegrableOn
(fun (z : CKN.Foundation.Parabolic.ParabolicPoint) =>
CKN.spatialGradientSq (parabolicRescaleVelocity x₀ t₀ (r k) u) (parabolicRescaleGradient x₀ t₀ (r k) Du)
z)
(CKN.spaceTimeSet (CKN.euclideanBall 0 R) (Set.Ioo a 0)) MeasureTheory.volume ∧ 2 * ∫ (z : CKN.Foundation.Parabolic.ParabolicPoint) in CKN.spaceTimeSet (CKN.euclideanBall 0 R) (Set.Ioo a 0), CKN.spatialGradientSq (parabolicRescaleVelocity x₀ t₀ (r k) u)
(parabolicRescaleGradient x₀ t₀ (r k) Du) z ≤ Cg ∧ MeasureTheory.MemLp (blowupPressure x₀ t₀ (r k) p) (3 / 2)
(MeasureTheory.volume.restrict (CKN.Foundation.Parabolic.vec3Ball 0 R ×ˢ Set.Ioo a 0)) ∧ ∫ (z : CKN.Foundation.Parabolic.Vec3 × ℝ) in CKN.Foundation.Parabolic.vec3Ball 0 R ×ˢ Set.Ioo a 0, |blowupPressure x₀ t₀ (r k) p z| ^ (3 / 2) ≤ Cp
The first step of the blow-up limit follows from the local hypotheses and the fixed pressure split.