Blowup Pressure Mass Source

Pressure bounds and convergence estimates for the rescaled sequence.

theorem ESS.blowupPressure_eventually_mass_bound_pastBox (p p₁ : CKN.Foundation.Parabolic.ParabolicPoint → ℝ) (hp₁ : MeasureTheory.AEStronglyMeasurable p₁ (MeasureTheory.volume.restrict (Set.univ ×ˢ Set.Ioo (-1) 0))) (M : ENNReal) (hM : M < ⊤) (hsource : ∀ᵐ (s : ℝ) ∂MeasureTheory.volume.restrict (Set.Ioo (-1) 0), MeasureTheory.AEStronglyMeasurable (fun (x : CKN.Foundation.Parabolic.Vec3) => p₁ (x, s)) MeasureTheory.volume ∧ MeasureTheory.eLpNorm (fun (x : CKN.Foundation.Parabolic.Vec3) => p₁ (x, s)) (3 / 2) MeasureTheory.volume ≤ M) (hp₂ : MeasureTheory.MemLp (fun (z : CKN.Foundation.Parabolic.Vec3 × ℝ) => p (z.1, z.2) - p₁ (z.1, z.2)) (3 / 2) ((MeasureTheory.volume.restrict (CKN.euclideanBall 0 1)).prod (MeasureTheory.volume.restrict (Set.Ioo (-1) 0)))) (hharm : ∀ᵐ (t : ℝ) ∂MeasureTheory.volume.restrict (Set.Ioo (-1) 0), CKN.Foundation.Heat.WeaklyHarmonicOn (CKN.euclideanBall 0 1) fun (x : CKN.Foundation.Parabolic.Vec3) => p (x, t) - p₁ (x, t)) (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) : ∃ (C : ℝ), 0 ≤ C ∧ ∀ᶠ (k : ℕ) in Filter.atTop, 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.ParabolicPoint) in CKN.Foundation.Parabolic.vec3Ball 0 R ×ˢ Set.Ioo a 0, |blowupPressure x₀ t₀ (r k) p z| ^ (3 / 2) ≤ C

The fixed pressure split bounds the rescaled pressure mass on every bounded open past cylinder.