| import OAI.Analysis.AncientFlat.Regularity.LowerWindowEstimate |
| import OAI.Analysis.AncientFlat.Evolution.TopSeminorm |
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|
| namespace OAI |
| noncomputable section |
| open Set Filter |
| open scoped ContDiff Laplacian |
| namespace ParabolicLowerOrder |
| open ParabolicInterior ParabolicInterpolation ParabolicLocalization |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] |
| [MeasurableSpace V] [BorelSpace V] |
| lemma exists_lower_window_scale {α δ K : ℝ} (hα : 0<α) (hδ : 0<δ) (hK : 0≤K) : |
| ∃ η : ℝ, 0<η ∧ η≤1/8 ∧ K*(4*η)^α≤δ := by |
| let D := K*(4:ℝ)^α+1 |
| have hD : 0<D := by dsimp [D]; positivity |
| obtain ⟨η,hη,hη1,hgap,hsmall⟩ := exists_interpolation_scale (1-α) (δ/D) 1 |
| (by linarith) (div_pos hδ hD) (by norm_num) |
| rw [show 1-(1-α)=α by ring] at hsmall |
| refine ⟨η,hη,by linarith,?_⟩ |
| have hn : 0≤η^α := Real.rpow_nonneg hη.le _ |
| have hsm : η^α≤δ/D := by linarith |
| rw [Real.mul_rpow (by norm_num : (0:ℝ)≤4) hη.le] |
| calc |
| _=(K*(4:ℝ)^α)*η^α := by ring |
| _≤D*η^α := by dsimp [D]; nlinarith |
| _≤D*(δ/D) := mul_le_mul_of_nonneg_left hsm hD.le |
| _=δ := by field_simp |
|
|
| theorem lower_nested_fourth_estimate (α κ K θ : ℝ) (hα : 0<α) (hα1 : α<1) |
| (hκ : 0<κ) (hK : 0≤K) (hθ : 0<θ) : |
| ∃ C : ℝ, 0<C ∧ ∀ (r R : ℝ), 1≤r → r<R → R≤3/2 → |
| ∀ (a : V × ℝ → ℝ) (b : V × ℝ → ℂ) (c : Coefficients V), |
| HolderBound α (cylinder R) a K → HolderBound α (cylinder R) b K → (∀ j, HolderBound α (cylinder R) (c j) K) → |
| (∀ p∈cylinder R, κ≤a p) → |
| ∀ (u ut f : ℝ → V → ℂ) (F U H : ℝ), 0≤F → 0≤U → 0≤H → |
| (∀ t, ContDiff ℝ 4 (u t)) → |
| (∀ p∈cylinder R, HasDerivAt (fun t => u t p.1) (ut p.2 p.1) p.2) → |
| (∃ M, 0≤M ∧ JetBound α (cylinder R) u ut M) → |
| HolderBound α (cylinder R) (fun p => f p.2 p.1) F → (∀ p∈cylinder R, ‖u p.2 p.1‖≤U) → |
| (∀ p∈cylinder R, ∀ z∈cylinder R, |
| ‖iteratedFDeriv ℝ 4 (u p.2) p.1-iteratedFDeriv ℝ 4 (u z.2) z.1‖≤H*pdist p z^α) → |
| (∀ p∈cylinder R, ut p.2 p.1+(a p : ℂ)*Δ (Δ (u p.2)) p.1+b p*u p.2 p.1+spatialLower c u p=f p.2 p.1) → |
| ∀ p∈cylinder r, ∀ z∈cylinder r, |
| ‖iteratedFDeriv ℝ 4 (u p.2) p.1-iteratedFDeriv ℝ 4 (u z.2) z.1‖≤ |
| (θ*H+(C/(R-r)^5)*(F+U))*pdist p z^α := by |
| obtain ⟨δ,C0,hδ,hC0,Hwin⟩ := lower_window_fourth_estimate (V := V) α κ K (θ/2) hα hα1 hκ hK (by linarith) |
| obtain ⟨η,hη,hη4,hsmall⟩ := exists_lower_window_scale hα hδ hK |
| refine ⟨2*C0/η^5,by positivity,?_⟩ |
| intro r R hr hrR hR a b c ha hb hc hella u ut f F U H hF hU hH hs hd hj hf hu hh he |
| let q := η*(R-r) |
| have hg : 0<R-r := sub_pos.mpr hrR |
| have hg1 : R-r≤1 := by linarith |
| have hq : 0<q := mul_pos hη hg |
| have hqη : q≤η := by dsimp [q]; nlinarith |
| have hq1 : q≤1 := by linarith |
| have hgap : 8*q≤R-r := by dsimp [q]; nlinarith |
| have hqsmall : K*(4*q)^α≤δ := |
| (mul_le_mul_of_nonneg_left (Real.rpow_le_rpow (by positivity) (by linarith : 4*q≤4*η) hα.le) hK).trans hsmall |
| have hr0 : 0≤r := by linarith |
| have hsub : (cylinder r : Set (V × ℝ))⊆cylinder R := cylinder_mono hr0 hrR.le |
| let B := θ/2*H+C0/(q^4*q^α)*(F+U) |
| have hB : 0≤B := by dsimp [B]; positivity |
| have Hw (p0 : V × ℝ) (hp0 : p0∈cylinder r) := |
| Hwin q hq hq1 hqsmall p0 (cylinder R) (hsub hp0) (small_lower_window_maps hr0 hq hgap p0 hp0) |
| a b c ha hb hc hella u ut f F U H hF hU hH hs hd hj hf hu hh he |
| have hn (p : V × ℝ) (hp : p∈cylinder r) : ‖iteratedFDeriv ℝ 4 (u p.2) p.1‖≤B*q^α := by |
| have hz : ((0:V),(0:ℝ))∈cylinder 1 := by norm_num [cylinder] |
| simpa only [window_zero] using (Hw p hp).1 (0,0) hz |
| have Hsemi : ∀ p∈cylinder r, ∀ z∈cylinder r, |
| ‖iteratedFDeriv ℝ 4 (u p.2) p.1-iteratedFDeriv ℝ 4 (u z.2) z.1‖≤(2*B)*pdist p z^α := by |
| intro p hp z hz |
| by_cases hnear : pdist p z<q |
| · obtain ⟨hctr,hip,hiz⟩ := close_pair_window hq p z hp hz hnear |
| have hc := (Hw (p.1,max p.2 z.2) hctr).2 (windowInv (p.1,max p.2 z.2) q p) hip |
| (windowInv (p.1,max p.2 z.2) q z) hiz |
| simp only [window_windowInv _ hq.ne'] at hc |
| exact hc.trans (mul_le_mul_of_nonneg_right (by linarith : B ≤ 2*B) |
| (Real.rpow_nonneg (pdist_nonneg p z) _)) |
| · have hdq : q≤pdist p z := le_of_not_gt hnear |
| calc |
| _≤‖iteratedFDeriv ℝ 4 (u p.2) p.1‖+‖iteratedFDeriv ℝ 4 (u z.2) z.1‖ := norm_sub_le _ _ |
| _≤B*q^α+B*q^α := add_le_add (hn p hp) (hn z hz) |
| _=(2*B)*q^α := by ring |
| _≤(2*B)*pdist p z^α := mul_le_mul_of_nonneg_left (Real.rpow_le_rpow hq.le hdq hα.le) (by positivity) |
| have hqden : q^5≤q^4*q^α := by |
| have := mul_le_mul_of_nonneg_left (Real.self_le_rpow_of_le_one hq.le hq1 hα1.le) (pow_nonneg hq.le 4) |
| simpa only [←pow_succ] using this |
| have hc : 2*(C0/(q^4*q^α))≤(2*C0/η^5)/(R-r)^5 := by |
| have hdiv : C0/(q^4*q^α)≤C0/q^5 := div_le_div_of_nonneg_left hC0.le (pow_pos hq _) hqden |
| calc |
| _≤2*(C0/q^5) := by gcongr |
| _=_ := by dsimp [q]; rw [mul_pow]; field_simp |
| have HB : 2*B≤θ*H+((2*C0/η^5)/(R-r)^5)*(F+U) := by |
| dsimp [B] |
| nlinarith [mul_le_mul_of_nonneg_right hc (add_nonneg hF hU)] |
| intro p hp z hz |
| exact (Hsemi p hp z hz).trans (mul_le_mul_of_nonneg_right HB (Real.rpow_nonneg (pdist_nonneg p z) _)) |
| end ParabolicLowerOrder |
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| end |
|
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| end OAI |
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