| import OAI.Analysis.AncientFlat.Regularity.ParabolicInterpolation |
| import OAI.Analysis.AncientFlat.Regularity.HolderProducts |
| import OAI.Analysis.AncientFlat.Evolution.TraceFour |
|
|
| namespace OAI |
| noncomputable section |
| open Set Filter |
| open scoped ContDiff Laplacian |
| namespace ParabolicInterior |
| open ParabolicInterpolation ClassicalIBP |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] |
| lemma finite_jet_of_fourth (α K U H R : ℝ) (hα : 0<α) (hα1 : α<1) |
| (hK : 0≤K) (hU : 0≤U) (hH : 0≤H) (I : Set ℝ) (hI : OrdConnected I) |
| (a : V × ℝ → ℝ) (b : V × ℝ → ℂ) (u ut f : ℝ → V → ℂ) |
| (ha : HolderBound α {p | p.2∈I} a K) (hb : HolderBound α {p | p.2∈I} b K) |
| (hf : HolderBound α {p | p.2∈I} (fun p => f p.2 p.1) K) |
| (hu : ∀ t∈I, ∀ x, ‖u t x‖≤U) |
| (h4 : HolderBound α {p | p.2∈I} (fun p => iteratedFDeriv ℝ 4 (u p.2) p.1) H) |
| (hs : ∀ t∈I, ContDiff ℝ 4 (u t)) |
| (hd : ∀ t∈I, ∀ x, HasDerivAt (fun t => u t x) (ut t x) t) |
| (he : ∀ t∈I, ∀ x, ut t x+(a (x,t) : ℂ)*Δ (Δ (u t)) x+b (x,t)*u t x=f t x) : |
| ∃ M : ℝ, 0≤M ∧ JetBound α {p | ‖p.1‖<R ∧ p.2∈I} u ut M := by |
| let T : (V [×4]→L[ℝ] ℂ) →L[ℝ] ℂ := traceFour |
| let B := H+K+K*‖T‖*H+K*U |
| have hB : 0≤B := by dsimp [B]; positivity |
| have hut (t) (ht : t∈I) (x : V) : ‖ut t x‖≤B := by |
| have heq : ut t x=f t x-(a (x,t) : ℂ)*T (iteratedFDeriv ℝ 4 (u t) x)-b (x,t)*u t x := by |
| have hh := he t ht x |
| rw [bilaplacian_eq_traceFour _ (hs t ht)] at hh |
| dsimp [T] |
| linear_combination hh |
| rw [heq] |
| have htrace : ‖T (iteratedFDeriv ℝ 4 (u t) x)‖≤‖T‖*H := |
| (T.le_opNorm _).trans (mul_le_mul_of_nonneg_left (h4.1 (x,t) ht) (norm_nonneg T)) |
| have hprod : ‖(a (x,t) : ℂ)*T (iteratedFDeriv ℝ 4 (u t) x)‖≤K*(‖T‖*H) := by |
| rw [norm_mul,Complex.norm_real] |
| exact mul_le_mul (ha.1 (x,t) ht) htrace (norm_nonneg _) hK |
| have hprod' : ‖b (x,t)*u t x‖≤K*U := by |
| rw [norm_mul]; exact mul_le_mul (hb.1 (x,t) ht) (hu t ht x) (norm_nonneg _) hK |
| calc |
| _ ≤ ‖f t x‖+‖(a (x,t) : ℂ)*T (iteratedFDeriv ℝ 4 (u t) x)‖+‖b (x,t)*u t x‖ := |
| (norm_sub_le ..).trans (add_le_add (norm_sub_le ..) le_rfl) |
| _ ≤ K+K*(‖T‖*H)+K*U := add_le_add (add_le_add (hf.1 (x,t) ht) hprod) hprod' |
| _ ≤ B := by dsimp [B]; nlinarith |
| let D := 110*(U+B) |
| have hD : 0≤D := by dsimp [D]; positivity |
| have Hlow (n : ℕ) (hn : n≤3) : HolderBound α {p : V × ℝ | ‖p.1‖<R ∧ p.2∈I} |
| (fun p => iteratedFDeriv ℝ n (u p.2) p.1) D := by |
| have hh := lower_parabolic_bound n hn u ut I hI R (R+12) 1 U B α zero_lt_one le_rfl |
| (by ring_nf; exact le_rfl) hU hB hα.le hα1.le hs (fun t ht x _ => hd t ht x) |
| (fun t ht x _ => hut t ht x) (fun t ht x _ => hu t ht x) |
| (fun t ht x _ => (h4.1 (x,t) ht).trans (by dsimp [B]; nlinarith [mul_nonneg (mul_nonneg hK (norm_nonneg T)) hH,mul_nonneg hK hU])) |
| simpa only [one_pow,Real.one_rpow,div_one,one_mul,D] using hh |
| have hh0 := Hlow 0 (by norm_num) |
| have hv : HolderBound α {p : V × ℝ | ‖p.1‖<R ∧ p.2∈I} (fun p => u p.2 p.1) D := by |
| constructor |
| · intro p hp; simpa only [norm_iteratedFDeriv_zero] using hh0.1 p hp |
| · intro p hp q hq |
| have hh := hh0.2 p hp q hq |
| simpa only [iteratedFDeriv_zero_eq_comp,Function.comp_apply,←map_sub,LinearIsometryEquiv.norm_map] using hh |
| have hm : {p : V × ℝ | ‖p.1‖<R ∧ p.2∈I}⊆{p : V × ℝ | p.2∈I} := fun p hp => hp.2 |
| have htrace := (h4.restrict hm).clm T |
| have hprod := (ha.restrict hm).smul htrace hK (mul_nonneg (norm_nonneg T) hH) |
| have hprod' := (hb.restrict hm).mul hv hK hD |
| have htime := ((hf.restrict hm).add hprod.neg).add hprod'.neg |
| let J := K+2*K*(‖T‖*H)+2*K*D |
| have hJ : 0≤J := by dsimp [J]; positivity |
| refine ⟨D+H+J,by positivity,?_,?_⟩ |
| · intro n hn |
| by_cases hn' : n≤3 |
| · exact (Hlow n hn').mono (by linarith) |
| · have hn4 : n=4 := by omega |
| subst n |
| exact (h4.restrict hm).mono (by linarith) |
| · have hh : HolderBound α {p : V × ℝ | ‖p.1‖<R ∧ p.2∈I} (fun p => ut p.2 p.1) J := by |
| constructor |
| · intro p hp |
| have heq : ut p.2 p.1=(f p.2 p.1+-a p • T (iteratedFDeriv ℝ 4 (u p.2) p.1))+-b p*u p.2 p.1 := by |
| have hh := he p.2 hp.2 p.1 |
| rw [bilaplacian_eq_traceFour _ (hs p.2 hp.2)] at hh |
| simp only [Complex.real_smul,Complex.ofReal_neg,neg_mul] |
| dsimp [T]; linear_combination hh |
| dsimp only |
| rw [heq] |
| simpa only [neg_smul,neg_mul,J] using htime.1 p hp |
| · intro p hp q hq |
| have heq (p : V × ℝ) (hp : p∈{p : V × ℝ | ‖p.1‖<R ∧ p.2∈I}) : |
| ut p.2 p.1=(f p.2 p.1+-(a p • T (iteratedFDeriv ℝ 4 (u p.2) p.1)))+-(b p*u p.2 p.1) := by |
| have hh := he p.2 hp.2 p.1 |
| rw [bilaplacian_eq_traceFour _ (hs p.2 hp.2)] at hh |
| simp only [Complex.real_smul] |
| dsimp [T]; linear_combination hh |
| dsimp only |
| rw [heq p hp,heq q hq] |
| exact htime.2 p hp q hq |
| exact hh.mono (by linarith) |
| end ParabolicInterior |
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| end |
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| end OAI |
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