| import OAI.Analysis.AncientFlat.Evolution.InitialWindows |
| import OAI.Analysis.AncientFlat.Evolution.InitialClosure |
| import OAI.Analysis.AncientFlat.Regularity.LowerInteriorSchauder |
| import OAI.Analysis.AncientFlat.Evolution.LowerInitialContinuation |
| import OAI.Analysis.AncientFlat.Evolution.LowerWindows |
|
|
| namespace OAI |
| noncomputable section |
| open Set Filter |
| open scoped ContDiff Laplacian |
| namespace ParabolicLowerOrder |
| open ParabolicInitial ParabolicInterior ParabolicLocalization |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] |
|
|
| theorem lower_initial_schauder (α κ K : ℝ) (hα : 0<α) (hα1 : α<1) (hκ : 0<κ) (hK : 0≤K) : |
| ∃ C : ℝ, 0<C ∧ ∀ h : ℝ, 0<h → h≤1 → ∀ (a : V × ℝ → ℝ) (b : V × ℝ → ℂ) (c : Coefficients V), |
| HolderBound α (slab 2 h) a K → HolderBound α (slab 2 h) b K → (∀ j, HolderBound α (slab 2 h) (c j) K) → |
| (∀ p∈slab 2 h, κ≤a p) → ∀ (u ut f : ℝ → V → ℂ) (F U : ℝ), 0≤F → 0≤U → |
| (∀ t : ℝ, ContDiff ℝ 4 (u t)) → |
| (∀ p∈slab 2 h, HasDerivWithinAt (fun t => u t p.1) (ut p.2 p.1) (Icc 0 h) p.2) → |
| (∀ x : V, ‖x‖<2 → u 0 x=0) → |
| (∃ M : ℝ, 0≤M ∧ JetBound α (slab 2 h) u ut M) → |
| HolderBound α (slab 2 h) (fun p => f p.2 p.1) F → |
| (∀ p∈slab 2 h, ‖u p.2 p.1‖≤U) → |
| (∀ p∈slab 2 h, 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) → |
| JetBound α (slab 1 h) u ut (C*(F+U)) := by |
| obtain ⟨Ce,hCe,HE⟩ := lower_initial_continuation (V := V) α K hα hα1 hK |
| obtain ⟨Ci,hCi,HI⟩ := lower_interior_schauder (V := V) α κ (2*K) hα hα1 hκ (by positivity) |
| let q : ℝ := 1/4 |
| let A := Ci*(1+2*Ce) |
| let C := 2*A/(q^4*q^α) |
| have hq : 0<q := by norm_num [q] |
| have hq1 : q≤1 := by norm_num [q] |
| have hA : 0<A := by dsimp [A]; positivity |
| have hC : 0<C := by dsimp [C]; positivity |
| refine ⟨C,hC,?_⟩ |
| intro h hh hh1 a b c ha hb hc hell u ut f F U hF hU hs hd hu0 hj hf hu he |
| obtain ⟨v,vt,g,hvu,hvs,hvd,hvj,hgf,hvuB,hve⟩ := HE h hh a b c ha hb hc u ut f F U hF hU hs hd hu0 hj hf hu he |
| have hsub : (initialStrip h : Set (V × ℝ))⊆initialClosedStrip h := initialStrip_subset_closed h |
| have hcoeffSub : (initialClosedStrip h : Set (V × ℝ))⊆{p : V × ℝ | ‖p.1‖<2 ∧ p.2∈Icc (-1/2:ℝ) h} := |
| fun p hp => ⟨by linarith [hp.1],hp.2⟩ |
| have ha' : HolderBound α (initialClosedStrip h) (initialCoeff a) (2*K) := |
| (holder_initialCoeff hα.le hK hh.le (by norm_num : (-1/2:ℝ)≤0) ha).restrict hcoeffSub |
| have hb' : HolderBound α (initialClosedStrip h) (initialCoeff b) (2*K) := |
| (holder_initialCoeff hα.le hK hh.le (by norm_num : (-1/2:ℝ)≤0) hb).restrict hcoeffSub |
| have hc' (j : Fin 3) : HolderBound α (initialClosedStrip h) (initialCoeff (c j)) (2*K) := |
| (holder_initialCoeff hα.le hK hh.le (by norm_num : (-1/2:ℝ)≤0) (hc j)).restrict hcoeffSub |
| have hell' (p : V × ℝ) (hp : p∈initialStrip h) : κ ≤ initialCoeff a p := by |
| by_cases ht : p.2<0 |
| · rw [initialCoeff_neg a p ht] |
| exact hell (p.1,0) ⟨by linarith [hp.1],le_rfl,hh.le⟩ |
| · rw [initialCoeff_pos a p (le_of_not_gt ht)] |
| exact hell p ⟨by linarith [hp.1],le_of_not_gt ht,hp.2.2.le⟩ |
| have hCF : 0≤Ce*F := mul_nonneg hCe.le hF |
| have hSum : 0≤F+U := add_nonneg hF hU |
| have hlocal (p0 : V × ℝ) (hp0 : p0∈openSlab 1 h) : |
| JetBound α (cylinder 1) (windowValue p0 q v) (windowTime p0 q vt) (A*(F+U)) := by |
| have hm : MapsTo (window p0 q) (cylinder 2) (initialStrip h) := initial_window_maps p0 hp0 |
| have hmc : MapsTo (window p0 q) (cylinder 2) (initialClosedStrip h) := fun p hp => hsub (hm hp) |
| have H := HI (fun p => initialCoeff a (window p0 q p)) (fun p => q^4 • initialCoeff b (window p0 q p)) |
| (windowCoeff p0 q (fun j => initialCoeff (c j))) |
| (holder_window hα.le hq hq1 p0 ha' hmc) |
| (holder_window_smul hα.le (by positivity : 0≤2*K) hq hq1 p0 4 hb' hmc) |
| (fun j => holder_window_smul hα.le (by positivity : 0≤2*K) hq hq1 p0 (3-j.val) (hc' j) hmc) |
| (fun p hp => hell' _ (hm hp)) |
| (windowValue p0 q v) (windowTime p0 q vt) (windowTime p0 q g) (Ce*F) (U+Ce*F) hCF (by positivity) |
| (windowValue_contDiff p0 q v hvs) |
| (fun p hp => window_hasDerivAt p0 q v vt p (hvd _ (hm hp))) |
| (by |
| obtain ⟨M,hM,HJ⟩ := hvj |
| exact ⟨M,hM,window_jet_bound hα.le hM hq hq1 p0 v vt hvs HJ hmc⟩) |
| (holder_window_smul hα.le hCF hq hq1 p0 4 hgf hmc) |
| (fun p hp => hvuB _ (hm hp)) |
| (fun p hp => lower_window_equation p0 q (initialCoeff a) (initialCoeff b) (fun j => initialCoeff (c j)) v vt g hvs p (hve _ (hm hp))) |
| have hbnd : Ci*(Ce*F+(U+Ce*F))≤A*(F+U) := by |
| dsimp [A] |
| nlinarith [mul_nonneg hCi.le hF,mul_nonneg (mul_nonneg hCi.le hCe.le) hU] |
| exact ⟨fun n hn => (H.1 n hn).mono hbnd,H.2.mono hbnd⟩ |
| have hopenV : JetBound α (openSlab 1 h) v vt (C*(F+U)) := by |
| have hEq : 2*(A*(F+U))/(q^4*q^α)=C*(F+U) := by dsimp [C]; ring |
| constructor |
| · intro n hn |
| rw [←hEq] |
| apply holder_of_initial_windows hα.le hq hq1 (mul_nonneg hA.le hSum) n hn |
| intro p0 hp0 |
| have H := (hlocal p0 hp0).1 n hn |
| have heq : (fun p : V × ℝ => iteratedFDeriv ℝ n (windowValue p0 q v p.2) p.1)= |
| (fun p => q^n • iteratedFDeriv ℝ n (v (window p0 q p).2) (window p0 q p).1) := |
| funext (window_jet p0 q v hvs n hn) |
| rwa [heq] at H |
| · rw [←hEq] |
| exact holder_of_initial_windows hα.le hq hq1 (mul_nonneg hA.le hSum) 4 le_rfl _ (fun p0 hp0 => (hlocal p0 hp0).2) |
| have hopen : JetBound α (openSlab 1 h) u ut (C*(F+U)) := by |
| constructor |
| · intro n hn |
| refine ⟨fun p hp => ?_,fun p hp s hs => ?_⟩ |
| · simpa only [(hvu p.2 hp.2.1).1] using (hopenV.1 n hn).1 p hp |
| · simpa only [(hvu p.2 hp.2.1).1,(hvu s.2 hs.2.1).1] using (hopenV.1 n hn).2 p hp s hs |
| · refine ⟨fun p hp => ?_,fun p hp s hs => ?_⟩ |
| · simpa only [(hvu p.2 hp.2.1).2] using hopenV.2.1 p hp |
| · simpa only [(hvu p.2 hp.2.1).2,(hvu s.2 hs.2.1).2] using hopenV.2.2 p hp s hs |
| obtain ⟨M,hM,HJ⟩ := hj |
| have hsmall : (slab 1 h : Set (V × ℝ))⊆slab 2 h := fun p hp => ⟨by linarith [hp.1],hp.2⟩ |
| exact ⟨fun n hn => holder_close_initial hα hh _ ((HJ.1 n hn).restrict hsmall) (hopen.1 n hn), |
| holder_close_initial hα hh _ (HJ.2.restrict hsmall) hopen.2⟩ |
| end ParabolicLowerOrder |
|
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| end |
|
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| end OAI |
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