| import OAI.Analysis.AncientFlat.Regularity.CutoffRegularity |
| import OAI.Analysis.AncientFlat.Regularity.HolderC0Extension |
| import OAI.Analysis.AncientFlat.Evolution.BCFTimeDerivative |
| import OAI.Analysis.AncientFlat.Regularity.FrozenLongSchauder |
|
|
| namespace OAI |
| noncomputable section |
| open Set Filter |
| open scoped ContDiff ZeroAtInfty BoundedContinuousFunction Laplacian |
| namespace ParabolicLocalization |
| open ParabolicInterior ParabolicSchauder ParametrixEvolution ClassicalIBP |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] |
| def cutoffResidual (c : ℝ) (u ut : ℝ → V → ℂ) (t : ℝ) (x : V) : ℂ := |
| cutoffDerivative u ut t x+c • Δ (Δ (cutoffValue u t)) x |
| lemma cutoffResidual_compact (c : ℝ) (u ut : ℝ → V → ℂ) |
| (hs : ∀ t, ContDiff ℝ 4 (u t)) (t : ℝ) : HasCompactSupport (cutoffResidual c u ut t) := by |
| apply HasCompactSupport.of_support_subset_isCompact (isCompact_closedBall (0:V) 2) |
| intro x hx |
| by_contra h |
| have hn : 2<‖x‖ := by simpa only [Metric.mem_closedBall,dist_zero_right,not_le] using h |
| have hz : cutoffResidual c u ut t x=0 := by |
| rw [cutoffResidual,cutoffDerivative_zero u ut t x (by linarith)] |
| have he := bilaplacian_eq_traceFour (cutoffValue u t) (cutoffValue_contDiff u t (hs t)) x |
| have hj : iteratedFDeriv ℝ 4 (cutoffValue u t) x=0 := cutoff_product_jet_zero 4 t (u t) x (by linarith) |
| rw [hj,map_zero] at he |
| rw [he,smul_zero,add_zero] |
| exact hx hz |
| lemma cutoffResidual_holder (c α A M : ℝ) (hα : 0≤α) (hα1 : α≤1) (hA : 0≤A) (hM : 0≤M) |
| (ha : ∀ n : ℕ, n≤4 → HolderBound α univ |
| (fun p : V × ℝ => iteratedFDeriv ℝ n (cutoff p.2) p.1) A) |
| (hat : HolderBound α univ (fun p : V × ℝ => cutoffTime p.2 p.1) A) |
| (u ut : ℝ → V → ℂ) (hs : ∀ t, ContDiff ℝ 4 (u t)) |
| (hj : JetBound α (cylinder 2) u ut M) : |
| HolderBound α {p : V × ℝ | p.2∈Icc (-2) 0} (fun p => cutoffResidual c u ut p.2 p.1) |
| (160*A*M+‖(c • traceFour : (V [×4]→L[ℝ] ℂ) →L[ℝ] ℂ)‖*(160*A*M)) := by |
| obtain ⟨_,Ht,Hj⟩ := cutoff_regular_holder α A M hα hα1 hA hM ha hat u ut hs hj.1 hj.2 |
| have H := Ht.add (Hj.clm (c • traceFour)) |
| have he (p : V × ℝ) : cutoffResidual c u ut p.2 p.1 = |
| cutoffDerivative u ut p.2 p.1+(c • traceFour : (V [×4]→L[ℝ] ℂ) →L[ℝ] ℂ) |
| (iteratedFDeriv ℝ 4 (cutoffValue u p.2) p.1) := by |
| simp only [cutoffResidual,bilaplacian_eq_traceFour _ (cutoffValue_contDiff u p.2 (hs p.2)),smul_apply] |
| simpa only [he] using H |
|
|
| theorem cutoff_source_evolves (α : ℝ) (hα : 0<α) (hα1 : α≤1) |
| (c : ℝ) (u ut : ℝ → V → ℂ) (hs : ∀ t, ContDiff ℝ 4 (u t)) |
| (hd : ∀ p∈cylinder 2, HasDerivAt (fun t => u t p.1) (ut p.2 p.1) p.2) |
| (hj : ∃ M : ℝ, 0≤M ∧ JetBound α (cylinder 2) u ut M) : |
| ∃ f : ℝ →ᵇ C₀(V,ℂ), |
| (∀ t∈Icc 0 2, ∀ x, f t x=cutoffResidual c u ut (t-2) x) ∧ |
| SourceEvolves (coefficientLine c) 0 f 0 2 0 (fun t => cutoffC0 u hs (t-2)) := by |
| obtain ⟨M,hM,hj⟩ := hj |
| obtain ⟨A,hA,ha,hat⟩ := cutoff_uniform_holder (V := V) α hα hα1 |
| obtain ⟨H0,Ht,H4⟩ := cutoff_regular_holder α A M hα.le hα1 hA.le hM ha hat u ut hs hj.1 hj.2 |
| obtain ⟨w,hw,_⟩ := holder_c0_extension hα (by positivity : 0≤160*A*M) (by norm_num : (-2:ℝ)≤0) |
| (fun p => cutoffValue u p.2 p.1) H0 (fun t _ => cutoff_product_compact t _) |
| obtain ⟨v,hv,_⟩ := holder_c0_extension hα (by positivity : 0≤160*A*M) (by norm_num : (-2:ℝ)≤0) |
| (fun p => cutoffDerivative u ut p.2 p.1) Ht (fun t _ => cutoffDerivative_compact u ut t) |
| have Hr := cutoffResidual_holder c α A M hα.le hα1 hA.le hM ha hat u ut hs hj |
| obtain ⟨q,hq,_⟩ := holder_c0_extension hα (by positivity) (by norm_num : (-2:ℝ)≤0) |
| (fun p => cutoffResidual c u ut p.2 p.1) Hr (fun t _ => cutoffResidual_compact c u ut hs t) |
| let f : ℝ →ᵇ C₀(V,ℂ) := q.compContinuous ⟨fun t => t-2,continuous_id.sub continuous_const⟩ |
| have hm (t : ℝ) (ht : t∈Icc 0 2) : t-2∈Icc (-2) 0 := ⟨by linarith [ht.1],by linarith [ht.2]⟩ |
| have hwe (t : ℝ) (ht : t∈Icc 0 2) : cutoffC0 u hs (t-2)=w (t-2) := by |
| ext x; exact (hw (t-2) (hm t ht) x).symm |
| refine ⟨f,fun t ht x => hq (t-2) (hm t ht) x,?_,?_,?_,?_,?_⟩ |
| · ext x |
| simp only [sub_eq_add_neg,zero_add,cutoffC0_apply,ZeroAtInftyContinuousMap.zero_apply] |
| change cutoff (-2) x • u (-2) x=0 |
| rw [cutoff_zero_time (by norm_num : (3:ℝ)/2≤|(-2:ℝ)|),zero_smul] |
| · exact ((w.continuous.comp (continuous_id.sub continuous_const)).continuousOn).congr |
| (fun t ht => hwe t ht) |
| · intro t _; exact cutoffValue_contDiff u (t-2) (hs (t-2)) |
| · intro t _ n hn |
| exact (cutoff_product_compact (t-2) (u (t-2))).iteratedFDeriv n |>.exists_bound_of_continuous |
| ((cutoffValue_contDiff u (t-2) (hs (t-2))).continuous_iteratedFDeriv (by exact_mod_cast hn)) |
| · intro t ht |
| let v' : ℝ → (V →ᵇ ℂ) := fun s => (v (s-2)).toBCF |
| have hv' : Continuous v' := ZeroAtInftyContinuousMap.isometry_toBCF.continuous.comp |
| (v.continuous.comp (continuous_id.sub continuous_const)) |
| have hpoint (s : ℝ) (hsi : s∈Ioo 0 2) (x : V) : |
| HasDerivAt (fun r => (cutoffC0 u hs (r-2)).toBCF x) (v' s x) s := by |
| have hsi' : s-2∈Ioo (-2) 0 := ⟨by linarith [hsi.1],by linarith [hsi.2]⟩ |
| have hh := (cutoffValue_pointwise_derivative u ut hd (s-2) hsi' x).scomp s |
| ((hasDerivAt_id s).sub_const 2) |
| have he := hv (s-2) (hm s ⟨hsi.1.le,hsi.2.le⟩) x |
| change HasDerivAt (fun r => cutoffValue u (r-2) x) (v (s-2) x) s |
| rw [he] |
| simpa only [Function.comp_def,id_eq,one_smul] using hh |
| refine ⟨v' t,hasDerivAt_bcf_of_pointwise _ v' hv' 0 2 t ht hpoint,fun x => ?_⟩ |
| have hti := hm t ⟨ht.1.le,ht.2.le⟩ |
| change v (t-2) x=f t x-(c:ℂ)*Δ (Δ (cutoffValue u (t-2))) x-0*cutoffValue u (t-2) x |
| change v (t-2) x=q (t-2) x-(c:ℂ)*Δ (Δ (cutoffValue u (t-2))) x-0*cutoffValue u (t-2) x |
| rw [hv (t-2) hti x,hq (t-2) hti x] |
| simp only [cutoffResidual,Complex.real_smul,zero_mul,sub_zero,add_sub_cancel_right] |
| end ParabolicLocalization |
|
|
| end |
|
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| end OAI |
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