| import OAI.Analysis.AncientFlat.Evolution.ShiftedEvolution | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter | |
| open scoped ContDiff ZeroAtInfty BoundedContinuousFunction Laplacian | |
| namespace ParametrixEvolution | |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] | |
| [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] | |
| variable {α β s T : ℝ} {A : C(ℝ,V →ᵇ Icc α β)} {b : C(ℝ,V →ᵇ ℂ)} | |
| {d : C₀(V,ℂ)} {u v : ℝ → C₀(V,ℂ)} | |
| omit [MeasurableSpace V] [BorelSpace V] in | |
| lemma Evolves.congr (hu : Evolves A b s T d u) (hsT : s≤T) | |
| (he : ∀ t ∈ Icc s T, v t=u t) : Evolves A b s T d v := by | |
| refine ⟨(he s ⟨le_rfl,hsT⟩).trans hu.1,hu.2.1.congr he,?_,?_,?_⟩ | |
| · intro t ht | |
| rw [he t ⟨ht.1.le,ht.2.le⟩] | |
| exact hu.2.2.1 t ht | |
| · intro t ht n hn | |
| rw [he t ⟨ht.1.le,ht.2.le⟩] | |
| exact hu.2.2.2.1 t ht n hn | |
| · intro t ht | |
| obtain ⟨p,hp,hpe⟩ := hu.2.2.2.2 t ht | |
| refine ⟨p,hp.congr_of_eventuallyEq ?_,?_⟩ | |
| · filter_upwards [Ioo_mem_nhds ht.1 ht.2] with r hr | |
| rw [he r ⟨hr.1.le,hr.2.le⟩] | |
| · rw [he t ⟨ht.1.le,ht.2.le⟩] | |
| exact hpe | |
| theorem exists_evolution_family (T α β C L B : ℝ) (hT : 0<T) (hT1 : T≤1) | |
| (hα : 0<α) (hC : 0≤C) (hL : 0≤L) (hB : 0≤B) | |
| (A : C(ℝ,V →ᵇ Icc α β)) (b : C(ℝ,V →ᵇ ℂ)) | |
| (hb : ∀ s ∈ Icc 0 T, ‖b s‖≤B) | |
| (hLip : ∀ s ∈ Icc 0 T, ∀ r ∈ Icc 0 T, ∀ x z : V, | |
| |(A s x).val-(A r z).val|≤L*(‖x-z‖+|s-r|)) | |
| (hLipb : ∀ s ∈ Icc 0 T, ∀ x z : V, ‖b s x-b s z‖≤L*‖x-z‖) | |
| (hA : ∀ s ∈ Icc 0 T, ContDiff ℝ ∞ (fun x => (A s x).val)) | |
| (hAj : ∀ s ∈ Icc 0 T, ∀ n : ℕ, n≤2 → ∀ x, | |
| ‖iteratedFDeriv ℝ n (fun y => (A s y).val) x‖≤C) : | |
| ∃ K : ℝ, 0≤K ∧ ∃ E : ℝ → ℝ → (C₀(V,ℂ) →L[ℂ] C₀(V,ℂ)), | |
| (∀ s ∈ Icc 0 T, E s s=ContinuousLinearMap.id ℂ C₀(V,ℂ)) ∧ | |
| (∀ s ∈ Icc 0 T, ∀ t ∈ Icc s T, ‖E t s‖≤K) ∧ | |
| (∀ s ∈ Icc 0 T, ∀ d : C₀(V,ℂ), Evolves A b s T d (fun t => E t s d)) ∧ | |
| (∀ s ∈ Icc 0 T, ∀ r ∈ Icc s T, ∀ t ∈ Icc r T, | |
| (E t r).comp (E r s)=E t s) := by | |
| classical | |
| obtain ⟨K,hK,hex⟩ := exists_uniform_evolves T α β L B hT hT1 hα hL hB A b hb hLip hLipb hA | |
| choose u hu hn using hex | |
| have huc (s : ℝ) (hs : s ∈ Icc 0 T) {d : C₀(V,ℂ)} {u v : ℝ → C₀(V,ℂ)} | |
| (hu : Evolves A b s T d u) (hv : Evolves A b s T d v) : | |
| ∀ t ∈ Icc s T, u t=v t := | |
| hu.unique hv hs.2 hα hC hB | |
| (fun t ht => hb t ⟨hs.1.trans ht.1,ht.2⟩) | |
| (fun t ht => (hA t ⟨hs.1.trans ht.1,ht.2⟩).of_le (by decide)) | |
| (fun t ht => hAj t ⟨hs.1.trans ht.1,ht.2⟩) | |
| let F (s : ℝ) (hs : s ∈ Icc 0 T) (t : ℝ) (ht : t ∈ Icc s T) : C₀(V,ℂ) →ₗ[ℂ] C₀(V,ℂ) := | |
| { toFun := fun d => u s hs d t | |
| map_add' := fun d e => huc s hs (hu s hs (d+e)) ((hu s hs d).add (hu s hs e)) t ht | |
| map_smul' := fun c d => huc s hs (hu s hs (c•d)) ((hu s hs d).smul c) t ht } | |
| let G (s : ℝ) (hs : s ∈ Icc 0 T) (t : ℝ) (ht : t ∈ Icc s T) : C₀(V,ℂ) →L[ℂ] C₀(V,ℂ) := | |
| (F s hs t ht).mkContinuous K (fun d => hn s hs d t ht) | |
| let E : ℝ → ℝ → (C₀(V,ℂ) →L[ℂ] C₀(V,ℂ)) := fun t s => | |
| if hs : s ∈ Icc 0 T then if ht : t ∈ Icc s T then G s hs t ht else 0 else 0 | |
| have hE (s : ℝ) (hs : s ∈ Icc 0 T) (t : ℝ) (ht : t ∈ Icc s T) (d : C₀(V,ℂ)) : | |
| E t s d=u s hs d t := by | |
| simp only [E,dite_eq_left hs,dite_eq_left ht,G,LinearMap.mkContinuous_apply,F] | |
| rfl | |
| have hsol (s : ℝ) (hs : s ∈ Icc 0 T) (d : C₀(V,ℂ)) : Evolves A b s T d (fun t => E t s d) := | |
| (hu s hs d).congr hs.2 (fun t ht => hE s hs t ht d) | |
| refine ⟨K,hK,E,?_,?_,hsol,?_⟩ | |
| · intro s hs | |
| apply ContinuousLinearMap.ext | |
| intro d | |
| exact (hE s hs s ⟨le_rfl,hs.2⟩ d).trans (hu s hs d).1 | |
| · intro s hs t ht | |
| rw [show E t s=G s hs t ht by simp only [E,dite_eq_left hs,dite_eq_left ht]] | |
| exact LinearMap.mkContinuous_norm_le _ hK _ | |
| · intro s hs r hr t ht | |
| have hr' : r ∈ Icc 0 T := ⟨hs.1.trans hr.1,hr.2⟩ | |
| apply ContinuousLinearMap.ext | |
| intro d | |
| exact huc r hr' (hsol r hr' (E r s d)) ((hsol s hs d).restrict hr.1) t ht | |
| end ParametrixEvolution | |
| end | |
| end OAI | |