| import OAI.Analysis.AncientFlat.Evolution.EvolutionAlgebra |
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|
| namespace OAI |
| noncomputable section |
| open Set Filter |
| open scoped ContDiff ZeroAtInfty BoundedContinuousFunction Laplacian |
| namespace ParametrixEvolution |
| open ParametrixUniqueness |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
| [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] |
| variable {α β s T C B : 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.restrict (hu : Evolves A b s T d u) {r : ℝ} (hr : s≤r) : |
| Evolves A b r T (u r) u := by |
| refine ⟨rfl,hu.2.1.mono (Icc_subset_Icc_left hr),?_,?_,?_⟩ |
| · intro t ht; exact hu.2.2.1 t ⟨hr.trans_lt ht.1,ht.2⟩ |
| · intro t ht; exact hu.2.2.2.1 t ⟨hr.trans_lt ht.1,ht.2⟩ |
| · intro t ht; exact hu.2.2.2.2 t ⟨hr.trans_lt ht.1,ht.2⟩ |
| lemma Evolves.zero_unique (hu : Evolves A b s T 0 u) (hsT : s<T) (hα : 0<α) |
| (hC : 0≤C) (hB : 0≤B) |
| (hb : ∀ t ∈ Icc s T, ‖b t‖≤B) |
| (hA : ∀ t ∈ Icc s T, ContDiff ℝ 2 (fun x => (A t x).val)) |
| (hAj : ∀ t ∈ Icc s T, ∀ n : ℕ, n≤2 → ∀ x, |
| ‖iteratedFDeriv ℝ n (fun y => (A t y).val) x‖≤C) : |
| ∀ t ∈ Icc s T, u t=0 := by |
| let : NormedAddCommGroup (V →ᵇ ℂ) := inferInstance |
| let : NormedSpace ℝ (V →ᵇ ℂ) := inferInstance |
| obtain ⟨K,hK0,hK⟩ := exists_energy_constant (V := V) α C hα hC |
| let w : ℝ → V →ᵇ ℂ := fun r => (u (s+r)).toBCF |
| have hwc : ContinuousOn w (Icc 0 (T-s)) := by |
| apply ZeroAtInftyContinuousMap.isometry_toBCF.continuous.comp_continuousOn |
| apply hu.2.1.comp (continuous_const.add continuous_id).continuousOn |
| intro t ht |
| change s+t ∈ Icc s T |
| exact ⟨by linarith [ht.1],by linarith [ht.2]⟩ |
| have hw0 : mass (w 0 : V → ℂ)=0 := by simp [w,hu.1,mass] |
| have hd (t : ℝ) (ht : t ∈ Ioo 0 (T-s)) : |
| ∃ e : ℝ, HasDerivAt (fun r => mass (w r : V → ℂ)) e t ∧ |
| e≤(2*K+2*B)*mass (w t : V → ℂ) := by |
| have hst : s+t ∈ Ioo s T := ⟨by linarith [ht.1],by linarith [ht.2]⟩ |
| have hst' : s+t ∈ Icc s T := ⟨hst.1.le,hst.2.le⟩ |
| obtain ⟨p,hp,hpe⟩ := hu.2.2.2.2 (s+t) hst |
| have hwt : HasDerivAt w p t := by |
| have ha : HasDerivAt (fun r : ℝ => s+r) 1 t := (hasDerivAt_id t).const_add s |
| convert hp.scomp t ha using 1 <;> first | rfl | simp only [one_smul] |
| have hj : BoundedJets 4 (w t : V → ℂ) := by |
| intro n hn |
| obtain ⟨M,hM⟩ := hu.2.2.2.1 (s+t) hst n hn |
| exact ⟨max M 0,le_max_right _ _,fun x => (hM x).trans (le_max_left _ _)⟩ |
| let a : V →ᵇ ℝ := (A (s+t)).comp Subtype.val (LipschitzWith.subtype_val (Icc α β)) |
| refine ⟨2*pair (w t) p,hasDerivAt_mass hwt,?_⟩ |
| exact energy_rhs_le hα hB a (b (s+t)) (w t) p (hb _ hst') (hA _ hst') |
| (hAj _ hst') (fun x => (A (s+t) x).property.1) (hu.2.2.1 _ hst) hj hK hpe |
| have he := nonneg_energy_eq_zero (sub_pos.mpr hsT) (continuousOn_mass hwc) hw0 |
| (fun t _ => mass_nonneg (w t : V → ℂ)) hd |
| intro t ht |
| have ht' : t-s ∈ Icc 0 (T-s) := ⟨by linarith [ht.1],by linarith [ht.2]⟩ |
| have hh := mass_eq_zero_bcf (he (t-s) ht') |
| apply ZeroAtInftyContinuousMap.toBCF_injective V ℂ |
| change (u t).toBCF=0 |
| simpa [w] using hh |
| lemma Evolves.unique (hu : Evolves A b s T d u) (hv : Evolves A b s T d v) |
| (hsT : s≤T) (hα : 0<α) (hC : 0≤C) (hB : 0≤B) |
| (hb : ∀ t ∈ Icc s T, ‖b t‖≤B) |
| (hA : ∀ t ∈ Icc s T, ContDiff ℝ 2 (fun x => (A t x).val)) |
| (hAj : ∀ t ∈ Icc s T, ∀ n : ℕ, n≤2 → ∀ x, |
| ‖iteratedFDeriv ℝ n (fun y => (A t y).val) x‖≤C) : |
| ∀ t ∈ Icc s T, u t=v t := by |
| let : NormedAddCommGroup C₀(V,ℂ) := inferInstance |
| let : NormedSpace ℂ C₀(V,ℂ) := inferInstance |
| rcases hsT.eq_or_lt with he|hlt |
| · subst T |
| intro t ht |
| have ht : t=s := le_antisymm ht.2 ht.1 |
| subst t |
| exact hu.1.trans hv.1.symm |
| · have hh : Evolves A b s T 0 (fun t => u t+(-1 : ℂ)•v t) := by |
| simpa only [neg_smul,one_smul,add_neg_cancel] using hu.add (hv.smul (-1)) |
| have hz := hh.zero_unique hlt hα hC hB hb hA hAj |
| intro t ht |
| simpa only [neg_smul,one_smul,←sub_eq_add_neg,sub_eq_zero] using hz t ht |
| end ParametrixEvolution |
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| end |
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| end OAI |
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