| import OAI.Analysis.AncientFlat.Evolution.ActualClassical |
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| namespace OAI |
| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped ContDiff |
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| namespace ParametrixVolterra |
| open scoped ZeroAtInfty BoundedContinuousFunction SchwartzMap Laplacian |
| open FrozenParametrix StrongWeightedVolterra |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
| [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] |
| variable {T α β : ℝ} |
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| theorem exists_classical_homogeneous_evolution (hα : 0 < α) |
| (A : C(Slab T,V →ᵇ Icc α β)) (b : C(Slab T,V →ᵇ ℂ)) |
| (hT₀ : 0 ≤ T) (hT : T ≤ 1) (L B : ℝ) (hL : 0 ≤ L) (hB : 0 ≤ B) |
| (hb : ∀ s, ‖b s‖ ≤ B) |
| (hLip : ∀ s r : Slab T, ∀ x z : V, |
| |(A s x).val-(A r z).val| ≤ L*(‖x-z‖+|s.val-r.val|)) |
| (hLipb : ∀ s : Slab T, ∀ x z : V, ‖b s x-b s z‖ ≤ L*‖x-z‖) |
| (hA : ∀ s, ContDiff ℝ ∞ (fun x => (A s x).val)) : |
| ∃ K : ℝ, 0 ≤ K ∧ ∀ d : C₀(V,ℂ), ∃ u : ℝ → C₀(V,ℂ), |
| u 0=d ∧ ContinuousOn u (Icc 0 T) ∧ |
| (∀ t ∈ Icc 0 T, ‖u t‖ ≤ K*‖d‖) ∧ |
| (∀ t ∈ Ioo 0 T, ContDiff ℝ 4 (fun x => u t x)) ∧ |
| (∀ (t : Time T), t.val < T → ∀ x : V, |
| HasDerivAt (fun r => u r x) |
| (-((A (now t) x).val : ℂ)*(Δ (Δ (fun y => u t.val y))) x |
| -b (now t) x*u t.val x) t.val) := by |
| obtain ⟨C,D,hC,hD,hs⟩ := exists_parametrix_correction_full hα A b hT₀ hT L B hL hB hb hLip |
| let M := frozenBound (V := V) (β := β) hα |
| let Q := ∫ v : UnitTime, MildVolterra.massWeight v |
| have hM : 0≤M := frozenBound_nonneg hα |
| have hQ : 0≤Q := integral_nonneg MildVolterra.massWeight_nonneg |
| refine ⟨M+Q*M*D*C,by positivity,fun d => ?_⟩ |
| obtain ⟨j,hi,hj,hjb⟩ := hs d 0 |
| have hj' : ∀ t : Time T, unweighted j t.val+∫ r in 0..t.val, |
| residualKernel hα A b hT₀ t r (unweighted j r) = |
| -actualResidual hα A b t (initial hT₀) d := by simpa using hj |
| let : NormedAddCommGroup (Time T →ᵇ C₀(V,ℂ)) := inferInstance |
| have hz : ‖(0 : Time T →ᵇ C₀(V,ℂ))‖ = 0 := norm_zero |
| have hjb' : ‖j‖ ≤ D*(C*‖d‖) := by |
| rw [hz,zero_add] at hjb |
| exact hjb |
| refine ⟨mildTime hα A hT₀ d j,mildTime_zero hα A hT₀ d j, |
| continuousOn_mildTime hα A hT₀ d j,fun t ht => ?_,fun t ht => ?_,fun t ht x => ?_⟩ |
| · have ht1 : t^(1/4 : ℝ)≤1 := Real.rpow_le_one ht.1 (ht.2.trans hT) (by positivity) |
| have h0 := norm_c0Time_le hα (A (initial hT₀)) t d |
| have h1 := norm_duhamel_le hα A hT₀ j t ht |
| calc |
| ‖mildTime hα A hT₀ d j t‖ ≤ M*‖d‖+t^(1/4 : ℝ)*(Q*M*‖j‖) := |
| (norm_add_le _ _).trans (add_le_add h0 h1) |
| _ ≤ M*‖d‖+Q*M*‖j‖ := add_le_add le_rfl |
| (mul_le_of_le_one_left (mul_nonneg (mul_nonneg hQ hM) (norm_nonneg j)) ht1) |
| _ ≤ M*‖d‖+Q*M*(D*(C*‖d‖)) := add_le_add le_rfl |
| (mul_le_mul_of_nonneg_left hjb' (mul_nonneg hQ hM)) |
| _ = _ := by ring |
| · exact contDiff_mildTime_of_correction hα A b hT₀ hT L B hL hB hb hLip hLipb d j hi hj' ⟨t,ht.1,ht.2.le⟩ |
| · exact hasDerivAt_mildTime_apply hα A b hT₀ hT L B hL hB hb hLip hLipb d j hi hj' hA t ht x |
| end ParametrixVolterra |
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| end |
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| end OAI |
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