| import OAI.Analysis.AncientFlat.Evolution.SourceEvolves |
| import OAI.Analysis.AncientFlat.Evolution.SourceResponse |
|
|
| namespace OAI |
| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped ContDiff ZeroAtInfty BoundedContinuousFunction Laplacian |
| namespace ParametrixVolterra |
| open FrozenParametrix StrongWeightedVolterra ParametrixEvolution |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] |
| [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] |
| variable {T α β : ℝ} |
| lemma correction_source_classical (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)) |
| (γ : ℝ) (hγ : 0<γ) (hγ1 : γ≤1) |
| (d : C₀(V,ℂ)) (l j : Time T →ᵇ C₀(V,ℂ)) |
| (hl : ∃ H : ℝ, 0 ≤ H ∧ ∀ t x y, ‖l t x-l t y‖ ≤ H*‖x-y‖^γ) |
| (hi : ∀ t : Time T, IntervalIntegrable |
| (fun r => residualKernel hα A b hT₀ t r (unweighted j r)) volume 0 t.val) |
| (hj : ∀ t : Time T, unweighted j t.val+∫ r in 0..t.val, |
| residualKernel hα A b hT₀ t r (unweighted j r) = |
| l t-actualResidual hα A b t (initial hT₀) d) : |
| (∀ t ∈ Ioo 0 T, ContDiff ℝ 4 (fun x => mildTime hα A hT₀ d j t x)) ∧ |
| (∀ t ∈ Ioo 0 T, ∀ n : ℕ, n≤4 → ∃ M : ℝ, ∀ x, |
| ‖iteratedFDeriv ℝ n (fun y => mildTime hα A hT₀ d j t y) x‖≤M) ∧ |
| (∀ t : Time T, t.val<T → ∃ v : V →ᵇ ℂ, |
| HasDerivAt (fun r => (mildTime hα A hT₀ d j r).toBCF) v t.val ∧ ∀ x, |
| v x=l t x-((A (now t) x).val : ℂ)*Δ (Δ (fun y => mildTime hα A hT₀ d j t.val y)) x- |
| b (now t) x*mildTime hα A hT₀ d j t.val x) := by |
| obtain ⟨H,hH,hl⟩ := hl |
| let η := min γ (1/4 : ℝ) |
| have hη : 0 < η := lt_min hγ (by norm_num) |
| have hηγ : η ≤ γ := min_le_left _ _ |
| have hηq : η ≤ 1/4 := min_le_right _ _ |
| have hη1 : η ≤ 1 := hηγ.trans hγ1 |
| have hholdl : ∀ t x y, ‖l t x-l t y‖ ≤ (H+2*‖l‖)*‖x-y‖^η := by |
| intro t x y |
| by_cases hd : ‖x-y‖ ≤ 1 |
| · apply (hl t x y).trans |
| apply mul_le_mul |
| · exact le_add_of_nonneg_right (by positivity) |
| · exact Real.rpow_le_rpow_of_exponent_ge' (norm_nonneg _) hd hη.le hηγ |
| · exact Real.rpow_nonneg (norm_nonneg _) _ |
| · positivity |
| · have h1 : 1 ≤ ‖x-y‖^η := Real.one_le_rpow (le_of_lt (lt_of_not_ge hd)) hη.le |
| have hx : ‖l t x‖ ≤ ‖l‖ := ((l t).toBCF.norm_coe_le_norm x).trans (l.norm_coe_le_norm t) |
| have hy : ‖l t y‖ ≤ ‖l‖ := ((l t).toBCF.norm_coe_le_norm y).trans (l.norm_coe_le_norm t) |
| calc |
| _ ≤ 2*‖l‖ := (norm_sub_le _ _).trans (by linarith) |
| _ ≤ H+2*‖l‖ := le_add_of_nonneg_left hH |
| _ ≤ _ := le_mul_of_one_le_right (by positivity) h1 |
| obtain ⟨J,hJ,hholder⟩ := source_correction_holder hα A b hT₀ hT L B hL hB hb hLip hLipb |
| η hη hηq d l j (H+2*‖l‖) (by positivity) hholdl hi hj |
| refine ⟨fun t ht => ?_,fun t ht n hn => ?_,fun t ht => ?_⟩ |
| · exact contDiff_mildTime_of_holder_gamma hα A hT₀ η hη hη1 L hL hLip d j J hJ hholder ⟨t,ht.1,ht.2.le⟩ |
| · exact bounded_iteratedFDeriv_mildTime_gamma hα A hT₀ η hη hη1 L hL hLip j J hJ hholder d ⟨t,ht.1,ht.2.le⟩ n hn |
| · refine ⟨(l t).toBCF+mildRHS hα A b hT₀ d j t, |
| hasDerivAt_source_mildTime hα A b hT₀ η hη hη1 L hL hLip d j l J hJ hholder hi hj hA t ht,?_⟩ |
| intro x |
| change l t x+(-(((A (now t) x).val : ℂ)*mildBilaplacian hα A hT₀ d j t x)-b (now t) x*mildTime hα A hT₀ d j t.val x) = _ |
| rw [mildBilaplacian_apply_gamma hα A hT₀ d j η hη hη1 L J hL hJ hLip hholder t x] |
| ring |
| end ParametrixVolterra |
| |
| end |
| |
| end OAI |
| |
