lean/OAI/Analysis/AncientFlat/Regularity/CorrectionSourceClassical.lean

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