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12dee50 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 | import OAI.Analysis.AncientFlat.Evolution.ActualClassical
namespace OAI
noncomputable section
open Set Filter MeasureTheory
open scoped ContDiff
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}
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
end
end OAI
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