| import OAI.Analysis.AncientFlat.Regularity.InitialJetModel |
| import OAI.Analysis.AncientFlat.Evolution.TimePaste |
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|
| namespace OAI |
| noncomputable section |
| open Set Filter |
| open scoped ContDiff ZeroAtInfty BoundedContinuousFunction Laplacian |
| namespace ParabolicInitial |
| open ParabolicInterior ParabolicSchauder ParametrixEvolution ClassicalIBP |
| variable {V : Type} [NormedAddCommGroup V] |
| def reflectTime (p : V × ℝ) : V × ℝ := (p.1,-p.2) |
| lemma pdist_reflect (p q : V × ℝ) : pdist (reflectTime p) (reflectTime q)=pdist p q := by |
| simp only [pdist,reflectTime,neg_sub_neg,abs_sub_comm] |
| lemma holder_reflect {E : Type*} [NormedAddCommGroup E] {α K a b R : ℝ} |
| {f : V × ℝ → E} (hf : HolderBound α {p : V × ℝ | ‖p.1‖<R ∧ p.2∈Icc a b} f K) : |
| HolderBound α {p : V × ℝ | ‖p.1‖<R ∧ p.2∈Icc (-b) (-a)} (fun p => f (reflectTime p)) K := by |
| have hm (p : V × ℝ) (hp : ‖p.1‖<R ∧ p.2∈Icc (-b) (-a)) : |
| reflectTime p∈{p : V × ℝ | ‖p.1‖<R ∧ p.2∈Icc a b} := |
| ⟨hp.1,by dsimp [reflectTime]; constructor <;> linarith [hp.2.1,hp.2.2]⟩ |
| exact ⟨fun p hp => hf.1 _ (hm p hp),fun p hp q hq => by |
| simpa only [pdist_reflect] using hf.2 _ (hm p hp) _ (hm q hq)⟩ |
|
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| variable [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] |
|
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| theorem exists_backward_model (α : ℝ) (hα : 0<α) (hα1 : α<1) : |
| ∃ C : ℝ, 0<C ∧ ∀ (g : C₀(V,ℂ)) (F : ℝ), 0≤F → ‖g‖≤F → |
| (∀ x y, ‖g x-g y‖≤F*‖x-y‖^α) → |
| ∃ v vt : ℝ → V → ℂ, (∀ x, v 0 x=0) ∧ (∀ x, vt 0 x=g x) ∧ |
| (∀ t∈Icc (-1/2:ℝ) 0, ContDiff ℝ 4 (v t)) ∧ |
| JetBound α {p : V × ℝ | ‖p.1‖<3 ∧ p.2∈Icc (-1/2:ℝ) 0} v vt (C*F) ∧ |
| (∀ t∈Icc (-1/2:ℝ) 0, ∀ x, HasDerivWithinAt (fun s => v s x) |
| (vt t x) (Icc (-1/2:ℝ) 0) t) := by |
| obtain ⟨C,hC,H⟩ := exists_initial_jet_model (V := V) α hα hα1 |
| refine ⟨C,hC,?_⟩ |
| intro g F hF hg hh |
| obtain ⟨u,hu0,hs,hj,hd⟩ := H g F hF hg hh |
| let v : ℝ → V → ℂ := fun t => -u (-t) |
| let vt : ℝ → V → ℂ := fun t x => g x-Δ (Δ (u (-t))) x |
| have ht (t : ℝ) (ht : t∈Icc (-1/2:ℝ) 0) : -t∈Icc 0 (1/2:ℝ) := by |
| constructor <;> linarith [ht.1,ht.2] |
| have hu0' : u 0=fun _ => (0:ℂ) := funext hu0 |
| refine ⟨v,vt,?_,?_,?_,⟨?_,?_⟩,?_⟩ |
| · intro x; simp [v,hu0] |
| · intro x |
| dsimp [vt] |
| rw [neg_zero,hu0'] |
| have hz : Δ (0 : V → ℂ)=0 := InnerProductSpace.laplacian_const (c := (0:ℂ)) |
| rw [InnerProductSpace.laplacian_const,hz] |
| simp |
| · intro t h; exact (hs _ (ht t h)).neg |
| · intro j hj4 |
| have H := (holder_reflect (hj.1 j hj4)).neg |
| simpa only [neg_zero,neg_div,neg_neg,reflectTime,Prod.fst,Prod.snd, |
| iteratedFDeriv_neg_apply,v,Pi.neg_apply] using H |
| · have H := holder_reflect hj.2 |
| simpa only [neg_zero,neg_div,reflectTime,Prod.fst,Prod.snd,vt] using H |
| · intro t h x |
| have H := (hd (-t) (ht t h) x).scomp t ((hasDerivAt_neg t).hasDerivWithinAt |
| (s := Icc (-1/2:ℝ) 0)) (fun s hs => ht s hs) |
| convert H.neg using 1 <;> first | rfl | simp [vt] |
| end ParabolicInitial |
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| end |
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| end OAI |
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