| import OAI.Analysis.AncientFlat.Evolution.InitialModel |
| import OAI.Analysis.AncientFlat.Regularity.WithinInterpolation |
| import OAI.Analysis.AncientFlat.Regularity.HolderProducts |
|
|
| 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] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] |
| [MeasurableSpace V] [BorelSpace V] |
|
|
| theorem exists_initial_jet_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) → |
| ∃ u : ℝ → V → ℂ, (∀ x, u 0 x=0) ∧ |
| (∀ t∈Icc 0 (1/2:ℝ), ContDiff ℝ 4 (u t)) ∧ |
| JetBound α {p : V × ℝ | ‖p.1‖<3 ∧ p.2∈Icc 0 (1/2:ℝ)} u |
| (fun t x => g x-Δ (Δ (u t)) x) (C*F) ∧ |
| (∀ t∈Icc 0 (1/2:ℝ), ∀ x, HasDerivWithinAt (fun s => u s x) |
| (g x-Δ (Δ (u t)) x) (Icc 0 (1/2:ℝ)) t) := by |
| obtain ⟨C,hC,H⟩ := exists_initial_model (V := V) α hα hα1 |
| obtain ⟨L,hL,HI⟩ := within_lower_jet_interpolation (V := V) (E := ℂ) α 1 3 4 hα hα1 |
| zero_lt_one (by norm_num) |
| let T := (traceFour : (V [×4]→L[ℝ] ℂ) →L[ℝ] ℂ) |
| let D := C+(1+L)*C+1+‖T‖*C |
| have hD : 0<D := by dsimp [D]; positivity |
| refine ⟨D,hD,?_⟩ |
| intro g F hF hg hh |
| obtain ⟨u,hu0,hs,h4,hd,hu,hut⟩ := H g F hF hg hh |
| have hlow := HI (fun t x => u t x) (fun t x => g x-Δ (Δ (fun y => u t y)) x) |
| 0 (1/2) (by norm_num) (C*F) (C*F) (mul_nonneg hC.le hF) (mul_nonneg hC.le hF) hs |
| (fun t ht x _ => hd t ht x) (fun t ht x _ => hut t ht x) (fun t ht x _ => hu t ht x) |
| (fun t ht x _ => h4.1 (x,t) ht) |
| have hG : HolderBound α {p : V × ℝ | ‖p.1‖<3 ∧ p.2∈Icc 0 (1/2:ℝ)} |
| (fun p => g p.1) F := by |
| refine ⟨fun p _ => (g.toBCF.norm_coe_le_norm p.1).trans hg,?_⟩ |
| intro p _ q _ |
| exact (hh p.1 q.1).trans (mul_le_mul_of_nonneg_left |
| (Real.rpow_le_rpow (norm_nonneg _) (le_add_of_nonneg_right |
| (Real.rpow_nonneg (abs_nonneg _) _)) hα.le) hF) |
| have hT := hG.add ((h4.restrict (fun _ ht => ht.2)).clm T).neg |
| have he (p : V × ℝ) (hp : ‖p.1‖<3 ∧ p.2∈Icc 0 (1/2:ℝ)) : |
| g p.1-Δ (Δ (fun y => u p.2 y)) p.1= |
| g p.1+-T (iteratedFDeriv ℝ 4 (fun y => u p.2 y) p.1) := by |
| rw [bilaplacian_eq_traceFour _ (hs p.2 hp.2)]; rfl |
| refine ⟨fun t x => u t x,fun x => by simp [hu0],hs,⟨?_,?_⟩,hd⟩ |
| · intro j hj |
| by_cases hj3 : j≤3 |
| · apply (hlow j hj3).mono |
| dsimp [D] |
| nlinarith [mul_nonneg (norm_nonneg T) hC.le,mul_nonneg (norm_nonneg T) (mul_nonneg hC.le hF)] |
| · have hj4 : j=4 := by omega |
| subst j |
| apply (h4.restrict (fun _ hp => hp.2)).mono |
| exact mul_le_mul_of_nonneg_right (by dsimp [D]; nlinarith [mul_nonneg (norm_nonneg T) hC.le, mul_nonneg hL.le hC.le]) hF |
| · have hh : F+‖T‖*(C*F)≤D*F := by dsimp [D]; nlinarith [mul_nonneg hC.le hF,mul_nonneg hL.le (mul_nonneg hC.le hF)] |
| refine ⟨fun p hp => ?_,fun p hp q hq => ?_⟩ |
| · dsimp only |
| rw [he p hp] |
| exact (hT.1 p hp).trans hh |
| · dsimp only |
| rw [he p hp,he q hq] |
| exact (hT.2 p hp q hq).trans (mul_le_mul_of_nonneg_right hh (Real.rpow_nonneg (pdist_nonneg p q) α)) |
| end ParabolicInitial |
|
|
| end |
|
|
| end OAI |
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|
