| import OAI.Analysis.AncientFlat.Evolution.MovingSource |
| import OAI.Analysis.AncientFlat.Evolution.SemigroupNormalize |
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|
| namespace OAI |
| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped ZeroAtInfty BoundedContinuousFunction Topology |
| namespace FlatSimilarity |
| open ParametrixEvolution |
| variable {V : Type} [NormedAddCommGroup V] |
| lemma continuous_normalizedSource (γ a : ℝ) (f : ℝ →ᵇ C₀(V,ℂ)) : |
| Continuous (fun t => (Real.exp (-γ*unitClamp t) : ℂ) • f (a+unitClamp t)) := by |
| exact (Complex.continuous_ofReal.comp |
| (Real.continuous_exp.comp (continuous_const.mul unitClamp_lipschitz.continuous))).smul |
| (f.continuous.comp (continuous_const.add unitClamp_lipschitz.continuous)) |
| def normalizedSource (γ a : ℝ) (hγ : 0≤γ) (f : ℝ →ᵇ C₀(V,ℂ)) : ℝ →ᵇ C₀(V,ℂ) := |
| BoundedContinuousFunction.ofNormedAddCommGroup |
| (fun t => (Real.exp (-γ*unitClamp t) : ℂ) • f (a+unitClamp t)) |
| (continuous_normalizedSource γ a f) ‖f‖ (fun t => by |
| rw [norm_smul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)] |
| have he : Real.exp (-γ*unitClamp t)≤1 := Real.exp_le_one_iff.mpr |
| (mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hγ) (unitClamp_mem t).1) |
| exact (mul_le_mul_of_nonneg_right he (norm_nonneg _)).trans |
| (by simpa using f.norm_coe_le_norm (a+unitClamp t))) |
| @[simp] lemma normalizedSource_apply (γ a : ℝ) (hγ : 0≤γ) (f : ℝ →ᵇ C₀(V,ℂ)) |
| {t : ℝ} (ht : t ∈ Icc (0:ℝ) 1) : |
| normalizedSource γ a hγ f t=(Real.exp (-γ*t) : ℂ) • f (a+t) := by |
| change (Real.exp (-γ*unitClamp t) : ℂ) • f (a+unitClamp t)=_ |
| rw [unitClamp_eq ht] |
| lemma normalizedSource_holder (γ a η : ℝ) (hγ : 0≤γ) (f : ℝ →ᵇ C₀(V,ℂ)) |
| (hf : ∃ H : ℝ, 0≤H ∧ ∀ t ∈ Icc a (a+1), ∀ x y, ‖f t x-f t y‖≤H*‖x-y‖^η) : |
| ∃ H : ℝ, 0≤H ∧ ∀ t ∈ Icc (0:ℝ) 1, ∀ x y, |
| ‖normalizedSource γ a hγ f t x-normalizedSource γ a hγ f t y‖≤H*‖x-y‖^η := by |
| obtain ⟨H,hH,hf⟩ := hf |
| refine ⟨H,hH,fun t ht x y => ?_⟩ |
| rw [normalizedSource_apply γ a hγ f ht] |
| change ‖(Real.exp (-γ*t) : ℂ) • f (a+t) x-(Real.exp (-γ*t) : ℂ) • f (a+t) y‖≤_ |
| rw [←smul_sub,norm_smul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)] |
| have he : Real.exp (-γ*t)≤1 := Real.exp_le_one_iff.mpr |
| (mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hγ) ht.1) |
| exact ((mul_le_mul_of_nonneg_right he (norm_nonneg _)).trans_eq (one_mul _)).trans |
| (hf _ ⟨by linarith [ht.1],by linarith [ht.2]⟩ x y) |
| end FlatSimilarity |
| namespace AncientMild |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] |
| lemma unnormalize_duhamel (γ : ℝ) (S : ℝ → E →L[ℂ] E) (f v : ℝ → E) (a t : ℝ) |
| (hv : v (a+t)=(Real.exp (γ*t) : ℂ) • S t (v a)+ |
| ∫ s in (0:ℝ)..t, (Real.exp (γ*(t-s)) : ℂ) • S (t-s) (f (a+s))) : |
| (Real.exp (-γ*t) : ℂ) • v (a+t)=S t (v a)+ |
| ∫ s in (0:ℝ)..t, S (t-s) ((Real.exp (-γ*s) : ℂ) • f (a+s)) := by |
| rw [hv,smul_add,smul_smul,←Complex.ofReal_mul,←Real.exp_add, |
| show -γ*t+γ*t=0 by ring,Real.exp_zero,Complex.ofReal_one,one_smul, |
| ←intervalIntegral.integral_smul] |
| congr 1 |
| apply intervalIntegral.integral_congr |
| intro s _ |
| dsimp only |
| rw [map_smul,smul_smul,←Complex.ofReal_mul,←Real.exp_add] |
| congr 2 |
| ring_nf |
| end AncientMild |
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|
| end |
|
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| end OAI |
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