File size: 3,430 Bytes
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.MovingSource
import OAI.Analysis.AncientFlat.Evolution.SemigroupNormalize
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
end
end OAI
|