File size: 3,590 Bytes
ef96466 | 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 67 68 69 70 | import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralParameterLogJets
import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralCanonicalParameterEquation
namespace OAI
/-! The actual canonical parameter derivative has bounded normalized jets. -/
open Set Filter Topology
namespace DefocusingNLS
local notation "E₄" => (ℂ × ℂ) × (ℂ × ℂ)
theorem canonical_circular_parameter_logJets (ν η b : ℂ) (n : ℕ) (hn : 1≤n)
(L : ℝ) (hX : HasRadialExterior ν n b L) (hb : b ≠ 0)
(c : ℂ × ℂ) (Y : ℂ → ℝ → E₄)
(hY : IsCanonicalHolomorphicColumn ν η b n L c Y) (z : ℂ) :
HasLogJetBound 0 (fun t => (deriv (fun lam => Y lam t) z).1.1) ∧
HasLogJetBound 0 (fun t => (deriv (fun lam => Y lam t) z).2.1) := by
let q := radialExteriorCanonical ν n b L
let D := fun t => deriv (fun lam => Y lam t) z
let νp := ν-2*z
let νm := star ν-2*z
let L₀ := max L 0
have hq (t : ℝ) (ht : L₀<t) : HasDerivAt q (radialExteriorODEField ν n t (q t)) t :=
((radialExteriorCanonical_spec hX).2.2 t ((le_max_left L 0).trans ht.le)).2
have hYe (t : ℝ) (ht : L₀<t) : HasDerivAt (Y z)
(circularLeadingField t (Y z t)+circularBoundedField νp νm η n (q t).1 (Y z t)) t :=
hY.1 z t ((le_max_right L 0).trans ht.le)
have hDe (t : ℝ) (ht : L₀<t) : HasDerivAt D
(circularLeadingField t (D t)+circularBoundedField νp νm η n (q t).1 (D t)+
circularPointSlopeCLM νp νm 0 (Y z t)) t := by
have h := canonical_circular_parameter_equation ν η b n hn L hX hb c Y hY z t
((le_max_right L 0).trans_lt ht)
rw [circularPointSlope_shift] at h
exact h
let Mq := ‖(b,(0 : ℂ))‖+1
let MY := ‖((c.1,(0 : ℂ)),(c.2,(0 : ℂ)))‖+1
obtain ⟨Uq,hUq⟩ := eventually_atTop.mp
((radialExteriorCanonical_spec hX).2.1.tendsto.norm.eventually
(gt_mem_nhds (show ‖(b,(0 : ℂ))‖<Mq by dsimp only [Mq]; linarith)))
obtain ⟨UY,hUY⟩ := eventually_atTop.mp
((hY.2.1 z).norm.eventually
(gt_mem_nhds (show ‖((c.1,(0 : ℂ)),(c.2,(0 : ℂ)))‖<MY by dsimp only [MY]; linarith)))
obtain ⟨C,hC⟩ := canonical_circular_parameter_bounded ν η b n hn L hX hb c Y hY z
let M := max Mq (max MY C)
let U := max 0 (max Uq UY)
have hM : 0≤M := (by dsimp only [Mq]; positivity : 0≤Mq).trans (le_max_left _ _)
have hbq (t : ℝ) (ht : U≤t) : ‖q t‖≤M :=
(hUq t (((le_max_left Uq UY).trans (le_max_right 0 _)).trans ht)).le.trans (le_max_left _ _)
have hbY (t : ℝ) (ht : U≤t) : ‖Y z t‖≤M :=
(hUY t (((le_max_right Uq UY).trans (le_max_right 0 _)).trans ht)).le.trans
((le_max_left MY C).trans (le_max_right _ _))
have hbD (t : ℝ) (ht : U≤t) : ‖D t‖≤M :=
(hC t ((le_max_left 0 _).trans ht)).trans
((le_max_right MY C).trans (le_max_right _ _))
obtain ⟨P,hP⟩ := canonical_parameter_polynomial_approximation ν η b n hn L hX hb c Y hY z
have hpos := spectralParameter_positive_derivatives ν νp νm η n q (Y z) D L₀ U M hM
hq hYe hDe hbq hbY hbD P (by
intro J
obtain ⟨A,hA,hAT⟩ := hP J
exact ⟨A,0,hA,hAT⟩)
have hs := spectralParameter_solution_contDiffOn ν νp νm η n q (Y z) D L₀ hq hYe hDe
exact ⟨HasLogJetBound.of_bounded_positive_decay _ L₀ U M hM hs.fst.fst
(fun t ht => (norm_fst_le _).trans ((norm_fst_le _).trans (hbD t ht))) hpos.1,
HasLogJetBound.of_bounded_positive_decay _ L₀ U M hM hs.snd.fst
(fun t ht => (norm_fst_le _).trans ((norm_snd_le _).trans (hbD t ht))) hpos.2⟩
end DefocusingNLS
end OAI
|