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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 | import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralAiryRiccati
import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralAiryMass
import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralRiccatiLimit
namespace OAI
/-! The real logarithmic slope of every nonzero-flux forbidden-side Airy
solution tends to the growing slope. -/
open Set Filter Topology
namespace DefocusingNLS
theorem spectralAirySlope_tendsto_one
(q : ℝ → ℂ × ℂ) (T : ℝ) (hT : 1≤T)
(hq : ContinuousOn q (Ici T))
(hD : ∀ t, T≤t → HasDerivAt q (spectralScalarField (-(t : ℂ)) (q t)) t)
(hJ : spectralScalarFlux (q T)≠0) :
Tendsto (spectralAirySlope q) atTop (𝓝 1) := by
obtain ⟨S,hTS,hm,hbound⟩ := spectralAiry_eventual_mass_bound q T hT hq hD hJ
let Q := fun t => spectralScalarMass (q t)
let P := fun t => spectralScalarMomentum (q t)
let J := spectralScalarFlux (q T)
let e := fun t => J^2/(Real.sqrt t*(Q t)^2)
let D := fun t => Real.sqrt t*(1-(spectralAirySlope q t)^2)-spectralAirySlope q t/(2*t)+e t
have hS : 1≤S := hT.trans hTS
have hmass (t : ℝ) (ht : S≤t) : 0<Q t := hm.trans_le (hbound t ht).2
have hsqrt (t : ℝ) (ht : S≤t) : 0<Real.sqrt t := Real.sqrt_pos.mpr (by linarith)
have hflux (t : ℝ) (ht : S≤t) : spectralScalarFlux (q t)=J :=
spectralScalarFlux_const T t q (fun x => -x) (hq.mono (fun _ hx => hx.1))
(fun x hx => by simpa only [Complex.ofReal_neg] using hD x hx.1.le) t ⟨hTS.trans ht,le_rfl⟩
have hqS : ContinuousOn q (Ici S) := hq.mono (Ici_subset_Ici.mpr hTS)
have hQc : ContinuousOn Q (Ici S) := by
have hh := hqS.fst.norm.pow 2
change ContinuousOn (fun t => ‖(q t).1‖^2) (Ici S) at hh
simpa only [Q,spectralScalarMass,Complex.normSq_eq_norm_sq] using hh
have hPc : ContinuousOn P (Ici S) := Complex.continuous_re.comp_continuousOn (hqS.fst.star.mul hqS.snd)
have hsc : ContinuousOn (spectralAirySlope q) (Ici S) :=
hPc.div (Real.continuous_sqrt.continuousOn.mul hQc)
(fun t ht => mul_ne_zero (hsqrt t ht).ne' (hmass t ht).ne')
have hd (t : ℝ) (ht : S≤t) : HasDerivAt (spectralAirySlope q) (D t) t := by
have hh := spectralAirySlope_hasDerivAt q t (by linarith) (hD t (hTS.trans ht)) (hmass t ht).ne'
simpa only [hflux t ht] using hh
have he0 (t : ℝ) (ht : S≤t) : 0≤e t := by
dsimp only [e]
exact div_nonneg (sq_nonneg J) (mul_nonneg (hsqrt t ht).le (sq_nonneg _))
have heC (t : ℝ) (ht : S≤t) : e t≤(J^2/(Q S)^2)/Real.sqrt t := by
have hs0 := hsqrt t ht
have hQt := hmass t ht
have hQs : 0<Q S := hm
have hQQ : (Q S)^2≤(Q t)^2 := pow_le_pow_left₀ hQs.le (hbound t ht).2 2
calc
e t ≤ J^2/(Real.sqrt t*(Q S)^2) := by
dsimp only [e]
exact div_le_div_of_nonneg_left (sq_nonneg J) (by positivity) (mul_le_mul_of_nonneg_left hQQ hs0.le)
_ = _ := by ring
exact spectralRiccati_tendsto_one (spectralAirySlope q) D e S (J^2/(Q S)^2) hS hsc hd
(fun t ht => div_nonneg (hbound t ht).1.le (mul_nonneg (hsqrt t ht).le (hmass t ht).le))
he0 heC (fun _ _ => rfl)
end DefocusingNLS
end OAI
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