openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Spectrum /SpectralAirySlopeLimit.lean
| 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 | |