openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Spectrum /SpectralAiryMass.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralAiryMomentum | |
| namespace OAI | |
| /-! A positive lower bound for the Airy mass after the momentum turns positive. -/ | |
| open Set | |
| namespace DefocusingNLS | |
| theorem spectralScalarMass_pos (q : ℂ × ℂ) (hJ : spectralScalarFlux q≠0) : | |
| 0<spectralScalarMass q := by | |
| rw [spectralScalarMass,Complex.normSq_eq_norm_sq] | |
| exact sq_pos_of_pos (norm_pos_iff.mpr (spectralScalarFlux_ne_zero_value q hJ)) | |
| theorem spectralAiry_eventual_mass_bound | |
| (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) : | |
| ∃ S : ℝ, T≤S ∧ 0<spectralScalarMass (q S) ∧ | |
| ∀ t, S≤t → 0<spectralScalarMomentum (q t) ∧ spectralScalarMass (q S)≤ spectralScalarMass (q t) := by | |
| obtain ⟨S,hTS,hP⟩ := spectralAiry_eventually_positive_momentum q T hT hq hD hJ | |
| have hJs : spectralScalarFlux (q S)≠0 := by | |
| rw [spectralScalarFlux_const T S 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) S ⟨hTS,le_rfl⟩] | |
| exact hJ | |
| refine ⟨S,hTS,spectralScalarMass_pos (q S) hJs,?_⟩ | |
| have hc : ContinuousOn (fun t => spectralScalarMass (q t)) (Ici S) := by | |
| have hh := ((hq.mono (Ici_subset_Ici.mpr hTS)).fst.norm.pow 2) | |
| change ContinuousOn (fun t => ‖(q t).1‖^2) (Ici S) at hh | |
| simpa only [spectralScalarMass,Complex.normSq_eq_norm_sq] using hh | |
| have hm : MonotoneOn (fun t => spectralScalarMass (q t)) (Ici S) := by | |
| apply monotoneOn_of_deriv_nonneg (convex_Ici S) hc | |
| · intro t ht | |
| have htS : S<t := by simpa only [interior_Ici,mem_Ioi] using ht | |
| exact (spectralScalarMass_hasDerivAt q (-(t : ℂ)) t (hD t (hTS.trans htS.le))).differentiableAt.differentiableWithinAt | |
| · intro t ht | |
| have htS : S<t := by simpa only [interior_Ici,mem_Ioi] using ht | |
| rw [(spectralScalarMass_hasDerivAt q (-(t : ℂ)) t (hD t (hTS.trans htS.le))).deriv] | |
| exact mul_nonneg (by norm_num) (hP t htS.le).le | |
| intro t ht | |
| exact ⟨hP t ht,hm (by simp only [mem_Ici,le_refl]) (by simpa only [mem_Ici] using ht) ht⟩ | |
| end DefocusingNLS | |
| end OAI | |