openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Spectrum /SpectralAnnulusClassicalFlux.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralAnnulusFluxPrimitive | |
| import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralForcedClassicalFlux | |
| namespace OAI | |
| /-! Exterior classical fluxes from tests supported outside the core. -/ | |
| open Set MeasureTheory | |
| open scoped ContDiff | |
| namespace DefocusingNLS | |
| theorem spectralSecond_forced_classical_annulus (ell : ℕ) (L R : ℝ) | |
| (hL : 0 ≤ L) (hR : 0 < R) | |
| (w a : SpectralHarmonicWeight R) (u : SpectralHarmonicPair ell R) (G : ℝ → ℂ) | |
| (hw : ContinuousOn w.density (Ioo L R)) (ha : ContinuousOn a.density (Ioo L R)) | |
| (hpos : ∀ x ∈ Ioo L R, 0 < w.density x) (hG : ContinuousOn G (Ioo L R)) | |
| (hF : LocallyIntegrableOn (spectralSecondFlux ell R w a u) (Ioo L R)) | |
| (he : ∀ φ : ℝ → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ → | |
| tsupport φ ⊆ Ioo L R → | |
| (∫ x, deriv φ x • spectralSecondFlux ell R w a u x) = -(∫ x, φ x • G x)) : | |
| DifferentiableOn ℝ (spectralHarmonicRepresentative ell R hR u.snd) (Ioo L R) ∧ | |
| ContinuousOn (deriv (spectralHarmonicRepresentative ell R hR u.snd)) (Ioo L R) ∧ | |
| ∀ x ∈ Ioo L R, HasDerivAt (spectralSecondClassicalFlux ell R hR w a u) (G x) x := by | |
| have hlocal (x : ℝ) (hx : x ∈ Ioo L R) : | |
| DifferentiableAt ℝ (spectralHarmonicRepresentative ell R hR u.snd) x ∧ | |
| ContinuousAt (deriv (spectralHarmonicRepresentative ell R hR u.snd)) x ∧ | |
| HasDerivAt (spectralSecondClassicalFlux ell R hR w a u) | |
| (G x) x := by | |
| let l := (L+x)/2 | |
| let r := (x+R)/2 | |
| have hLl : L < l := by dsimp only [l]; linarith [hx.1] | |
| have hl : 0 < l := hL.trans_lt hLl | |
| have hxl : l < x := by dsimp only [l]; linarith [hx.1] | |
| have hxr : x < r := by dsimp only [r]; linarith [hx.2] | |
| have hr : r < R := by dsimp only [r]; linarith [hx.2] | |
| have hs : Ioo l r ⊆ Ioo L R := fun t ht => ⟨hLl.trans ht.1,ht.2.trans hr⟩ | |
| obtain ⟨P,hP,hflux⟩ := spectralAnnulusFlux_primitive L R l r hLl (hxl.trans hxr) hr | |
| (spectralSecondFlux ell R w a u) G | |
| hF hG he | |
| have hPc : ContinuousOn P (Ioo l r) := fun t ht => (hP t ht).continuousAt.continuousWithinAt | |
| have hd (t : ℝ) (ht : t ∈ Ioo l r) := spectralSecondFlux_hasDerivAt ell R l r hR | |
| hl hr.le w a u P (hw.mono hs) (ha.mono hs) (fun s hs' => hpos s (hs hs')) hPc hflux t ht | |
| have heq (t : ℝ) (ht : t ∈ Ioo l r) : | |
| spectralSecondClassicalFlux ell R hR w a u t=P t := | |
| spectralSecondClassicalFlux_eq ell R hR w a u P t (hl.trans ht.1).ne' | |
| (hpos t (hs ht)).ne' (hd t ht) | |
| refine ⟨(hd x ⟨hxl,hxr⟩).differentiableAt,?_,?_⟩ | |
| · have hDc := spectralSecondDerivativeValue_continuousOn ell R l r hR hl hr.le | |
| w a u P (hw.mono hs) (ha.mono hs) (fun s hs' => hpos s (hs hs')) hPc | |
| exact (hDc.congr (fun t ht => (hd t ht).deriv)).continuousAt | |
| (Ioo_mem_nhds hxl hxr) | |
| · apply (hP x ⟨hxl,hxr⟩).congr_of_eventuallyEq | |
| filter_upwards [Ioo_mem_nhds hxl hxr] with t ht | |
| exact heq t ht | |
| exact ⟨fun x hx => (hlocal x hx).1.differentiableWithinAt, | |
| fun x hx => (hlocal x hx).2.1.continuousWithinAt,fun x hx => (hlocal x hx).2.2⟩ | |
| end DefocusingNLS | |
| end OAI | |