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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 | 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
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