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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 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 | import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralFirstBalance
namespace OAI
/-! Retain the exact outer-boundary terms in both scalar variational equations. -/
open MeasureTheory
open scoped SchwartzMap
namespace DefocusingNLS
theorem spectralSecondTest_boundaryBalance (ell : ℕ) (R : ℝ) (hR : 0 < R)
(w a : SpectralHarmonicWeight R) (u : SpectralHarmonicPair ell R) (c ζ : ℂ)
(B : ℂ × ℂ →L[ℂ] ℂ × ℂ) (f : 𝓢(ℝ,ℂ))
(he : spectralHarmonicPairComplexForm ell R w u (spectralSecondTest ell R f)=
inner ℂ (spectralLowerOrderOperator ell R hR
(spectralRadialWeightMultiplier R w) (spectralRadialWeightMultiplier R a) c ζ B
(spectralHarmonicObservation ell R hR u)) (spectralSecondTest ell R f)) :
(∫ r, star (deriv f r)*(w.density r • spectralHarmonicDerivative ell R u.snd r)
∂radialPressureMeasure R) +
(((ell : ℝ)*(ell+10) : ℝ) : ℂ)*
(∫ r, star (f r)*(w.density r • spectralHarmonicValue ell R u.snd r)
∂spectralAngularMeasure R) +
(c-ζ)*(∫ r, star (f r)*(w.density r • spectralHarmonicValue ell R u.fst r)
∂radialPressureMeasure R) +
(∫ r, star (deriv f r)*(a.density r • spectralHarmonicValue ell R u.fst r)
∂radialPressureMeasure R)=star (f R)*(B (spectralHarmonicPairTraces ell R hR u)).2 := by
have h := spectralSecondTest_equation ell R hR w u
(spectralRadialWeightMultiplier R w) (spectralRadialWeightMultiplier R a)
c ζ B (spectralHarmonicObservation ell R hR u) f he
rw [spectralHarmonicScalar_pairing] at h
have hv (v : SpectralRadialL2 R) :
inner ℂ (spectralHarmonicValue ell R (spectralHarmonicSmoothEmbedding ell R f))
(spectralRadialWeightMultiplier R w v)=
∫ r, star (f r)*(w.density r • v r) ∂radialPressureMeasure R :=
spectralL2ComplexMultiplier_pairing_test _ _ _ _ _ _ _ _
(spectralHarmonicValue_smooth_ae ell R f)
have hd (v : SpectralRadialL2 R) :
inner ℂ (spectralHarmonicDerivative ell R (spectralHarmonicSmoothEmbedding ell R f))
(spectralRadialWeightMultiplier R a v)=
∫ r, star (deriv f r)*(a.density r • v r) ∂radialPressureMeasure R :=
spectralL2ComplexMultiplier_pairing_test _ _ _ _ _ _ _ _
(spectralHarmonicDerivative_smooth_ae ell R f)
rw [hv,hv,hd] at h
change _ =
(∫ r, star (f r)*(w.density r • spectralHarmonicValue ell R u.snd r)
∂radialPressureMeasure R)-
(c-ζ)*(∫ r, star (f r)*(w.density r • spectralHarmonicValue ell R u.fst r)
∂radialPressureMeasure R)-
(∫ r, star (deriv f r)*(a.density r • spectralHarmonicValue ell R u.fst r)
∂radialPressureMeasure R)+
star (f R)*(B (spectralHarmonicPairTraces ell R hR u)).2 at h
linear_combination h
theorem spectralFirstTest_boundaryBalance (ell : ℕ) (R : ℝ) (hR : 0 < R)
(w a : SpectralHarmonicWeight R) (u : SpectralHarmonicPair ell R) (c ζ : ℂ)
(B : ℂ × ℂ →L[ℂ] ℂ × ℂ) (f : 𝓢(ℝ,ℂ))
(he : spectralHarmonicPairComplexForm ell R w u (spectralFirstTest ell R f)=
inner ℂ (spectralLowerOrderOperator ell R hR
(spectralRadialWeightMultiplier R w) (spectralRadialWeightMultiplier R a) c ζ B
(spectralHarmonicObservation ell R hR u)) (spectralFirstTest ell R f)) :
(∫ r, star (deriv f r)*(w.density r • spectralHarmonicDerivative ell R u.fst r)
∂radialPressureMeasure R) +
(((ell : ℝ)*(ell+10) : ℝ) : ℂ)*
(∫ r, star (f r)*(w.density r • spectralHarmonicValue ell R u.fst r)
∂spectralAngularMeasure R) -
(c-ζ)*(∫ r, star (f r)*(w.density r • spectralHarmonicValue ell R u.snd r)
∂radialPressureMeasure R) -
(∫ r, star (deriv f r)*(a.density r • spectralHarmonicValue ell R u.snd r)
∂radialPressureMeasure R)=star (f R)*(B (spectralHarmonicPairTraces ell R hR u)).1 := by
have h := spectralFirstTest_equation ell R hR w u
(spectralRadialWeightMultiplier R w) (spectralRadialWeightMultiplier R a)
c ζ B (spectralHarmonicObservation ell R hR u) f he
rw [spectralHarmonicScalar_pairing] at h
have hv (v : SpectralRadialL2 R) :
inner ℂ (spectralHarmonicValue ell R (spectralHarmonicSmoothEmbedding ell R f))
(spectralRadialWeightMultiplier R w v)=
∫ r, star (f r)*(w.density r • v r) ∂radialPressureMeasure R :=
spectralL2ComplexMultiplier_pairing_test _ _ _ _ _ _ _ _
(spectralHarmonicValue_smooth_ae ell R f)
have hd (v : SpectralRadialL2 R) :
inner ℂ (spectralHarmonicDerivative ell R (spectralHarmonicSmoothEmbedding ell R f))
(spectralRadialWeightMultiplier R a v)=
∫ r, star (deriv f r)*(a.density r • v r) ∂radialPressureMeasure R :=
spectralL2ComplexMultiplier_pairing_test _ _ _ _ _ _ _ _
(spectralHarmonicDerivative_smooth_ae ell R f)
rw [hv,hv,hd] at h
change _ =
(∫ r, star (f r)*(w.density r • spectralHarmonicValue ell R u.fst r)
∂radialPressureMeasure R)+
(c-ζ)*(∫ r, star (f r)*(w.density r • spectralHarmonicValue ell R u.snd r)
∂radialPressureMeasure R)+
(∫ r, star (deriv f r)*(a.density r • spectralHarmonicValue ell R u.snd r)
∂radialPressureMeasure R)+
star (f R)*(B (spectralHarmonicPairTraces ell R hR u)).1 at h
linear_combination h
end DefocusingNLS
end OAI
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