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fac6777 | 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 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.RadialAngularTransport
namespace OAI
/-! The angular part of the radial spectral form and its exact testing identity. -/
open Set
open scoped ContDiff
namespace DefocusingNLS
open ProfileCertificate
noncomputable def radialAngularForm (n : ℕ) (z : ProfileMatchingBall)
(R : ℝ) (f g : ℝ → ℝ) : ℝ :=
∫ r in (0 : ℝ)..R, radialAngularDensity n z r*f r*g r
noncomputable def radialAngularVirial (n : ℕ) (z : ProfileMatchingBall)
(R : ℝ) (f : ℝ → ℝ) : ℝ :=
-((6-2*radialShootingA n)/2)*radialAngularForm n z R f f+
∫ r in (0 : ℝ)..R, radialMatchedVelocityRatio n z r*radialAngularDensity n z r*(f r)^2
noncomputable def radialAngularBoundary (n : ℕ) (z : ProfileMatchingBall)
(R : ℝ) (f : ℝ → ℝ) : ℝ :=
(1/2 : ℝ)*radialAngularDensity n z R*radialMatchedVelocity n z R*(f R)^2
theorem radialAngularForm_diag (n : ℕ) (z : ProfileMatchingBall) (R : ℝ) (f : ℝ → ℝ) :
radialAngularForm n z R f f=∫ r in (0 : ℝ)..R, radialAngularDensity n z r*(f r)^2 := by
unfold radialAngularForm
apply intervalIntegral.integral_congr
intro r _
ring
theorem radialAngularForm_symm (n : ℕ) (z : ProfileMatchingBall) (R : ℝ) (f g : ℝ → ℝ) :
radialAngularForm n z R f g=radialAngularForm n z R g f := by
unfold radialAngularForm
apply intervalIntegral.integral_congr
intro r _
ring
theorem radialAngularForm_nonneg (n : ℕ) (z : ProfileMatchingBall) (R : ℝ) (hR : 0 ≤ R)
(f : ℝ → ℝ) : 0 ≤ radialAngularForm n z R f f := by
rw [radialAngularForm_diag]
apply intervalIntegral.integral_nonneg hR
intro r hr
have hr0 := hr.1
unfold radialAngularDensity
positivity
theorem radialAngularForm_spectralTest (n : ℕ) (z : ProfileMatchingBall)
(hX : HasRadialExterior (radialShootingNu (n+radialInnerShootingThreshold) z)
(n+radialInnerShootingThreshold) (radialShootingM z) (Real.log innerBoundaryRadius))
(hz : radialMatchingMap n z=0) (R s t : ℝ) (hR : 0 ≤ R)
(f g : ℝ → ℝ) (hf : ContDiff ℝ 1 f) (hg : Continuous g) :
(∫ r in (0 : ℝ)..R,
radialSpectralTest n z s t f g r*radialAngularDensity n z r*f r)=
s*radialAngularForm n z R f f+t*radialAngularForm n z R f g+
radialAngularVirial n z R f+radialAngularBoundary n z R f := by
have hK := radialAngularDensity_continuous n z hX hz
have hW := radialMatchedVelocity_continuousOn n z hX hz R
have hA : IntervalIntegrable (fun r => radialAngularDensity n z r*f r*f r)
MeasureTheory.volume 0 R := ((hK.mul hf.continuous).mul hf.continuous).intervalIntegrable 0 R
have hB : IntervalIntegrable (fun r => radialAngularDensity n z r*f r*g r)
MeasureTheory.volume 0 R := ((hK.mul hf.continuous).mul hg).intervalIntegrable 0 R
have hC : IntervalIntegrable (fun r =>
radialAngularDensity n z r*radialMatchedVelocity n z r*f r*deriv f r)
MeasureTheory.volume 0 R :=
(((hK.continuousOn.mul hW).mul hf.continuous.continuousOn).mul
hf.continuous_deriv_one.continuousOn).intervalIntegrable_of_Icc hR
have he : (∫ r in (0 : ℝ)..R,
radialSpectralTest n z s t f g r*radialAngularDensity n z r*f r)=
s*radialAngularForm n z R f f+t*radialAngularForm n z R f g+
∫ r in (0 : ℝ)..R, radialAngularDensity n z r*radialMatchedVelocity n z r*f r*deriv f r := by
calc
_ = ∫ r in (0 : ℝ)..R,
(s*(radialAngularDensity n z r*f r*f r)+t*(radialAngularDensity n z r*f r*g r))+
radialAngularDensity n z r*radialMatchedVelocity n z r*f r*deriv f r := by
apply intervalIntegral.integral_congr
intro r _
unfold radialSpectralTest
ring
_ = _ := by
rw [intervalIntegral.integral_add ((hA.const_mul s).add (hB.const_mul t)) hC,
intervalIntegral.integral_add (hA.const_mul s) (hB.const_mul t),
intervalIntegral.integral_const_mul,intervalIntegral.integral_const_mul]
rfl
have ht := radialAngular_transport n z hX hz R hR f hf
rw [he]
unfold radialAngularVirial radialAngularBoundary
rw [radialAngularForm_diag]
linarith
end DefocusingNLS
end OAI
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