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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 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.RadialVelocityCalculus
namespace OAI
/-! Regular weighted angular coefficients at the radial origin. -/
open Set
namespace DefocusingNLS
open ProfileCertificate
noncomputable def radialAngularDensity (n : ℕ) (z : ProfileMatchingBall) (r : ℝ) : ℝ :=
r^9*‖radialMatchedProfile n z r‖^2
noncomputable def radialAngularSlope (n : ℕ) (z : ProfileMatchingBall) (r : ℝ) : ℝ :=
9*r^8*‖radialMatchedProfile n z r‖^2+
2*r^9*(star (radialMatchedProfile n z r)*deriv (radialMatchedProfile n z) r).re
noncomputable def radialMatchedVelocityRatio (n : ℕ) (z : ProfileMatchingBall) : ℝ → ℝ :=
radialVelocityRatio (6-2*radialShootingA n) (fun r => ‖radialMatchedProfile n z r‖)
theorem radialAngularDensity_continuous (n : ℕ) (z : ProfileMatchingBall)
(hX : HasRadialExterior (radialShootingNu (n+radialInnerShootingThreshold) z)
(n+radialInnerShootingThreshold) (radialShootingM z) (Real.log innerBoundaryRadius))
(hz : radialMatchingMap n z=0) : Continuous (radialAngularDensity n z) := by
exact (continuous_id.pow 9).mul
((radialMatchedProfile_differentiable n z hX hz).continuous.norm.pow 2)
theorem radialAngularDensity_hasDerivAt (n : ℕ) (z : ProfileMatchingBall)
(hX : HasRadialExterior (radialShootingNu (n+radialInnerShootingThreshold) z)
(n+radialInnerShootingThreshold) (radialShootingM z) (Real.log innerBoundaryRadius))
(hz : radialMatchingMap n z=0) (r : ℝ) :
HasDerivAt (radialAngularDensity n z) (radialAngularSlope n z r) r := by
have h := ((hasDerivAt_id r).pow 9).mul
(radialComplexNormSq_hasDerivAt (radialMatchedProfile n z) r
(radialMatchedProfile_differentiable n z hX hz r))
convert h using 1
· funext t
simp only [radialAngularDensity,Pi.mul_apply,Pi.pow_apply,id_eq,Complex.sq_norm]
· simp only [radialAngularSlope,Pi.pow_apply,id_eq,Nat.cast_ofNat,Nat.reduceSub,mul_one,
Complex.sq_norm]
ring
theorem radialMatchedVelocityRatio_continuousOn (n : ℕ) (z : ProfileMatchingBall)
(hX : HasRadialExterior (radialShootingNu (n+radialInnerShootingThreshold) z)
(n+radialInnerShootingThreshold) (radialShootingM z) (Real.log innerBoundaryRadius))
(hz : radialMatchingMap n z=0) (R : ℝ) :
ContinuousOn (radialMatchedVelocityRatio n z) (Icc 0 R) := by
have hA := (radialMatchedProfile_differentiable n z hX hz).continuous.norm
exact ((continuous_const.mul (continuous_radialAverage _ (hA.pow 2))).continuousOn.div
(hA.pow 2).continuousOn (by
intro r hr
exact pow_ne_zero 2 (norm_ne_zero_iff.mpr
(radialMatchedProfile_ne_zero n z hX r hr.1))))
theorem radialMatchedVelocity_eq_mul_ratio (n : ℕ) (z : ProfileMatchingBall) (r : ℝ) :
radialMatchedVelocity n z r=r*radialMatchedVelocityRatio n z r := rfl
theorem radialAngularFlux_hasDerivAt (n : ℕ) (z : ProfileMatchingBall)
(hX : HasRadialExterior (radialShootingNu (n+radialInnerShootingThreshold) z)
(n+radialInnerShootingThreshold) (radialShootingM z) (Real.log innerBoundaryRadius))
(hz : radialMatchingMap n z=0) (r : ℝ) (hr : 0 < r) :
HasDerivAt (fun t => radialAngularDensity n z t*radialMatchedVelocity n z t)
((6-2*radialShootingA n-2*radialMatchedVelocityRatio n z r)*radialAngularDensity n z r) r := by
have h := (radialAngularDensity_hasDerivAt n z hX hz r).mul
(radialMatchedVelocity_differentiableAt n z hX hz r hr).hasDerivAt
apply h.congr_deriv
apply mul_left_cancel₀ (pow_ne_zero 2 hr.ne')
have hM := radialMassSlope_velocity n z hX hz r hr.le
have hK : r^2*radialAngularDensity n z r=radialMassDensity n z r := by
unfold radialAngularDensity radialMassDensity
ring
have hS : r^2*radialAngularSlope n z r=
radialMassSlope n z r-2*r*radialAngularDensity n z r := by
unfold radialAngularSlope radialMassSlope radialAngularDensity
ring
calc
r^2*(radialAngularSlope n z r*radialMatchedVelocity n z r+
radialAngularDensity n z r*deriv (radialMatchedVelocity n z) r) =
(r^2*radialAngularSlope n z r)*radialMatchedVelocity n z r+
(r^2*radialAngularDensity n z r)*deriv (radialMatchedVelocity n z) r := by ring
_ = (6-2*radialShootingA n)*radialMassDensity n z r-
2*r*radialAngularDensity n z r*radialMatchedVelocity n z r := by
rw [hS,hK]
linear_combination hM
_ = r^2*((6-2*radialShootingA n-2*radialMatchedVelocityRatio n z r)*
radialAngularDensity n z r) := by
rw [← hK,radialMatchedVelocity_eq_mul_ratio]
ring
end DefocusingNLS
end OAI
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