openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Profile /RadialAngularDensity.lean
| 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 | |