openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Profile /RadialAngularTransport.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Profile.RadialAngularDensity | |
| import OAI.MathematicalPhysics.DefocusingNLS.Profile.RadialScalarBoundary | |
| namespace OAI | |
| /-! Angular potential transport, written with the regular density r^9 |Q|². -/ | |
| open Set | |
| open scoped ContDiff | |
| namespace DefocusingNLS | |
| open ProfileCertificate | |
| theorem radialAngular_transport (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) | |
| (f : ℝ → ℝ) (hf : ContDiff ℝ 1 f) : | |
| 2*(∫ r in (0 : ℝ)..R, radialAngularDensity n z r*radialMatchedVelocity n z r*f r*deriv f r)= | |
| radialAngularDensity n z R*radialMatchedVelocity n z R*(f R)^2- | |
| (6-2*radialShootingA n)*(∫ r in (0 : ℝ)..R, radialAngularDensity n z r*(f r)^2)+ | |
| 2*(∫ r in (0 : ℝ)..R, | |
| radialMatchedVelocityRatio n z r*radialAngularDensity n z r*(f r)^2) := by | |
| let A := fun r => radialAngularDensity n z r*(f r)^2 | |
| let B := fun r => radialAngularDensity n z r*radialMatchedVelocity n z r*f r*deriv f r | |
| let C := fun r => radialMatchedVelocityRatio n z r*radialAngularDensity n z r*(f r)^2 | |
| have hK := radialAngularDensity_continuous n z hX hz | |
| have hW := radialMatchedVelocity_continuousOn n z hX hz R | |
| have hv := radialMatchedVelocityRatio_continuousOn n z hX hz R | |
| have hA : Continuous A := hK.mul (hf.continuous.pow 2) | |
| have hB : ContinuousOn B (Icc 0 R) := | |
| ((hK.continuousOn.mul hW).mul hf.continuous.continuousOn).mul hf.continuous_deriv_one.continuousOn | |
| have hC : ContinuousOn C (Icc 0 R) := | |
| (hv.mul hK.continuousOn).mul (hf.continuous.pow 2).continuousOn | |
| have hAi := hA.intervalIntegrable (μ := MeasureTheory.volume) 0 R | |
| have hBi := hB.intervalIntegrable_of_Icc (μ := MeasureTheory.volume) hR | |
| have hCi := hC.intervalIntegrable_of_Icc (μ := MeasureTheory.volume) hR | |
| have hder (r : ℝ) (hr : r ∈ Ioo 0 R) : | |
| HasDerivAt (fun t => radialAngularDensity n z t*radialMatchedVelocity n z t*(f t)^2) | |
| ((6-2*radialShootingA n)*A r-2*C r+2*B r) r := by | |
| have h := (radialAngularFlux_hasDerivAt n z hX hz r hr.1).mul | |
| (((hf.differentiable (by norm_num) r).hasDerivAt).pow 2) | |
| apply h.congr_deriv | |
| dsimp [A,B,C] | |
| norm_num only [Nat.cast_ofNat,Nat.reduceSub,pow_one] | |
| ring | |
| have he := intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le hR | |
| ((hK.continuousOn.mul hW).mul (hf.continuous.pow 2).continuousOn) hder | |
| (((hAi.const_mul (6-2*radialShootingA n)).sub (hCi.const_mul 2)).add (hBi.const_mul 2)) | |
| simp only [Pi.mul_apply,Pi.pow_apply,radialAngularDensity,zero_pow (by norm_num : (9 : ℕ)≠0),zero_mul,sub_zero] at he | |
| rw [intervalIntegral.integral_add ((hAi.const_mul _).sub (hCi.const_mul 2)) (hBi.const_mul 2), | |
| intervalIntegral.integral_sub (hAi.const_mul _) (hCi.const_mul 2), | |
| intervalIntegral.integral_const_mul,intervalIntegral.integral_const_mul, | |
| intervalIntegral.integral_const_mul] at he | |
| change 2*(∫ r in (0 : ℝ)..R, B r)= | |
| radialAngularDensity n z R*radialMatchedVelocity n z R*(f R)^2- | |
| (6-2*radialShootingA n)*(∫ r in (0 : ℝ)..R, A r)+ | |
| 2*(∫ r in (0 : ℝ)..R, C r) | |
| dsimp only [radialAngularDensity] | |
| linarith | |
| end DefocusingNLS | |
| end OAI | |