openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Spectrum /SpectralBoundaryFlux.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralBoundaryBalance | |
| import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralFirstFlux | |
| namespace OAI | |
| /-! Ordinary radial integration retains the boundary trace in the weak flux identity. -/ | |
| open MeasureTheory | |
| open scoped SchwartzMap | |
| namespace DefocusingNLS | |
| theorem spectralSecondBoundaryBalance_to_flux (ell : ℕ) (R : ℝ) (hR : 0 < R) | |
| (w a : SpectralHarmonicWeight R) (u : SpectralHarmonicPair ell R) (c ζ : ℂ) | |
| (f : 𝓢(ℝ,ℂ)) (β : ℂ) | |
| (hbal : | |
| (∫ 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)=β) : | |
| (∫ r in (0 : ℝ)..R, star (deriv f r)*spectralSecondFlux ell R w a u r)= | |
| β-(∫ r in (0 : ℝ)..R, star (f r)*spectralSecondSource ell R w u c ζ r) := by | |
| let J₁ := fun r => star (deriv f r)*(w.density r • spectralHarmonicDerivative ell R u.snd r) | |
| let J₂ := fun r => star (deriv f r)*(a.density r • spectralHarmonicValue ell R u.fst r) | |
| let J₃ := fun r => star (f r)*(w.density r • spectralHarmonicValue ell R u.snd r) | |
| let J₄ := fun r => star (f r)*(w.density r • spectralHarmonicValue ell R u.fst r) | |
| let η : ℂ := (((ell : ℝ)*(ell+10) : ℝ) : ℂ) | |
| have hd (v : SpectralRadialL2 R) (b : SpectralHarmonicWeight R) : | |
| Integrable (fun r => star (deriv f r)*(b.density r • v r)) (radialPressureMeasure R) := | |
| spectralWeightedTest_integrable _ _ b.radial_measurable b.bound b.radial_bound v | |
| (spectralHarmonicDerivative ell R (spectralHarmonicSmoothEmbedding ell R f)) | |
| (deriv f) (spectralHarmonicDerivative_smooth_ae ell R f) | |
| have hv (v : SpectralRadialL2 R) : | |
| Integrable (fun r => star (f r)*(w.density r • v r)) (radialPressureMeasure R) := | |
| spectralWeightedTest_integrable _ _ w.radial_measurable w.bound w.radial_bound v | |
| (spectralHarmonicValue ell R (spectralHarmonicSmoothEmbedding ell R f)) | |
| f (spectralHarmonicValue_smooth_ae ell R f) | |
| have hi₁ : IntervalIntegrable (fun r => (r : ℂ)^11*J₁ r) volume 0 R := | |
| (intervalIntegrable_iff_integrableOn_Icc_of_le hR.le).mpr | |
| (spectralRadial_integrableOn R J₁ (hd _ w)) | |
| have hi₂ : IntervalIntegrable (fun r => (r : ℂ)^11*J₂ r) volume 0 R := | |
| (intervalIntegrable_iff_integrableOn_Icc_of_le hR.le).mpr | |
| (spectralRadial_integrableOn R J₂ (hd _ a)) | |
| have hi₄ : IntervalIntegrable (fun r => (r : ℂ)^11*J₄ r) volume 0 R := | |
| (intervalIntegrable_iff_integrableOn_Icc_of_le hR.le).mpr | |
| (spectralRadial_integrableOn R J₄ (hv _)) | |
| have hi₃ : IntervalIntegrable (fun r => η*((r : ℂ)^9*J₃ r)) volume 0 R := by | |
| have hi := (intervalIntegrable_iff_integrableOn_Icc_of_le hR.le).mpr | |
| (spectralAngular_integrableOn R _ (spectralAngularTest_integrable ell R w u.snd f)) | |
| convert hi using 1 | |
| funext r | |
| dsimp only [η,J₃] | |
| ring | |
| have hflux : (∫ r in (0 : ℝ)..R, star (deriv f r)*spectralSecondFlux ell R w a u r)= | |
| (∫ r in (0 : ℝ)..R, (r : ℂ)^11*J₁ r)+ | |
| ∫ r in (0 : ℝ)..R, (r : ℂ)^11*J₂ r := by | |
| rw [← intervalIntegral.integral_add hi₁ hi₂] | |
| apply intervalIntegral.integral_congr | |
| intro r _ | |
| dsimp only [spectralSecondFlux,J₁,J₂] | |
| ring | |
| have hsource : (∫ r in (0 : ℝ)..R, star (f r)*spectralSecondSource ell R w u c ζ r)= | |
| η*(∫ r in (0 : ℝ)..R, (r : ℂ)^9*J₃ r)+ | |
| (c-ζ)*(∫ r in (0 : ℝ)..R, (r : ℂ)^11*J₄ r) := by | |
| rw [← intervalIntegral.integral_const_mul,← intervalIntegral.integral_const_mul, | |
| ← intervalIntegral.integral_add hi₃ (hi₄.const_mul (c-ζ))] | |
| apply intervalIntegral.integral_congr | |
| intro r _ | |
| dsimp only [spectralSecondSource,J₃,J₄,η] | |
| ring | |
| rw [hflux,hsource] | |
| change (∫ r, J₁ r ∂radialPressureMeasure R)+ | |
| η*(∫ r, J₃ r ∂spectralAngularMeasure R)+ | |
| (c-ζ)*(∫ r, J₄ r ∂radialPressureMeasure R)+ | |
| (∫ r, J₂ r ∂radialPressureMeasure R)=β at hbal | |
| rw [spectral_radial_complex_integral R hR.le, | |
| spectral_angular_complex_integral R hR.le, | |
| spectral_radial_complex_integral R hR.le, | |
| spectral_radial_complex_integral R hR.le] at hbal | |
| linear_combination hbal | |
| theorem spectralSecond_boundaryFlux (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 in (0 : ℝ)..R, star (deriv f r)*spectralSecondFlux ell R w a u r)= | |
| star (f R)*(B (spectralHarmonicPairTraces ell R hR u)).2- | |
| (∫ r in (0 : ℝ)..R, star (f r)*spectralSecondSource ell R w u c ζ r) := | |
| spectralSecondBoundaryBalance_to_flux ell R hR w a u c ζ f _ | |
| (spectralSecondTest_boundaryBalance ell R hR w a u c ζ B f he) | |
| theorem spectralFirstBoundaryBalance_to_flux (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 in (0 : ℝ)..R, star (deriv f r)* | |
| spectralSecondFlux ell R w (spectralNegWeight a) (spectralSwapPair ell R u) r)= | |
| star (f R)*(B (spectralHarmonicPairTraces ell R hR u)).1- | |
| (∫ r in (0 : ℝ)..R, star (f r)* | |
| spectralSecondSource ell R w (spectralSwapPair ell R u) (-c) (-ζ) r) := by | |
| have hb := spectralFirstTest_boundaryBalance ell R hR w a u c ζ B f he | |
| apply spectralSecondBoundaryBalance_to_flux ell R hR w (spectralNegWeight a) | |
| (spectralSwapPair ell R u) (-c) (-ζ) f | |
| (star (f R)*(B (spectralHarmonicPairTraces ell R hR u)).1) | |
| change (∫ 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 | |
| simp only [neg_smul,mul_neg,integral_neg] | |
| linear_combination hb | |
| end DefocusingNLS | |
| end OAI | |