openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Profile /RadialAngularEstimates.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Profile.RadialAngularForms | |
| import OAI.MathematicalPhysics.DefocusingNLS.Profile.RadialOutgoingBoundary | |
| namespace OAI | |
| /-! Positive angular energy, radial monotonicity, and vanishing outgoing angular trace. -/ | |
| open Set Filter Asymptotics | |
| open scoped ContDiff | |
| namespace DefocusingNLS | |
| open ProfileCertificate | |
| theorem radialAngularForm_mono (n : ℕ) (z : ProfileMatchingBall) | |
| (hX : HasRadialExterior (radialShootingNu (n+radialInnerShootingThreshold) z) | |
| (n+radialInnerShootingThreshold) (radialShootingM z) (Real.log innerBoundaryRadius)) | |
| (hz : radialMatchingMap n z=0) (f : ℝ → ℝ) (hf : Continuous f) : | |
| MonotoneOn (fun R => radialAngularForm n z R f f) (Ici 0) := by | |
| have hK := (radialAngularDensity_continuous n z hX hz).mul (hf.pow 2) | |
| intro R hR S _hS hRS | |
| have hpos : 0 ≤ ∫ r in R..S, radialAngularDensity n z r*(f r)^2 := by | |
| apply intervalIntegral.integral_nonneg hRS | |
| intro r hr | |
| have hr0 := hR.trans hr.1 | |
| unfold radialAngularDensity | |
| positivity | |
| have he := intervalIntegral.integral_add_adjacent_intervals | |
| (hK.intervalIntegrable (μ := MeasureTheory.volume) 0 R) | |
| (hK.intervalIntegrable (μ := MeasureTheory.volume) R S) | |
| simp only [Pi.mul_apply,Pi.pow_apply] at he | |
| change radialAngularForm n z R f f ≤ radialAngularForm n z S f f | |
| rw [radialAngularForm_diag,radialAngularForm_diag] | |
| linarith | |
| theorem radialAngularForm_coercive (n : ℕ) (z : ProfileMatchingBall) | |
| (R σ : ℝ) (hR : 0 ≤ R) (hσ : 4 ≤ σ) (f : ℝ → ℝ) | |
| (hv : ∀ r ∈ Icc 0 R, 0 ≤ radialMatchedVelocityRatio n z r) : | |
| (3/4 : ℝ)*radialAngularForm n z R f f ≤ | |
| σ*radialAngularForm n z R f f+radialAngularVirial n z R f := by | |
| have hA := radialAngularForm_nonneg n z R hR f | |
| have ha := (radialShootingA_bounds n (profileMatchingParameter z)).1 | |
| have hV : 0 ≤ ∫ r in (0 : ℝ)..R, | |
| radialMatchedVelocityRatio n z r*radialAngularDensity n z r*(f r)^2 := by | |
| apply intervalIntegral.integral_nonneg hR | |
| intro r hr | |
| have hr0 := hr.1 | |
| have hvr := hv r hr | |
| unfold radialAngularDensity | |
| positivity | |
| have hcoef : 1 ≤ σ-(6-2*radialShootingA n)/2 := by linarith | |
| have hmul := mul_le_mul_of_nonneg_right hcoef hA | |
| unfold radialAngularVirial | |
| nlinarith | |
| theorem radialMatched_angular_density_bigO : | |
| ∀ᶠ n in atTop, ∀ z : ProfileMatchingBall, | |
| radialAngularDensity n z =O[atTop] (fun r : ℝ => r^(9 : ℝ)) := by | |
| obtain ⟨d,_hd,hW⟩ := radialMatched_uniform_weight_comparison | |
| filter_upwards [hW] with n hn z | |
| refine IsBigO.of_bound 144 ?_ | |
| filter_upwards [eventually_gt_atTop (0 : ℝ)] with r hr | |
| have hμ := (hn z r hr.le).2 | |
| have ha := (radialShootingA_bounds n (profileMatchingParameter z)).1 | |
| have hρ : radialHardyWeight (radialShootingA n) r ≤ 1 := by | |
| unfold radialHardyWeight | |
| exact Real.rpow_le_one_of_one_le_of_nonpos (by nlinarith [sq_nonneg r]) (by linarith) | |
| have hp : ‖radialMatchedProfile n z r‖^2 ≤ 144 := by | |
| change ‖radialMatchedProfile n z r‖^2 ≤ 144*radialHardyWeight (radialShootingA n) r at hμ | |
| linarith | |
| have hK : 0 ≤ radialAngularDensity n z r := by | |
| unfold radialAngularDensity | |
| positivity | |
| simp only [Real.norm_eq_abs,Real.rpow_ofNat,abs_of_nonneg hK, | |
| abs_of_nonneg (pow_nonneg hr.le 9)] | |
| unfold radialAngularDensity | |
| nlinarith [mul_le_mul_of_nonneg_left hp (pow_nonneg hr.le 9)] | |
| theorem radialAngularBoundary_tendsto (n : ℕ) (z : ProfileMatchingBall) | |
| (hX : HasRadialExterior (radialShootingNu (n+radialInnerShootingThreshold) z) | |
| (n+radialInnerShootingThreshold) (radialShootingM z) (Real.log innerBoundaryRadius)) | |
| (hz : radialMatchingMap n z=0) (f : ℝ → ℝ) (hf : HasRadialEnergyDecay f) | |
| (hK : radialAngularDensity n z =O[atTop] (fun r : ℝ => r^(9 : ℝ))) : | |
| Tendsto (fun R => radialAngularBoundary n z R f) atTop (nhds 0) := by | |
| have hw : radialMatchedVelocity n z =O[atTop] (fun r : ℝ => r^(1 : ℝ)) := by | |
| refine IsBigO.of_bound 1 ?_ | |
| filter_upwards [radialMatchedVelocity_eventually_bound n z hX hz, | |
| eventually_gt_atTop (0 : ℝ)] with r hwr hr | |
| simpa only [Real.norm_eq_abs,Real.rpow_one,abs_of_pos hr,one_mul] using hwr | |
| have hKw := IsBigO.mul_atTop_rpow_of_isBigO_rpow 9 1 10 hK hw (by norm_num) | |
| have hff := IsBigO.mul_atTop_rpow_of_isBigO_rpow (-8) (-8) (-16) hf.1 hf.1 (by norm_num) | |
| have hT := IsBigO.mul_atTop_rpow_of_isBigO_rpow 10 (-16) (-4) hKw hff (by norm_num) | |
| have ht := (hT.trans_tendsto (tendsto_rpow_neg_atTop (by norm_num : (0 : ℝ)<4))).const_mul | |
| (1/2 : ℝ) | |
| simp only [mul_zero] at ht | |
| apply ht.congr' | |
| filter_upwards [] with R | |
| simp only [Pi.mul_apply,radialAngularBoundary] | |
| ring | |
| end DefocusingNLS | |
| end OAI | |