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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 94 95 96 97 98 99 100 101 102 | 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
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