File size: 2,400 Bytes
ef96466 | 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 | import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralIntegralGronwall
namespace OAI
/-! Remove the same-channel action before applying Gronwall. This keeps the
integral of the WKB residual, rather than the length of the whole shell. -/
open Set MeasureTheory
namespace DefocusingNLS
theorem spectral_action_gronwall (a b A : ℝ) (hab : a≤ b) (X g H : ℝ → ℝ)
(hX : ContinuousOn X (Icc a b)) (hg : ContinuousOn g (Icc a b))
(hH : ContinuousOn H (Icc a b)) (hg0 : ∀ r ∈ Icc a b, 0≤ g r)
(hbound : ∀ r ∈ Icc a b,
X r≤ A*Real.exp (H r)+∫ t in a..r, g t*Real.exp (H r-H t)*X t) :
∀ r ∈ Icc a b, X r≤ A*Real.exp (H r+∫ t in a..r, g t) := by
let Y := fun r => Real.exp (-H r)*X r
have hY : ContinuousOn Y (Icc a b) :=
(Real.continuous_exp.comp_continuousOn hH.neg).mul hX
have hprod (r t : ℝ) :
Real.exp (-H r)*(g t*Real.exp (H r-H t)*X t)=g t*Y t := by
have he : Real.exp (-H r)*Real.exp (H r-H t)=Real.exp (-H t) := by
rw [← Real.exp_add]
congr 1
ring
dsimp only [Y]
calc
_ = g t*(Real.exp (-H r)*Real.exp (H r-H t))*X t := by ring
_ = _ := by rw [he]; ring
have hYbound (r : ℝ) (hr : r ∈ Icc a b) : Y r≤ A+∫ t in a..r, g t*Y t := by
have hb := mul_le_mul_of_nonneg_left (hbound r hr) (Real.exp_pos (-H r)).le
have he : Real.exp (-H r)*Real.exp (H r)=1 := by
rw [← Real.exp_add,neg_add_cancel,Real.exp_zero]
change Y r≤ _ at hb
calc
Y r ≤ Real.exp (-H r)*(A*Real.exp (H r)+∫ t in a..r, g t*Real.exp (H r-H t)*X t) := hb
_ = A+∫ t in a..r, g t*Y t := by
rw [mul_add,← mul_assoc, mul_comm (Real.exp (-H r)) A,mul_assoc,he,mul_one,
← intervalIntegral.integral_const_mul]
congr 1
exact intervalIntegral.integral_congr (fun t _ => hprod r t)
intro r hr
have hh := spectral_integral_gronwall a b A hab Y g hY hg hg0 hYbound r hr
have hm := mul_le_mul_of_nonneg_left hh (Real.exp_pos (H r)).le
have he : Real.exp (H r)*Real.exp (-H r)=1 := by
rw [← Real.exp_add,add_neg_cancel,Real.exp_zero]
change Real.exp (H r)*(Real.exp (-H r)*X r)≤ _ at hm
rw [← mul_assoc,he,one_mul] at hm
calc
X r ≤ Real.exp (H r)*(A*Real.exp (∫ t in a..r, g t)) := hm
_ = _ := by rw [← mul_assoc,mul_comm (Real.exp (H r)) A,mul_assoc,← Real.exp_add]
end DefocusingNLS
end OAI
|