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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 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 | import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralPerturbedTransfer
import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralScalarReverseTransfer
namespace OAI
/-! A bounded approximate oscillatory frame controls true solutions from their
terminal data. The bound uses the integrated residual, uniformly over every
subinterval. -/
open Set MeasureTheory
namespace DefocusingNLS
theorem spectral_bounded_frame_terminal_bound
(R E C c J : ℝ) (hC : 0 ≤ C) (hc : 0 < c)
(D U q : ℝ → ℂ × ℂ) (V e : ℝ → ℂ) (W : ℂ) (k : ℝ → ℝ)
(hDc : ContinuousOn D (Icc R E)) (hUc : ContinuousOn U (Icc R E))
(hqc : ContinuousOn q (Icc R E)) (hVc : ContinuousOn V (Icc R E))
(hec : ContinuousOn e (Icc R E)) (hkc : ContinuousOn k (Icc R E))
(hk : ∀ t ∈ Icc R E, 0 < k t) (hW : c ≤ ‖W‖)
(hdet : ∀ t ∈ Icc R E, spectralScalarWronskian (D t) (U t) = W)
(hD : ∀ t ∈ Ioo R E, HasDerivAt D (spectralScalarField (V t) (D t)) t)
(hU : ∀ t ∈ Ioo R E, HasDerivAt U (spectralScalarField (V t) (U t)) t)
(hq : ∀ t ∈ Ioo R E, HasDerivAt q
(spectralScalarField (V t) (q t)+(0,e t*(q t).1)) t)
(hDb : ∀ t ∈ Icc R E, spectralShellNorm (k t) (D t) ≤ C)
(hUb : ∀ t ∈ Icc R E, spectralShellNorm (k t) (U t) ≤ C)
(hJ : (∫ t in R..E, ‖e t‖/(k t)^2) ≤ J) :
∀ r ∈ Icc R E, spectralShellNorm (k r) (q r) ≤
(2*(2*C^2/c)*Real.exp ((2*C^2/c)*J))*spectralShellNorm (k E) (q E) := by
let A := 2*C^2/c
have hA : 0 ≤ A := by dsimp only [A]; positivity
intro r hr
have hsub : Icc r E ⊆ Icc R E := Icc_subset_Icc hr.1 le_rfl
have hRE : R ≤ E := hr.1.trans hr.2
have hE : E ∈ Icc R E := ⟨hRE,le_rfl⟩
have hg : ContinuousOn (fun t => ‖e t‖/(k t)^2) (Icc R E) :=
hec.norm.div (hkc.pow 2) (fun t ht => pow_ne_zero _ (hk t ht).ne')
have hi : (∫ t in r..E, ‖e t‖/(k t)^2) ≤ J := by
apply le_trans _ hJ
exact intervalIntegral.integral_mono_interval hr.1 hr.2 le_rfl
(Filter.Eventually.of_forall (fun t => by positivity))
(ContinuousOn.intervalIntegrable_of_Icc hRE hg)
have hforward (u : ℝ → ℂ × ℂ) (huc : ContinuousOn u (Icc r E))
(hu : ∀ t ∈ Ioo r E, HasDerivAt u (spectralScalarField (V t-e t) (u t)) t) :
spectralShellNorm (k E) (u E) ≤
(A*Real.exp (A*J))*spectralShellNorm (k r) (u r) := by
have hu' (t : ℝ) (ht : t ∈ Ioo r E) : HasDerivAt u
(spectralScalarField (V t) (u t)+(0,e t*(u t).1)) t := by
apply (hu t ht).congr_deriv
apply Prod.ext
· simp only [spectralScalarField,Prod.fst_add,add_zero]
· dsimp only [spectralScalarField,Prod.snd_add]
ring
have ht := spectral_perturbed_transfer r E C c hr.2 hC hc D U u V e W k
(fun _ => 0) (hDc.mono hsub) (hUc.mono hsub) huc (hec.mono hsub)
(hkc.mono hsub) continuousOn_const (fun t ht => hk t (hsub ht))
(fun _ _ _ _ _ => le_rfl) hW (fun t ht => hdet t (hsub ht))
(fun t ht => hD t ⟨hr.1.trans_lt ht.1,ht.2⟩)
(fun t ht => hU t ⟨hr.1.trans_lt ht.1,ht.2⟩) hu'
(fun t ht => by simpa only [Real.exp_zero,mul_one] using hDb t (hsub ht))
(fun t ht => by simpa only [neg_zero,Real.exp_zero,mul_one] using hUb t (hsub ht))
E ⟨hr.2,le_rfl⟩
have hI : (∫ t in r..E, (2*C^2/c)*‖e t‖/(k t)^2) ≤ A*J := by
simp only [mul_div_assoc,intervalIntegral.integral_const_mul]
simpa only [A,mul_div_assoc] using mul_le_mul_of_nonneg_left hi hA
calc
_ ≤ A*spectralShellNorm (k r) (u r)*
Real.exp (∫ t in r..E, (2*C^2/c)*‖e t‖/(k t)^2) := by
simpa only [A,zero_sub,neg_zero,add_zero,zero_add] using ht
_ ≤ A*spectralShellNorm (k r) (u r)*Real.exp (A*J) :=
mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr hI)
(mul_nonneg hA (spectralShellNorm_nonneg (k r) (hk r hr).le _))
_ = _ := by ring
have hq' (t : ℝ) (ht : t ∈ Ioo r E) :
HasDerivAt q (spectralScalarField (V t-e t) (q t)) t := by
apply (hq t ⟨hr.1.trans_lt ht.1,ht.2⟩).congr_deriv
apply Prod.ext
· simp only [spectralScalarField,Prod.fst_add,add_zero]
· dsimp only [spectralScalarField,Prod.snd_add]
ring
have hrev := spectralScalar_reverse_transfer r E (k r) (k E) (A*Real.exp (A*J))
hr.2 (hk r hr) (hk E hE) (fun t => V t-e t) ((hVc.sub hec).mono hsub)
hforward q (hqc.mono hsub) hq'
simpa only [A,mul_assoc] using hrev
end DefocusingNLS
end OAI
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