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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 | import OAI.MathematicalPhysics.DefocusingNLS.Profile.RadialTerminalUniqueness
import Mathlib.Topology.UniformSpace.UniformConvergence
namespace OAI
/-! Stable backward propagation of spectral columns across a fixed annulus. -/
open Set Filter Topology
namespace DefocusingNLS
local notation "E₄" => (ℂ × ℂ) × (ℂ × ℂ)
theorem spectral_backward_difference_bound (L R K ε : ℝ) (hK : 0 < K) (hε : 0 ≤ ε)
(f g f' g' : ℝ → E₄) (hf : ContinuousOn f (Icc L R)) (hg : ContinuousOn g (Icc L R))
(hfd : ∀ t ∈ Icc L R, HasDerivAt f (f' t) t)
(hgd : ∀ t ∈ Icc L R, HasDerivAt g (g' t) t)
(hb : ∀ t ∈ Icc L R, ‖f' t-g' t‖ ≤ K*‖f t-g t‖+ε)
(r : ℝ) (hr : r ∈ Icc L R) :
‖f r-g r‖ ≤ (‖f R-g R‖+ε/K)*Real.exp (K*(R-L)) := by
let V := fun t => f (R-t)-g (R-t)
let D := fun t => -(f' (R-t)-g' (R-t))
have hm (t : ℝ) (ht : t ∈ Icc 0 (R-L)) : R-t ∈ Icc L R := by
constructor <;> linarith [ht.1,ht.2]
have hV : ContinuousOn V (Icc 0 (R-L)) := (hf.sub hg).comp
(continuous_const.sub continuous_id).continuousOn hm
have hd : ∀ t ∈ Ico 0 (R-L), HasDerivWithinAt V (D t) (Ici t) t := by
intro t ht
have h := ((hfd (R-t) (hm t ⟨ht.1,ht.2.le⟩)).sub
(hgd (R-t) (hm t ⟨ht.1,ht.2.le⟩))).scomp t ((hasDerivAt_id t).const_sub R)
simpa only [V,D,Function.comp_def,neg_one_smul] using! h.hasDerivWithinAt
have hbound : ∀ t ∈ Ico 0 (R-L), ‖D t‖ ≤ K*‖V t‖+ε := by
intro t ht
simpa only [D,V,norm_neg] using hb (R-t) (hm t ⟨ht.1,ht.2.le⟩)
have h := norm_le_gronwallBound_of_norm_deriv_right_le hV hd
(le_refl ‖V 0‖) hbound (R-r) ⟨by linarith [hr.2],by linarith [hr.1]⟩
simp only [V,sub_zero,sub_sub_cancel,gronwallBound_of_K_ne_0 hK.ne'] at h
calc
‖f r-g r‖ ≤ ‖f R-g R‖*Real.exp (K*(R-r)) +
ε/K*(Real.exp (K*(R-r))-1) := h
_ ≤ (‖f R-g R‖+ε/K)*Real.exp (K*(R-r)) := by
have := div_nonneg hε hK.le
nlinarith
_ ≤ _ := mul_le_mul_of_nonneg_left
(Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left (by linarith [hr.1]) hK.le))
(add_nonneg (norm_nonneg _) (div_nonneg hε hK.le))
theorem spectral_backward_uniform_limit (L R K : ℝ) (hK : 0 < K)
(f f' : ℕ → ℝ → E₄) (g g' : ℝ → E₄) (ε : ℕ → ℝ)
(hε : Tendsto ε atTop (𝓝 0))
(hend : Tendsto (fun n => f n R) atTop (𝓝 (g R)))
(hc : ∀ᶠ n in atTop, ContinuousOn (f n) (Icc L R)) (hgc : ContinuousOn g (Icc L R))
(hfd : ∀ᶠ n in atTop, ∀ t ∈ Icc L R, HasDerivAt (f n) (f' n t) t)
(hgd : ∀ t, t ∈ Icc L R → HasDerivAt g (g' t) t)
(hb : ∀ᶠ n in atTop, 0 ≤ ε n ∧
∀ t ∈ Icc L R, ‖f' n t-g' t‖ ≤ K*‖f n t-g t‖+ε n) :
TendstoUniformlyOn f g atTop (Icc L R) := by
have he : Tendsto (fun n => ‖f n R-g R‖) atTop (𝓝 0) :=
tendsto_iff_norm_sub_tendsto_zero.mp hend
have hrate : Tendsto (fun n => (‖f n R-g R‖+ε n/K)*Real.exp (K*(R-L)))
atTop (𝓝 0) := by
simpa only [zero_div,add_zero,zero_mul] using
(he.add (hε.div_const K)).mul_const (Real.exp (K*(R-L)))
rw [Metric.tendstoUniformlyOn_iff]
intro δ hδ
filter_upwards [hb,hrate.eventually (gt_mem_nhds hδ),hc,hfd] with n hn hr hcn hdn t ht
rw [dist_comm,dist_eq_norm]
exact (spectral_backward_difference_bound L R K (ε n) hK hn.1 (f n) g (f' n) g'
hcn hgc hdn hgd hn.2 t ht).trans_lt hr
end DefocusingNLS
end OAI
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