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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 | import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralCanonicalColumnLimit
import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralFreeSlowIdentification
namespace OAI
/-! The canonical first column converges to the explicit free angular H column. -/
open Filter Topology Set Polynomial
namespace DefocusingNLS
local notation "E₄" => (ℂ × ℂ) × (ℂ × ℂ)
theorem canonical_circular_first_H_limit
(ν m νp νm : ℕ → ℂ) (ν₀ m₀ νm₀ q : ℂ) (ell : ℕ)
(hν : Tendsto ν atTop (𝓝 ν₀)) (hm : Tendsto m atTop (𝓝 m₀))
(hp : Tendsto νp atTop (𝓝 ((ell : ℂ)-2*q)))
(hn : Tendsto νm atTop (𝓝 νm₀)) (hq : -1 < q.re)
(δ L : ℝ) (hδ : 0 < δ) (hsmall : ‖m₀‖+2*δ < 1)
(hX : ∀ᶠ n in atTop, HasRadialExterior (ν n) n (m n) L)
(W : ℕ → ℝ → E₄)
(hW : ∀ᶠ n in atTop, ∀ t, 0 ≤ t → HasDerivAt (W n)
(circularLeadingField t (W n t)+circularBoundedField (νp n) (νm n)
((ell*(ell+10) : ℕ) : ℂ) n (radialExteriorCanonical (ν n) n (m n) L t).1
(W n t)) t)
(heW : ∀ᶠ n in atTop, ∀ J : ℕ, ∃ k : ℕ, J ≤ k ∧ ∃ u : CircularTailSpace,
∀ t, 0 ≤ t → W n t=circularPolynomialJet
(spectralOutgoingPolynomial (νp n) (νm n) ((ell*(ell+10) : ℕ) : ℂ) n
(radialExteriorExpansion (ν n) n (m n) k) (1,0) k) t+
circularUnweight (2*(k : ℝ)) u t) :
∃ T : ℝ, 0 ≤ T ∧ L ≤ T ∧
TendstoUniformlyOn W (spectralFreeFirstColumn ell q) atTop (Ici T) := by
let η : ℂ := ((ell*(ell+10) : ℕ) : ℂ)
let νp₀ : ℂ := (ell : ℂ)-2*q
obtain ⟨j,hj⟩ := exists_nat_gt (max (radialExteriorMatrixBound ν₀)
(circularFieldBound νp₀ νm₀ η 1 0))
have hj₁ : radialExteriorMatrixBound ν₀ < 2*((j+1 : ℕ) : ℝ) := by
have hh := (le_max_left (radialExteriorMatrixBound ν₀)
(circularFieldBound νp₀ νm₀ η 1 0)).trans_lt hj
push_cast
linarith [Nat.cast_nonneg (α := ℝ) j]
have hj₂ : circularFieldBound νp₀ νm₀ η 1 0 < 2*((j+1 : ℕ) : ℝ) := by
have hh := (le_max_right (radialExteriorMatrixBound ν₀)
(circularFieldBound νp₀ νm₀ η 1 0)).trans_lt hj
push_cast
linarith [Nat.cast_nonneg (α := ℝ) j]
obtain ⟨T,ρ,hT,hLT,_hρ,_hρ1,v,w,_hvw,hlim,_hlimn,_hlim₀,hactual,hfree⟩ :=
exists_canonical_circular_solution_limit ν m νp νm ν₀ m₀ νp₀ νm₀ η
hν hm hp hn δ L hδ hsmall hX (j+1) hj₁ hj₂ (1,0)
let U := fun n => spectralOutgoingPolynomial (νp n) (νm n) η n
(radialExteriorExpansion (ν n) n (m n) (j+1)) (1,0) (j+1)
let Y := fun n t => circularPolynomialJet (U n) t+
circularUnweight (2*((j+1 : ℕ) : ℝ)) (v n) t
let Y₀ := fun t => circularPolynomialJet
(spectralOutgoingPolynomial νp₀ νm₀ η 1 0 (1,0) (j+1)) t+
circularUnweight (2*((j+1 : ℕ) : ℝ)) w t
have heq : ∀ᶠ n in atTop, ∀ t, T ≤ t → Y n t=W n t := by
filter_upwards [hactual,hW,heW,eventually_ge_atTop 1] with n han hwn hen hnn
exact circular_expansion_matches_allOrders (ν n) (νp n) (νm n) η (m n) n hnn
(1,0) (j+1) (v n) ρ T hT
(fun t => (radialExteriorCanonical (ν n) n (m n) L t).1) (Y n) (W n)
(fun t ht => (han.2 t ht).1) (fun t ht => (han.2 t ht).2)
(fun t ht => hwn t (hT.trans ht)) (fun _ _ => rfl) hen han.1
let S := max T (max 0 (Real.log 4/2))
have hH : ∀ t, S ≤ t → Y₀ t=spectralFreeFirstColumn ell q t :=
spectralFreeFirstColumn_eq_correction ell q νm₀ hq j w T hj₂ hfree
refine ⟨S,(le_max_left 0 _).trans (le_max_right T _),
hLT.trans (le_max_left T _),?_⟩
rw [Metric.tendstoUniformlyOn_iff]
intro ε hε
filter_upwards [(Metric.tendstoUniformlyOn_iff.mp hlim) ε hε,heq] with n hn he t ht
have htT : T ≤ t := (le_max_left T _).trans ht
rw [← he t htT,← hH t ht]
exact hn t (hT.trans htT)
end DefocusingNLS
end OAI
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