File size: 3,816 Bytes
d5b562f | 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 | import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.BoundedInnerNull
namespace OAI
/-! # The exact nonprincipal energy error vanishes on locally null sequences -/
open Filter Topology MeasureTheory
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
local notation "T" => UnitAddTorus (Fin 12)
noncomputable local instance compactEnergyErrorMeasure : MeasureSpace UnitAddCircle := ⟨AddCircle.haarAddCircle⟩
local instance compactEnergyErrorProbability : IsProbabilityMeasure (volume : Measure UnitAddCircle) :=
inferInstanceAs (IsProbabilityMeasure AddCircle.haarAddCircle)
variable (a Q M : ℝ) (N : ℕ) (ha : 0 < a) (ha1 : a < 1)
(hN : 8 < ((N + 1 : ℕ) : ℝ)) (L : ℕ → ℝ) (hL : ∀ n, 1 ≤ L n)
(hLinf : Tendsto L atTop atTop) (q : ℕ → FourierL2)
(hq : ExpandingCompactApproximation a (N + 1 : ℕ) ha1 hN L hL q) (hqb : ∀ n, ‖q n‖ ≤ Q)
(f : ℕ → FourierL2) (hf : ∀ n, ‖f n‖ ≤ M)
(hlocal : ∀ R ε : ℝ, 0 < ε → ∀ᶠ n in atTop, ∀ y : E, ‖y‖ ≤ R →
‖expandingTorusFunction a (N + 1 : ℕ) (L n) (f n) (euclideanToTorus ((L n)⁻¹ • y))‖ < ε)
(m : ℕ) (hm : 0 < m) (P : ℝ) (hP : 0 ≤ P)
(hQB : ∀ n (x : T), ‖expandingUnitTorusFunction a (N + 1 : ℕ) (L n) (q n) x‖ ^ (2 * m) ≤ P)
include hLinf hq hqb hf hlocal hm in
theorem tendsto_expandingOrderedCommutator (j : Fin (N + 1) → Fin 12) :
Tendsto (fun n => expandingOrderedCommutator a (L n) (N + 1) ha ha1 hN (hL n)
m (q n) P hP (hQB n) j (f n)) atTop (𝓝 0) := by
have ht := tendsto_expandingOddFourierCommutator a Q M N ha ha1 hN L hL hLinf q hq hqb f hf hlocal m hm j
have hi := (torusFourierIsometry.continuous.tendsto 0).comp ht
simp only [Function.comp_def, map_zero] at hi
have he (n : ℕ) : torusFourierIsometry (expandingOddFourierCommutator a (L n) (N + 1)
ha ha1 hN (hL n) m (q n) (f n) j) =
expandingOrderedCommutator a (L n) (N + 1) ha ha1 hN (hL n) m (q n) P hP (hQB n) j (f n) :=
torusOddCommutator_identification a (L n) (N + 1) ha ha1 hN (hL n) m (q n) P hP (hQB n) (f n) j
simpa only [he] using hi
include hLinf hq hqb hf hlocal hm in
theorem tendsto_expandingCompactEnergy_error :
Tendsto (fun n =>
inner ℝ (f n) (expandingLinearizedPotential a (N + 1 : ℕ) (L n) ha ha1 hN (hL n) m (q n) (f n)) -
∑ j : Fin (N + 1) → Fin 12,
inner ℝ (expandingOrderedPhysicalEnergy a (L n) (N + 1) (hL n) j (f n))
(torusPrincipalPotential m (expandingUnitTorusFunction a (N + 1 : ℕ) (L n) (q n))
(expandingUnitTorusFunction a (N + 1 : ℕ) (L n) (q n)).continuous P hP (hQB n)
(expandingOrderedPhysicalEnergy a (L n) (N + 1) (hL n) j (f n)))) atTop (𝓝 0) := by
have hlo := tendsto_expandingLinearized_lowEnergy a Q M N ha ha1 hN L hL hLinf q hq hqb f hf hlocal m hm
have hlow := tendsto_inner_of_bounded_null
(fun n => expandingLowEnergy a (N + 1 : ℕ) (L n) (hL n) (f n)) _ M
(fun n => (expandingLowEnergy_norm_le a _ _ _ _).trans (hf n)) hlo
have htop (j : Fin (N + 1) → Fin 12) := tendsto_inner_of_bounded_null
(fun n => expandingOrderedPhysicalEnergy a (L n) (N + 1) (hL n) j (f n)) _ M
(fun n => (expandingOrderedPhysicalEnergy_norm a _ _ _ _ _).le.trans
((expandingOrderedFourierEnergy_norm_le a _ _ _ _ _).trans (hf n)))
(tendsto_expandingOrderedCommutator a Q M N ha ha1 hN L hL hLinf q hq hqb f hf hlocal m hm P hP hQB j)
have ht := hlow.add (tendsto_finsetSum Finset.univ (fun j _ => htop j))
simp only [Finset.sum_const_zero, add_zero] at ht
have he (n : ℕ) := expandingLinearized_energy_split a (L n) (N + 1) ha ha1 hN (hL n)
m (q n) P hP (hQB n) (f n)
dsimp only at he
convert ht using 1
funext n
linarith [he n]
end DefocusingNLS
end OAI
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