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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 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingNonprincipalObservation
namespace OAI
/-! # Dissipative energy with a compact physical observation on large tori -/
open Filter Topology MeasureTheory
namespace DefocusingNLS
local notation "T" => UnitAddTorus (Fin 12)
local notation "Radius" => {L : ℝ // 1 ≤ L}
noncomputable local instance energyObservationMeasure : MeasureSpace UnitAddCircle := ⟨AddCircle.haarAddCircle⟩
local instance energyObservationProbability : IsProbabilityMeasure (volume : Measure UnitAddCircle) :=
inferInstanceAs (IsProbabilityMeasure AddCircle.haarAddCircle)
theorem exists_expandingEnergy_observation (a Q : ℝ) (N : ℕ)
(ha : 0 < a) (ha1 : a < 1) (hN : 8 < ((N + 1 : ℕ) : ℝ)) (hQ : 0 ≤ Q)
(q : Radius → FourierL2) (hqb : ∀ L, ‖q L‖ ≤ Q)
(hq : ∀ (L : ℕ → Radius), Tendsto (fun n => (L n).1) atTop atTop →
ExpandingCompactApproximation a (N + 1 : ℕ) ha1 hN
(fun n => (L n).1) (fun n => (L n).2) (fun n => q (L n)))
(m : ℕ) (hm : 0 < m) (P : ℝ) (hP : 0 ≤ P)
(hQB : ∀ (L : Radius) (x : T), ‖expandingUnitTorusFunction a (N + 1 : ℕ) L.1 (q L) x‖ ^ (2 * m) ≤ P)
(hgap : 0 < (N + 1 : ℕ) + 2 * a - 6 - 2 * ((2 * (m : ℝ) + 1) * P)) :
∃ c : ℝ, 0 < c ∧ ∃ n : ℕ, ∃ C : ℝ, 0 ≤ C ∧
∀ (L : Radius), (n : ℝ) ≤ L.1 → ∀ f : FourierL2,
-a * ‖expandingLowEnergy a (N + 1 : ℕ) L.1 L.2 f‖ ^ 2 +
(6 - 2 * a - (N + 1 : ℕ)) * ‖expandingHighEnergy a (N + 1 : ℕ) L.1 L.2 f‖ ^ 2 +
2 * inner ℝ f (expandingLinearizedPotential a (N + 1 : ℕ) L.1 ha ha1 hN L.2 m (q L) f) ≤
-c * ‖f‖ ^ 2 + C * ‖expandingPhysicalBall a (N + 1 : ℕ) L.1 n ha ha1 hN L.2 f‖ ^ 2 := by
let B := (2 * (m : ℝ) + 1) * P
let g := (N + 1 : ℕ) + 2 * a - 6 - 2 * B
let γ := min a g
have hγ : 0 < γ := lt_min ha hgap
obtain ⟨n, C, hC, herror⟩ := exists_expandingNonprincipal_observation a Q N ha ha1 hN hQ q hqb hq
m hm P hP hQB (γ / 4) (by positivity)
refine ⟨γ / 2, half_pos hγ, n, 2 * C, by positivity, ?_⟩
intro L hL f
have he := herror L hL f
have hp := expandingOrderedPrincipal_energy_le a L.1 (N + 1) m L.2
(expandingUnitTorusFunction a (N + 1 : ℕ) L.1 (q L))
(expandingUnitTorusFunction a (N + 1 : ℕ) L.1 (q L)).continuous P hP (hQB L) f
have hnorm := expandingEnergy_norm_sq a (N + 1 : ℕ) L.1 L.2 f
have hl := mul_nonneg (sub_nonneg.mpr (min_le_left a g))
(sq_nonneg ‖expandingLowEnergy a (N + 1 : ℕ) L.1 L.2 f‖)
have hh := mul_nonneg (sub_nonneg.mpr (min_le_right a g))
(sq_nonneg ‖expandingHighEnergy a (N + 1 : ℕ) L.1 L.2 f‖)
change 0 ≤ (a - γ) * _ at hl
change 0 ≤ (g - γ) * _ at hh
unfold expandingNonprincipalEnergy at he
dsimp only [B, g] at hh
nlinarith
end DefocusingNLS
end OAI
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