openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Nonlinear /ContinuousTorusSequence.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.ContinuousTorusDiscreteData | |
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.StableSmallRemainderContinuity | |
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffRemainderContinuity | |
| namespace OAI | |
| /-! # Continuous stable endpoint sequences for the expanding torus -/ | |
| open scoped NNReal | |
| universe u | |
| namespace DefocusingNLS | |
| local notation "Radius" => {L : ℝ // 1 ≤ L} | |
| theorem exists_continuous_cutoffStable_sequence_threshold | |
| {F : Type*} | |
| [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] | |
| (a δ B : ℝ) (ha : 0 < a) (hδ : 0 < δ) (hB : 0 ≤ B) | |
| (T : ℝ≥0) (hratio : 2 * Real.exp (-(2 + a) * T / 2) ≤ 1) | |
| (A : Radius → FourierL2 →L[ℝ] FourierL2) | |
| (hAc : Continuous (fun q : Radius × FourierL2 => A q.1 q.2)) | |
| (hA : HasContinuousTorusLinearStep (F := F) T A) : | |
| ∃ C : ℝ, 0 < C ∧ ∃ ρ : ℝ, 0 < ρ ∧ 2 * ρ ≤ δ ∧ ∃ L₀ : ℝ, | |
| ∃ ζ : Radius → ℕ → F →L[ℝ] FourierL2, | |
| ∃ π : Radius → ℕ → FourierL2 →L[ℝ] F, | |
| (∀ n, ContinuousOn (fun L => ζ L n) {L : Radius | L₀ ≤ L.1}) ∧ | |
| (∀ n, Continuous (fun p : Radius × FourierL2 => π p.1 n p.2)) ∧ | |
| (∀ L : Radius, L₀ ≤ L.1 → ∀ n, ‖ζ L n‖ ≤ 1 ∧ ‖π L n‖ ≤ C) ∧ | |
| (∀ L : Radius, L₀ ≤ L.1 → ∀ n v, π L n (ζ L n v) = v) ∧ | |
| ∀ (P : Type u) [TopologicalSpace P], ∀ L : P → Radius, | |
| Continuous L → (∀ p, L₀ ≤ (L p).1) → | |
| ∀ h : P → ℕ → {v : FourierL2 // ‖v‖ ≤ δ} → FourierL2, | |
| (∀ n, Continuous (fun q : P × {v : FourierL2 // ‖v‖ ≤ δ} => h q.1 n q.2)) → | |
| (∀ p n d, 0 < d → d ≤ δ → ∀ v u, ‖v.1‖ ≤ d → ‖u.1‖ ≤ d → | |
| ‖h p n v - h p n u‖ ≤ | |
| B * (d + (expandingDiscreteRadius (L p) T n).1 ^ (-2 - a)) * ‖v.1 - u.1‖) → | |
| (∀ p n v, v.1 = 0 → ‖h p n v‖ ≤ | |
| B * (expandingDiscreteRadius (L p) T n).1 ^ (-2 - a)) → | |
| ∀ w : P → FourierL2, Continuous w → (∀ p, ‖w p‖ ≤ ρ / 4) → | |
| (∀ p, π (L p) 0 (w p) = 0) → | |
| ∃ z : P → ℕ → FourierL2, | |
| (∀ n, Continuous (fun p => z p n)) ∧ | |
| (∀ p, stableFrameProjection (ζ (L p) 0) (π (L p) 0) (z p 0) = w p) ∧ | |
| (∀ p n, z p (n + 1) = A (expandingDiscreteRadius (L p) T n) (z p n) + | |
| cutoffRemainderExtension δ (h p) n (z p n)) ∧ | |
| (∀ p n, ‖z p n‖ ≤ (2 * ρ) * (1 / 2 : ℝ) ^ n) := by | |
| obtain ⟨C, hC, hlinear⟩ := continuous_torusLinearStep_discrete T A hA | |
| obtain ⟨σ, Llip, hσ, hσδ, hLlip, hlip⟩ := exists_cutoffRemainder_small_lipschitz | |
| a B δ (1 / (128 * (C + 1))) ha hB hδ (by positivity) | |
| let ρ := σ / 2 | |
| have hρ : 0 < ρ := by dsimp [ρ]; positivity | |
| have h2ρ : 2 * ρ = σ := by dsimp [ρ]; ring | |
| obtain ⟨Llin, ζ, π, D, R, hinv, hR, hζc, hπc, hdata⟩ := hlinear (1 / 128) (by norm_num) | |
| obtain ⟨Lforce, _, hforce⟩ := exists_cutoffResidual_small_scale a (C * B) (ρ / 8) ha (by positivity) | |
| let L₀ := max Llin (max Llip Lforce) | |
| have hL₀lip : Llip ≤ L₀ := (le_max_left _ _).trans (le_max_right _ _) | |
| have hL₀lin : Llin ≤ L₀ := le_max_left _ _ | |
| have hL₀force : Lforce ≤ L₀ := (le_max_right _ _).trans (le_max_right _ _) | |
| have hL₀pos : 0 < L₀ := lt_of_lt_of_le zero_lt_one (hLlip.trans hL₀lip) | |
| let ε := B * L₀ ^ (-2 - a) | |
| have hε : 0 ≤ ε := mul_nonneg hB (Real.rpow_nonneg hL₀pos.le _) | |
| have hεsmall : C * ε ≤ ρ / 8 := by | |
| simpa only [ε, ← mul_assoc] using hforce L₀ hL₀force | |
| refine ⟨C, hC, ρ, hρ, by rw [h2ρ]; exact hσδ, L₀, ζ, π, ?_, hπc, | |
| (fun L hL n => ⟨(hdata L (hL₀lin.trans hL)).2.1 n, | |
| (hdata L (hL₀lin.trans hL)).2.2.1 n⟩), | |
| (fun L hL => (hdata L (hL₀lin.trans hL)).1), ?_⟩ | |
| · intro n | |
| apply (hζc n).mono | |
| intro L hL | |
| exact hL₀lin.trans hL | |
| intro P _ L hLc hL h hhc hl hz w hwc hwb hker | |
| have hd (p : P) := hdata (L p) (hL₀lin.trans (hL p)) | |
| have hZc (n : ℕ) : Continuous (fun p => ζ (L p) n) := | |
| (hζc n).comp_continuous hLc (fun p => hL₀lin.trans (hL p)) | |
| have hZjoint (n : ℕ) : Continuous (fun q : P × F => ζ (L q.1) n q.2) := | |
| ((hZc n).comp continuous_fst).clm_apply continuous_snd | |
| have hPjoint (n : ℕ) : Continuous (fun q : P × FourierL2 => π (L q.1) n q.2) := | |
| (hπc n).comp ((hLc.comp continuous_fst).prodMk continuous_snd) | |
| have hAjoint (n : ℕ) : Continuous | |
| (fun q : P × FourierL2 => A (expandingDiscreteRadius (L q.1) T n) q.2) := | |
| hAc.comp ((((continuous_expandingDiscreteRadius T n).comp hLc).comp | |
| continuous_fst).prodMk continuous_snd) | |
| have hpow (p : P) : (L p).1 ^ (-2 - a) ≤ L₀ ^ (-2 - a) := | |
| Real.rpow_le_rpow_of_nonpos hL₀pos (hL p) (by linarith) | |
| have hsmall (p : P) : ‖w p‖ + 2 * (C * ε) ≤ ρ / 2 := by | |
| linarith [hwb p] | |
| have hLip : ∀ p n v u, ‖v‖ ≤ 2 * ρ → ‖u‖ ≤ 2 * ρ → | |
| ‖cutoffRemainderExtension δ (h p) n v - cutoffRemainderExtension δ (h p) n u‖ ≤ | |
| (1 / (128 * (C + 1))) * ‖v - u‖ := by | |
| intro p n v u hv hu | |
| have hh := cutoffRemainderExtension_lipschitz a (L p).1 T δ σ B | |
| (lt_of_lt_of_le zero_lt_one (L p).2) T.2 ha hσ hσδ hB (h p) (hl p) | |
| n v u (by rwa [← h2ρ]) (by rwa [← h2ρ]) | |
| exact hh.trans (mul_le_mul_of_nonneg_right | |
| (hlip (L p).1 (hL₀lip.trans (hL p))) (norm_nonneg _)) | |
| have hZero : ∀ p n, ‖cutoffRemainderExtension δ (h p) n 0‖ ≤ | |
| ε * (Real.exp (-(2 + a) * T / 2)) ^ n := by | |
| intro p n | |
| apply (cutoffRemainderExtension_zero_bound a (L p).1 T δ B | |
| (lt_of_lt_of_le zero_lt_one (L p).2) hδ.le (h p) (hz p) n).trans | |
| exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left (hpow p) hB) (by positivity) | |
| obtain ⟨z, hzc, hz0, hzstep, hzn⟩ := exists_continuous_stableGraph_small_remainder | |
| (fun p => ζ (L p)) (fun p => π (L p)) | |
| (fun p n => A (expandingDiscreteRadius (L p) T n)) D R hR hinv | |
| (fun p => (hd p).1) (fun p => (hd p).2.2.2.2.1) (fun p => (hd p).2.2.2.2.2.1) | |
| hZjoint hPjoint hAjoint C ρ ε (Real.exp (-(2 + a) * T / 2)) hC hρ.le hε | |
| (Real.exp_nonneg _) hratio (fun p => (hd p).2.1) (fun p => (hd p).2.2.1) | |
| (fun p => (hd p).2.2.2.1) (fun p => (hd p).2.2.2.2.2.2) | |
| (fun p => cutoffRemainderExtension δ (h p)) | |
| (continuous_cutoffRemainderExtension_on_ball δ (2 * ρ) (h2ρ ▸ hσδ) h hhc) | |
| hLip hZero w hwc hker hsmall | |
| refine ⟨z, hzc, hz0, hzstep, fun p n => (hzn p n).trans ?_⟩ | |
| apply mul_le_mul_of_nonneg_right _ (by positivity) | |
| linarith [hsmall p] | |
| end DefocusingNLS | |
| end OAI | |