openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Linear /ExpandingCoordinateSequence.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingIdentifiedCompactness | |
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingCoordinateMap | |
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.HomogeneousRealCoordinateEvolution | |
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.WeakLinearCoordinates | |
| namespace OAI | |
| /-! # Finite-coordinate defects vanish along every bounded expanding sequence -/ | |
| open Set Filter Topology | |
| open scoped SchwartzMap | |
| namespace DefocusingNLS | |
| local notation "E" => EuclideanSpace ℝ (Fin 12) | |
| variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F] | |
| attribute [local irreducible] expandingProfileTrajectory | |
| theorem expandingProfile_coordinate_sequence (a b k T Q M C₀ : ℝ) | |
| (ha : 0 < a) (ha1 : a < 1) (hk : 10 < k) (hT : 0 ≤ T) | |
| (hQ : 0 ≤ Q) (hM : 0 ≤ M) (hC₀ : 0 ≤ C₀) (m : ℕ) | |
| (χ : 𝓢(E, ℂ)) (ρ : ℝ) (hρ : 0 < ρ) (hχ : ∀ y : E, ‖y‖ ≤ ρ → χ y = 1) | |
| (hCb : ∀ (L : ℝ) (hL : 1 ≤ L) (f : FourierL2), | |
| ‖homogeneousLocalizationCLM a k L ha ha1 (by linarith) hL χ f‖ ≤ C₀ * ‖f‖) | |
| (L : ℕ → ℝ) (hL : ∀ n, 1 ≤ L n) (hLinf : Tendsto L atTop atTop) | |
| (q : ℕ → C(Icc (0 : ℝ) T, FourierL2)) (hq : ∀ n t, ‖q n t‖ ≤ Q) | |
| (f : ℕ → FourierL2) (hf : ∀ n, ‖f n‖ ≤ M) (Q₀ : HomogeneousY a k) | |
| (π : HomogeneousY a k →L[ℝ] F) (A : F →L[ℝ] F) | |
| (hA : ∀ u, π (homogeneousLinearizedTrajectory a b k T ha ha1 (by linarith) hT m Q₀ u | |
| ⟨T, hT, le_rfl⟩) = A (π u)) | |
| (hqp : ∀ (s : Icc (0 : ℝ) T) (y : E), | |
| Tendsto (fun n => expandingPhysicalContinuous a k (expandingRadius (L n) s) | |
| ha ha1 (by linarith) ((hL n).trans (expandingRadius_ge (L n) s (hL n) s.2.1)) (q n s) y) | |
| atTop (𝓝 (homogeneousPhysicalCLM a k ha ha1 (by linarith) Q₀ y))) : | |
| ∀ ε : ℝ, 0 < ε → ∀ᶠ n in atTop, | |
| ‖expandingCoordinates a k ha ha1 (by linarith) χ π | |
| (expandingRadiusCurve (L n) T (hL n) ⟨T, hT, le_rfl⟩) | |
| (expandingProfileTrajectory a b k (L n) T ha ha1 (by linarith) (hL n) hT | |
| m Q hQ (q n) (hq n) (f n) ⟨T, hT, le_rfl⟩) - | |
| A (expandingCoordinates a k ha ha1 (by linarith) χ π ⟨L n, hL n⟩ (f n))‖ < ε := by | |
| have hk8 : 8 < k := by linarith | |
| intro ε hε | |
| by_contra h | |
| obtain ⟨ψ, hψ, hbad⟩ := extraction_of_frequently_atTop (show ∃ᶠ n in atTop, | |
| ε ≤ ‖expandingCoordinates a k ha ha1 hk8 χ π | |
| (expandingRadiusCurve (L n) T (hL n) ⟨T, hT, le_rfl⟩) | |
| (expandingProfileTrajectory a b k (L n) T ha ha1 hk8 (hL n) hT | |
| m Q hQ (q n) (hq n) (f n) ⟨T, hT, le_rfl⟩) - | |
| A (expandingCoordinates a k ha ha1 hk8 χ π ⟨L n, hL n⟩ (f n))‖ by | |
| simpa only [not_eventually, not_lt] using h) | |
| obtain ⟨φ, _hφ, _v, u₀, _hu₀, _hv, _hphysical, hweak⟩ := | |
| expandingProfile_exists_identified_limit a b k T Q M C₀ ha ha1 hk hT hQ hM hC₀ m χ ρ hρ hχ hCb | |
| (fun n => L (ψ n)) (fun n => hL (ψ n)) (hLinf.comp hψ.tendsto_atTop) | |
| (fun n => q (ψ n)) (fun n => hq (ψ n)) (fun n => f (ψ n)) (fun n => hf (ψ n)) Q₀ | |
| (fun t y => (hqp t y).comp hψ.tendsto_atTop) | |
| let σ := fun n => ψ (φ n) | |
| have hfina := tendsto_finite_coordinates_of_weak π _ _ (hweak ⟨T, hT, le_rfl⟩) | |
| have hinit := tendsto_finite_coordinates_of_weak π _ _ (hweak ⟨0, le_rfl, hT⟩) | |
| simp only [expandingProfileTrajectory_initial, expandingRadius, zero_div, Real.exp_zero, | |
| mul_one, homogeneousLinearizedTrajectory_initial] at hinit | |
| have htarget := A.continuous.continuousAt.tendsto.comp hinit | |
| have hdefect := hfina.sub htarget | |
| rw [hA, sub_self] at hdefect | |
| have hnorm := hdefect.norm | |
| have hsmall := hnorm.eventually (gt_mem_nhds (show ‖(0 : F)‖ < ε by simpa using hε)) | |
| obtain ⟨j, hj⟩ := hsmall.exists | |
| exact (not_lt_of_ge (hbad (φ j))) hj | |
| end DefocusingNLS | |
| end OAI | |