openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Nonlinear /CutoffCoordinateTransfer.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingCoordinateSequence | |
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffStableTransfer | |
| namespace OAI | |
| /-! # Uniform finite-coordinate transfer for the sampled cutoff profile -/ | |
| open Set Filter Topology | |
| open scoped SchwartzMap ContDiff | |
| namespace DefocusingNLS | |
| local notation "E" => EuclideanSpace ℝ (Fin 12) | |
| local notation "Radius" => {L : ℝ // 1 ≤ L} | |
| variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F] | |
| theorem sampledCutoffProfile_uniform_coordinate_defect (a b k T Q C₀ : ℝ) | |
| (ha : 0 < a) (ha1 : a < 1) (hk : 10 < k) (hT : 0 ≤ T) | |
| (hQ : 0 ≤ Q) (hC₀ : 0 ≤ C₀) (m : ℕ) | |
| (χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ)) | |
| (hχzero : ∀ y : E, 1 ≤ ‖y‖ → χ y = 0) | |
| (hχone : ∀ y : E, ‖y‖ ≤ 1 / 2 → χ y = 1) | |
| (Qp : E → ℂ) (hQp : ContDiff ℝ ∞ Qp) (Q₀ : HomogeneousY a k) | |
| (hQ₀ : ∀ y, homogeneousPhysicalCLM a k ha ha1 (by linarith) Q₀ y = Qp y) | |
| (hqb : ∀ L : Radius, ‖cutoffProfileCoefficient a k ha1 (by linarith) χ hχ Qp hQp L‖ ≤ Q) | |
| (hCb : ∀ (L : ℝ) (hL : 1 ≤ L) (f : FourierL2), | |
| ‖homogeneousLocalizationCLM a k L ha ha1 (by linarith) hL χ f‖ ≤ C₀ * ‖f‖) | |
| (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)) : | |
| ∀ ε : ℝ, 0 < ε → ∃ L₀ : ℝ, ∀ L : Radius, L₀ ≤ L.1 → ∀ f : FourierL2, ‖f‖ ≤ 1 → | |
| ‖expandingCoordinates a k ha ha1 (by linarith) χ π | |
| (expandingRadiusCurve L.1 T L.2 ⟨T, hT, le_rfl⟩) | |
| (expandingProfileTrajectory a b k L.1 T ha ha1 (by linarith) L.2 hT m Q hQ | |
| (sampledCutoffProfilePath a k L.1 T ha ha1 (by linarith) L.2 χ hχ Qp hQp) | |
| (fun t => hqb (expandingRadiusCurve L.1 T L.2 t)) f ⟨T, hT, le_rfl⟩) - | |
| A (expandingCoordinates a k ha ha1 (by linarith) χ π L f)‖ < ε := by | |
| classical | |
| have hk8 : 8 < k := by linarith | |
| intro ε hε | |
| by_contra h | |
| push Not at h | |
| choose L hL f hf hbad using fun n : ℕ => h (n : ℝ) | |
| have hLinf : Tendsto (fun n => (L n).1) atTop atTop := | |
| tendsto_atTop_mono hL tendsto_natCast_atTop_atTop | |
| let q := fun n => sampledCutoffProfilePath a k (L n).1 T ha ha1 hk8 (L n).2 χ hχ Qp hQp | |
| have hq (n : ℕ) (t : Icc (0 : ℝ) T) : ‖q n t‖ ≤ Q := | |
| hqb (expandingRadiusCurve (L n).1 T (L n).2 t) | |
| have hqp (t : Icc (0 : ℝ) T) (y : E) : | |
| Tendsto (fun n => expandingPhysicalContinuous a k (expandingRadius (L n).1 t) | |
| ha ha1 hk8 ((L n).2.trans (expandingRadius_ge (L n).1 t (L n).2 t.2.1)) (q n t) y) | |
| atTop (𝓝 (homogeneousPhysicalCLM a k ha ha1 hk8 Q₀ y)) := by | |
| rw [hQ₀] | |
| exact tendsto_sampledCutoffProfilePath_physical a k T ha ha1 hk8 χ hχ hχzero hχone | |
| Qp hQp (fun n => (L n).1) (fun n => (L n).2) hLinf t y | |
| have hs := expandingProfile_coordinate_sequence a b k T Q 1 C₀ ha ha1 hk hT hQ | |
| zero_le_one hC₀ m χ (1 / 2) (by norm_num) hχone hCb | |
| (fun n => (L n).1) (fun n => (L n).2) hLinf q hq f hf Q₀ π A hA hqp ε hε | |
| obtain ⟨n, hn⟩ := hs.exists | |
| exact (not_lt_of_ge (hbad n)) hn | |
| end DefocusingNLS | |
| end OAI | |