openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Nonlinear /CutoffProfileCoordinateLimit.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffProfilePhysicalLimit | |
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.HomogeneousPhysicalWeakLimit | |
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.HomogeneousLocalizationPhysical | |
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.WeakLinearCoordinates | |
| namespace OAI | |
| /-! # Finite coordinates of the actual sampled profile converge to its whole-space coordinates -/ | |
| open 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 cutoffProfile_finite_coordinate_limit (a k : ℝ) (ha : 0 < a) | |
| (ha1 : a < 1) (hk : 8 < k) (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) | |
| (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) | |
| (q : HomogeneousY a k) | |
| (hq : ∀ y, homogeneousPhysicalCLM a k ha ha1 hk q y = Q y) | |
| (B : ℝ) (_hB : 0 ≤ B) | |
| (hb : ∀ (L : ℝ) (hL : 1 ≤ L), | |
| ‖physicalSchwartzTorusSamplingCLM a k L ha1 hk hL | |
| (cutoffProfileSchwartz L (by linarith) χ hχ Q hQ)‖ ≤ B) | |
| (L : ℕ → Radius) (hLinf : Tendsto (fun n => (L n).1) atTop atTop) | |
| (π : HomogeneousY a k →L[ℝ] F) : | |
| Tendsto (fun n => π (homogeneousLocalizationCLM a k (L n).1 ha ha1 hk (L n).2 χ | |
| (physicalSchwartzTorusSamplingCLM a k (L n).1 ha1 hk (L n).2 | |
| (cutoffProfileSchwartz (L n).1 (by linarith [(L n).2]) χ hχ Q hQ)))) | |
| atTop (𝓝 (π q)) := by | |
| obtain ⟨D, hD, hd⟩ := exists_homogeneousLocalization_bound a k ha ha1 hk χ | |
| apply tendsto_finite_coordinates_of_weak π _ q | |
| apply homogeneousY_weak_of_physical_limit a k (D * B) ha ha1 hk | |
| · intro n | |
| exact (hd (L n).1 (L n).2 _).trans (mul_le_mul_of_nonneg_left (hb (L n).1 (L n).2) hD) | |
| · intro y | |
| rw [hq] | |
| apply tendsto_const_nhds.congr' | |
| filter_upwards [hLinf.eventually (eventually_ge_atTop (2 * ‖y‖))] with n hn | |
| rw [homogeneousLocalization_physical, expandingPhysicalLocalization] | |
| have hc : χ ((L n).1⁻¹ • y) = 1 := by | |
| apply hχone | |
| rw [norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr (by linarith [(L n).2]))] | |
| exact (inv_mul_le_iff₀ (by linarith [(L n).2] : 0 < (L n).1)).mpr (by linarith) | |
| rw [hc, one_mul] | |
| simpa only [expandingPhysicalContinuous_apply, physicalSchwartzTorusSamplingCLM_apply] using | |
| (sampledCutoffProfile_physical_eq a k (L n).1 ha ha1 hk (L n).2 χ hχ | |
| hχzero hχone Q hQ y hn).symm | |
| end DefocusingNLS | |
| end OAI | |