openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Nonlinear /CutoffAnnulusUniform.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.SchwartzProfileSampling | |
| namespace OAI | |
| /-! # Uniform norm bounds for every annulus of the physical cutoff profile -/ | |
| open scoped SchwartzMap ContDiff | |
| namespace DefocusingNLS | |
| local notation "E" => EuclideanSpace ℝ (Fin 12) | |
| theorem cutoffOuterAnnulus_decay (a L R : ℝ) (ha : 0 < a) | |
| (hL : 0 < L) (hR : 0 < R) (hR2 : R ≤ 2 * L) : | |
| L ^ (-a) ≤ 2 ^ a * R ^ (-a) := by | |
| have hp : (R / L) ^ a ≤ (2 : ℝ) ^ a := | |
| Real.rpow_le_rpow (by positivity) ((div_le_iff₀ hL).mpr hR2) ha.le | |
| have he : (R / L) ^ a * R ^ (-a) = L ^ (-a) := by | |
| rw [Real.div_rpow hR.le hL.le, Real.rpow_neg hR.le, Real.rpow_neg hL.le] | |
| field_simp | |
| exact he.symm.trans_le (mul_le_mul_of_nonneg_right hp (Real.rpow_nonneg hR.le _)) | |
| theorem exists_cutoffAnnulus_uniform_bound (a k : ℝ) | |
| (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) | |
| (χ : 𝓢(E, ℂ)) (hχzero : ∀ y : E, 1 ≤ ‖y‖ → χ y = 0) : | |
| ∃ (N : ℕ) (C : ℝ), 0 ≤ C ∧ ∀ (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) | |
| (D : ℝ), 0 ≤ D → | |
| (∀ n ≤ N, ∀ y : E, y ≠ 0 → | |
| ‖iteratedFDeriv ℝ n Q y‖ ≤ D * ‖y‖ ^ (-2 * a - (n : ℝ))) → | |
| ∀ (L R : ℝ) (hL : 1 ≤ L) (hR : 1 ≤ R), | |
| ‖schwartzTorusSample a k L ha1 hk hL (radianFourierKernel | |
| (schwartzPhysicalDilation a R (by linarith) | |
| (cutoffProfileAnnulus a R L χ homogeneousAnnulusCutoff | |
| homogeneousAnnulusCutoff_hasCompactSupport Q hQ)))‖ ≤ C * D * R ^ (-a) := by | |
| obtain ⟨N₁, C₁, hC₁, hb₁⟩ := exists_cutoffProfileAnnulus_sampling_bound a k ha ha1 hk | |
| χ homogeneousAnnulusCutoff homogeneousAnnulusCutoff_hasCompactSupport homogeneousAnnulusCutoff_support | |
| obtain ⟨N₂, C₂, hC₂, hb₂⟩ := exists_outerCutoffAnnulus_sampling_bound a k ha ha1 hk | |
| χ homogeneousAnnulusCutoff homogeneousAnnulusCutoff_hasCompactSupport homogeneousAnnulusCutoff_support | |
| let C := max C₁ (2 ^ a * C₂) | |
| have hC : 0 ≤ C := hC₁.trans (le_max_left _ _) | |
| refine ⟨max N₁ N₂, C, hC, ?_⟩ | |
| intro Q hQ D hD hsymbol L R hL hR | |
| by_cases hRL : R ≤ L | |
| · exact (hb₁ Q hQ D hD (fun n hn => hsymbol n (hn.trans (le_max_left _ _))) | |
| R L hR hRL).trans (by gcongr; exact le_max_left _ _) | |
| · by_cases hR2 : R ≤ 2 * L | |
| · have h := hb₂ Q hQ D hD (fun n hn => hsymbol n (hn.trans (le_max_right _ _))) | |
| L R hL (le_of_lt (lt_of_not_ge hRL)) hR2 | |
| calc | |
| _ ≤ C₂ * D * L ^ (-a) := h | |
| _ ≤ C₂ * D * (2 ^ a * R ^ (-a)) := mul_le_mul_of_nonneg_left | |
| (cutoffOuterAnnulus_decay a L R ha (by linarith) (by linarith) hR2) (mul_nonneg hC₂ hD) | |
| _ = (2 ^ a * C₂) * D * R ^ (-a) := by ring | |
| _ ≤ C * D * R ^ (-a) := by gcongr; exact le_max_right _ _ | |
| · rw [cutoffProfileAnnulus_zero_of_large_radius a R L (by linarith) | |
| (le_of_lt (lt_of_not_ge hR2)) χ hχzero Q hQ] | |
| change ‖(physicalSchwartzTorusSamplingLinear a k L ha1 hk hL) 0‖ ≤ _ | |
| rw [map_zero, norm_zero] | |
| positivity | |
| end DefocusingNLS | |
| end OAI | |