openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Nonlinear /CutoffPhysicalTranslation.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffTranslationTaylor | |
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffDirectionalError | |
| namespace OAI | |
| /-! # Translation expansion with the actual physical profile direction -/ | |
| open scoped SchwartzMap LineDeriv ContDiff | |
| namespace DefocusingNLS | |
| local notation "E" => EuclideanSpace ℝ (Fin 12) | |
| theorem cutoffProfile_physical_translation_taylor (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) | |
| (Q : E → ℂ) (hQ : ContDiff ℝ ∞ Q) | |
| (hsymbol : ∀ n : ℕ, ∃ D : ℝ, 0 ≤ D ∧ ∀ y : E, y ≠ 0 → | |
| ‖iteratedFDeriv ℝ n Q y‖ ≤ D * ‖y‖ ^ (-2 * a - (n : ℝ))) : | |
| ∃ C : ℝ, 0 ≤ C ∧ ∀ (L : ℝ) (hL : 1 ≤ L) (v : E), | |
| let ψ := cutoffProfileSchwartz L (by linarith) χ hχ Q hQ | |
| ‖sobolevTranslation (euclideanToTorus ((1 / L) • v)) | |
| (physicalSchwartzTorusSamplingCLM a k L ha1 hk hL ψ) - | |
| physicalSchwartzTorusSamplingCLM a k L ha1 hk hL ψ - | |
| physicalSchwartzTorusSamplingCLM a k L ha1 hk hL | |
| (cutoffProfileSchwartz L (by linarith) χ hχ (cartesianDerivative v Q) | |
| (cartesianDerivative_contDiff Q hQ v))‖ ≤ C * (‖v‖ ^ 2 + ‖v‖ / L) := by | |
| obtain ⟨B, hB, hb⟩ := cutoffProfile_translation_taylor a k ha ha1 hk χ hχ hχzero Q hQ hsymbol | |
| obtain ⟨D, hD, hd⟩ := cutoff_directional_error_bound a k ha ha1 hk χ hχ hχzero Q hQ hsymbol | |
| refine ⟨B + D, add_nonneg hB hD, ?_⟩ | |
| intro L hL v ψ | |
| let u := sobolevTranslation (euclideanToTorus ((1 / L) • v)) | |
| (physicalSchwartzTorusSamplingCLM a k L ha1 hk hL ψ) - | |
| physicalSchwartzTorusSamplingCLM a k L ha1 hk hL ψ | |
| let d := physicalSchwartzTorusSamplingCLM a k L ha1 hk hL (∂_{v} ψ) | |
| let f := physicalSchwartzTorusSamplingCLM a k L ha1 hk hL | |
| (cutoffProfileSchwartz L (by linarith) χ hχ (cartesianDerivative v Q) | |
| (cartesianDerivative_contDiff Q hQ v)) | |
| change ‖u - f‖ ≤ _ | |
| calc | |
| _ ≤ ‖u - d‖ + ‖d - f‖ := norm_sub_le_norm_sub_add_norm_sub u d f | |
| _ ≤ B * ‖v‖ ^ 2 + D / L * ‖v‖ := add_le_add (hb L hL v) (hd L hL v) | |
| _ ≤ _ := by | |
| have h1 := mul_nonneg hB (show 0 ≤ ‖v‖ / L by positivity) | |
| have h2 := mul_nonneg hD (sq_nonneg ‖v‖) | |
| simp only [div_eq_mul_inv] at * | |
| nlinarith | |
| end DefocusingNLS | |
| end OAI | |