openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Nonlinear /CutoffMixedTaylor.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.FixedCutoffProfileIdentity | |
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.FixedCutoffProfileTaylor | |
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffPhysicalTranslation | |
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.SchwartzSampleLipschitz | |
| import OAI.MathematicalPhysics.DefocusingNLS.Profile.CartesianTransportSymbol | |
| namespace OAI | |
| /-! # Uniform joint time and translation expansion in the physical symmetry directions -/ | |
| open scoped SchwartzMap ContDiff | |
| namespace DefocusingNLS | |
| local notation "E" => EuclideanSpace ℝ (Fin 12) | |
| local notation "Radius" => {L : ℝ // 1 ≤ L} | |
| theorem cutoffProfile_mixed_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 : Radius), 2 ≤ L.1 → ∀ (t : ℝ) (v : E), | |
| |t| ≤ 2 * Real.log 2 → | |
| ‖sobolevTranslation (euclideanToTorus ((1 / L.1) • v)) | |
| (fixedCutoffProfile a k ha ha1 hk L χ hχ Q hQ (expandingRadius L.1 t)) - | |
| fixedCutoffProfile a k ha ha1 hk L χ hχ Q hQ L.1 - | |
| t • fixedCutoffProfile a k ha ha1 hk L χ hχ (cartesianTransport Q) | |
| (cartesianTransport_contDiff Q hQ) L.1 - | |
| fixedCutoffProfile a k ha ha1 hk L χ hχ (cartesianDerivative v Q) | |
| (cartesianDerivative_contDiff Q hQ v) L.1‖ ≤ | |
| C * (|t| ^ 2 + ‖v‖ ^ 2 + ‖v‖ / L.1) := by | |
| have hT := cartesianTransport_global_symbol a Q hQ hsymbol | |
| have hTT := cartesianTransport_global_symbol a (cartesianTransport Q) | |
| (cartesianTransport_contDiff Q hQ) hT | |
| obtain ⟨B, hB, hb⟩ := cutoffProfile_sampling_of_global_symbol a k ha ha1 hk χ hχ hχzero | |
| (cartesianTransport (cartesianTransport Q)) | |
| (cartesianTransport_contDiff _ (cartesianTransport_contDiff Q hQ)) hTT | |
| obtain ⟨A, hA, hspace⟩ := cutoffProfile_physical_translation_taylor a k ha ha1 hk χ hχ | |
| hχzero Q hQ hsymbol | |
| obtain ⟨D, hD, hshift⟩ := cutoffProfile_translation_lipschitz a k ha ha1 hk χ hχ hχzero | |
| (cartesianTransport Q) (cartesianTransport_contDiff Q hQ) hT | |
| let B' := (2 ^ a + 2 ^ (k - 6)) * B | |
| have hB' : 0 ≤ B' := by dsimp [B']; positivity | |
| refine ⟨B' + A + D, by positivity, ?_⟩ | |
| intro L hL t v ht | |
| let u := fixedCutoffProfile a k ha ha1 hk L χ hχ Q hQ (expandingRadius L.1 t) | |
| let q := fixedCutoffProfile a k ha ha1 hk L χ hχ Q hQ L.1 | |
| let d := fixedCutoffProfile a k ha ha1 hk L χ hχ (cartesianTransport Q) | |
| (cartesianTransport_contDiff Q hQ) L.1 | |
| let f := fixedCutoffProfile a k ha ha1 hk L χ hχ (cartesianDerivative v Q) | |
| (cartesianDerivative_contDiff Q hQ v) L.1 | |
| let S := (sobolevTranslation (euclideanToTorus ((1 / L.1) • v))).toContinuousLinearMap | |
| have htime : ‖u - q - t • d‖ ≤ B' * |t| ^ 2 := | |
| fixedCutoffProfile_logRadius_taylor a k ha ha1 hk L hL χ hχ Q hQ B hB | |
| (fun R => hb R.1 R.2) t ht | |
| have htrans : ‖S q - q - f‖ ≤ A * (‖v‖ ^ 2 + ‖v‖ / L.1) := by | |
| simpa only [S, LinearIsometry.coe_toContinuousLinearMap, q, f, fixedCutoffProfile_self] using hspace L.1 L.2 v | |
| have htd : ‖S d - d‖ ≤ D * ‖v‖ := by | |
| simpa only [S, LinearIsometry.coe_toContinuousLinearMap, d, fixedCutoffProfile_self] using hshift L.1 L.2 v | |
| have he : S u - q - t • d - f = | |
| S (u - q - t • d) + (S q - q - f) + t • (S d - d) := by | |
| simp only [map_sub, ContinuousLinearMap.map_smul_of_tower, smul_sub] | |
| abel | |
| change ‖S u - q - t • d - f‖ ≤ _ | |
| rw [he] | |
| calc | |
| _ ≤ ‖S (u - q - t • d)‖ + ‖S q - q - f‖ + ‖t • (S d - d)‖ := | |
| (norm_add_le _ _).trans (add_le_add (norm_add_le _ _) le_rfl) | |
| _ = ‖u - q - t • d‖ + ‖S q - q - f‖ + |t| * ‖S d - d‖ := by | |
| rw [norm_smul, Real.norm_eq_abs] | |
| simp only [S, LinearIsometry.coe_toContinuousLinearMap, LinearIsometry.norm_map] | |
| _ ≤ B' * |t| ^ 2 + A * (‖v‖ ^ 2 + ‖v‖ / L.1) + |t| * (D * ‖v‖) := | |
| add_le_add (add_le_add htime htrans) (mul_le_mul_of_nonneg_left htd (abs_nonneg _)) | |
| _ ≤ _ := by | |
| have hm := mul_nonneg hD (sq_nonneg (|t| - ‖v‖)) | |
| have hp : 0 ≤ ‖v‖ / L.1 := by positivity | |
| have h1 := mul_nonneg hB' (add_nonneg (sq_nonneg ‖v‖) hp) | |
| have h2 := mul_nonneg hA (sq_nonneg |t|) | |
| have h3 := mul_nonneg hD hp | |
| nlinarith | |
| end DefocusingNLS | |
| end OAI | |