openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Nonlinear /CutoffCoordinateInverse.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.FiniteInvariantInverse | |
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffCoordinateOperator | |
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffProfileEndpoint | |
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.TorusStableBlocks | |
| namespace OAI | |
| /-! # The finite inverse and the actual torus coordinate defect -/ | |
| open scoped SchwartzMap ContDiff NNReal | |
| namespace DefocusingNLS | |
| local notation "E" => EuclideanSpace ℝ (Fin 12) | |
| local notation "Radius" => {L : ℝ // 1 ≤ L} | |
| def HasContractingTorusCoordinateTransfer {F : Type*} | |
| [NormedAddCommGroup F] [NormedSpace ℝ F] | |
| (a k : ℝ) (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (χ : 𝓢(E, ℂ)) | |
| (π : HomogeneousY a k →L[ℝ] F) | |
| (T : ℝ≥0) (A : Radius → FourierL2 →L[ℝ] FourierL2) : Prop := | |
| ∃ D R : π.range →L[ℝ] π.range, (∀ v, D (R v) = v) ∧ ‖R‖ ≤ 1 ∧ | |
| ∀ ε : ℝ, 0 < ε → ∃ L₀ : ℝ, ∀ L : Radius, L₀ ≤ L.1 → | |
| ‖(expandingCoordinates a k ha ha1 hk χ π.rangeRestrict | |
| (torusEndpointRadius T L)).comp (A L) - | |
| D.comp (expandingCoordinates a k ha ha1 hk χ π.rangeRestrict L)‖ ≤ ε | |
| theorem cutoffProfile_coordinate_transfer_of_inverse {F : Type*} | |
| [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F] | |
| (a b k : ℝ) (ha : 0 < a) (ha1 : a < 1) (hk : 10 < k) (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) (q : HomogeneousY a k) | |
| (hq : ∀ y, homogeneousPhysicalCLM a k ha ha1 (by linarith) q y = Qp y) | |
| (Q : ℝ) (hQ : 0 ≤ Q) | |
| (hqb : ∀ L : Radius, | |
| ‖cutoffProfileCoefficient a k ha1 (by linarith) χ hχ Qp hQp L‖ ≤ Q) | |
| (π : HomogeneousY a k →L[ℝ] F) (T : ℝ≥0) | |
| (hI : HasContractingCoordinateInverse π | |
| (homogeneousLinearizedStep a b k ha ha1 (by linarith) m q T)) : | |
| HasContractingTorusCoordinateTransfer a k ha ha1 (by linarith) χ π T | |
| (cutoffProfileEndpoint a b k ha ha1 (by linarith) m χ hχ Qp hQp Q hQ hqb T) := by | |
| obtain ⟨D, R, hDR, hR, hD⟩ := hI | |
| refine ⟨D, R, hDR, hR, ?_⟩ | |
| have hk8 : 8 < k := by linarith | |
| obtain ⟨C₀, hC₀, hCb⟩ := exists_homogeneousLocalization_bound a k ha ha1 hk8 χ | |
| apply sampledCutoffProfile_coordinate_operator_norm a b k T Q C₀ ha ha1 hk T.2 hQ hC₀ | |
| m χ hχ hχzero hχone Qp hQp q hq hqb hCb π.rangeRestrict D | |
| intro u | |
| have h := hD u | |
| rwa [homogeneousLinearizedStep_eq_slab a b k ha ha1 hk8 m q T T.2 T le_rfl] at h | |
| end DefocusingNLS | |
| end OAI | |