openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Nonlinear /CutoffProfileEndpoint.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffStableTransfer | |
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingProfileOperator | |
| namespace OAI | |
| /-! # Endpoint operators for the actual sampled cutoff profile -/ | |
| open scoped SchwartzMap ContDiff NNReal | |
| namespace DefocusingNLS | |
| local notation "E" => EuclideanSpace ℝ (Fin 12) | |
| local notation "Radius" => {L : ℝ // 1 ≤ L} | |
| noncomputable def cutoffProfileEndpoint (a b k : ℝ) | |
| (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) | |
| (m : ℕ) (χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ)) | |
| (Qp : E → ℂ) (hQp : ContDiff ℝ ∞ Qp) (Q : ℝ) (hQ : 0 ≤ Q) | |
| (hqb : ∀ L : Radius, ‖cutoffProfileCoefficient a k ha1 hk χ hχ Qp hQp L‖ ≤ Q) | |
| (T : ℝ≥0) (L : Radius) : FourierL2 →L[ℝ] FourierL2 := | |
| expandingProfileEndpoint a b k L.1 T ha ha1 hk L.2 T.2 m Q hQ | |
| (sampledCutoffProfilePath a k L.1 T ha ha1 hk L.2 χ hχ Qp hQp) | |
| (fun t => hqb (expandingRadiusCurve L.1 T L.2 t)) | |
| @[simp] theorem cutoffProfileEndpoint_apply (a b k : ℝ) | |
| (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) | |
| (m : ℕ) (χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ)) | |
| (Qp : E → ℂ) (hQp : ContDiff ℝ ∞ Qp) (Q : ℝ) (hQ : 0 ≤ Q) | |
| (hqb : ∀ L : Radius, ‖cutoffProfileCoefficient a k ha1 hk χ hχ Qp hQp L‖ ≤ Q) | |
| (T : ℝ≥0) (L : Radius) (f : FourierL2) : | |
| cutoffProfileEndpoint a b k ha ha1 hk m χ hχ Qp hQp Q hQ hqb T L f = | |
| expandingProfileTrajectory a b k L.1 T ha ha1 hk L.2 T.2 m Q hQ | |
| (sampledCutoffProfilePath a k L.1 T ha ha1 hk L.2 χ hχ Qp hQp) | |
| (fun t => hqb (expandingRadiusCurve L.1 T L.2 t)) f ⟨T, T.2, le_rfl⟩ := rfl | |
| theorem cutoffProfileEndpoint_uniform_bound (a b k : ℝ) | |
| (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) | |
| (m : ℕ) (χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ)) | |
| (Qp : E → ℂ) (hQp : ContDiff ℝ ∞ Qp) (Q : ℝ) (hQ : 0 ≤ Q) | |
| (hqb : ∀ L : Radius, ‖cutoffProfileCoefficient a k ha1 hk χ hχ Qp hQp L‖ ≤ Q) | |
| (T : ℝ≥0) : ∃ C : ℝ, 0 ≤ C ∧ ∀ L : Radius, | |
| ‖cutoffProfileEndpoint a b k ha ha1 hk m χ hχ Qp hQp Q hQ hqb T L‖ ≤ C := by | |
| obtain ⟨K, hK, hb⟩ := exists_expandingProfileTrajectory_bound a b k ha ha1 hk m Q hQ | |
| refine ⟨(K + 1) * Real.exp ((K + 1) * T), by positivity, ?_⟩ | |
| intro L | |
| apply ContinuousLinearMap.opNorm_le_bound _ (by positivity) | |
| intro f | |
| rw [cutoffProfileEndpoint_apply] | |
| exact ((expandingProfileTrajectory a b k L.1 T ha ha1 hk L.2 T.2 m Q hQ | |
| (sampledCutoffProfilePath a k L.1 T ha ha1 hk L.2 χ hχ Qp hQp) | |
| (fun t => hqb (expandingRadiusCurve L.1 T L.2 t)) f).norm_coe_le_norm ⟨T, T.2, le_rfl⟩).trans | |
| (hb L.1 T L.2 T.2 _ _ f) | |
| end DefocusingNLS | |
| end OAI | |