openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Nonlinear /CutoffEndpointContinuity.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffProfileEndpoint | |
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffPathContinuity | |
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingProfileContinuity | |
| namespace OAI | |
| /-! # Strong continuity of the actual cutoff linear endpoint -/ | |
| open scoped SchwartzMap ContDiff NNReal | |
| namespace DefocusingNLS | |
| local notation "E" => EuclideanSpace ℝ (Fin 12) | |
| local notation "Radius" => {L : ℝ // 1 ≤ L} | |
| attribute [local irreducible] cutoffProfileEndpoint expandingProfileTrajectory | |
| theorem continuous_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) : | |
| Continuous (fun p : Radius × FourierL2 => | |
| cutoffProfileEndpoint a b k ha ha1 hk m χ hχ Qp hQp Q hQ hqb T p.1 p.2) := by | |
| have hq := (continuous_sampledCutoffProfilePath a k T ha ha1 hk χ hχ Qp hQp).comp | |
| (continuous_fst : Continuous (fun p : Radius × FourierL2 => p.1)) | |
| have hc := continuous_expandingProfileTrajectory_family a b k T Q ha ha1 hk T.2 m hQ | |
| (fun p : Radius × FourierL2 => p.1) continuous_fst | |
| (fun p : Radius × FourierL2 => sampledCutoffProfilePath a k p.1.1 T ha ha1 hk p.1.2 χ hχ Qp hQp) | |
| hq (fun p t => hqb (expandingRadiusCurve p.1.1 T p.1.2 t)) | |
| (fun p : Radius × FourierL2 => p.2) continuous_snd | |
| simpa only [cutoffProfileEndpoint_apply, Function.comp_def] using | |
| (continuous_eval_const ⟨T, T.2, le_rfl⟩).comp hc | |
| end DefocusingNLS | |
| end OAI | |