openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Nonlinear /CutoffLinearizedPropagator.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Profile.RadialMatchedProfileBounds | |
| import OAI.MathematicalPhysics.DefocusingNLS.Nonlinear.CutoffMovingPath | |
| import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingProfilePropagator | |
| namespace OAI | |
| /-! # Uniform finite-slab evolution for the actual cutoff profile | |
| The matched profile's proved symbol bounds give a single norm bound on | |
| all of its expanding-torus samples. The actual linearized potential then | |
| has bounded real-linear propagators uniformly in the starting radius. | |
| -/ | |
| open Set | |
| open scoped SchwartzMap ContDiff | |
| namespace DefocusingNLS | |
| open ProfileCertificate | |
| local notation "E" => EuclideanSpace ℝ (Fin 12) | |
| theorem exists_matchedCutoffProfile_uniform_bound (n : ℕ) (z : ProfileMatchingBall) | |
| (hX : HasRadialExterior (radialShootingNu (n + radialInnerShootingThreshold) z) | |
| (n + radialInnerShootingThreshold) (radialShootingM z) (Real.log innerBoundaryRadius)) | |
| (hz : radialMatchingMap n z = 0) (k : ℝ) (hk : 8 < k) | |
| (χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ)) | |
| (hχzero : ∀ y : E, 1 ≤ ‖y‖ → χ y = 0) : | |
| let ha := (radialShootingA_bounds n (profileMatchingParameter z)).1 | |
| let ha1 := (radialShootingA_bounds n (profileMatchingParameter z)).2 | |
| let hQ := radialMatchedCartesian_contDiff n z hX hz | |
| ∃ R : ℝ, 0 ≤ R ∧ ∀ (L T : ℝ) (hL : 1 ≤ L) (t : Icc (0 : ℝ) T), | |
| ‖sampledCutoffProfilePath (radialShootingA n) k L T ha ha1 hk hL | |
| χ hχ (radialMatchedCartesian n z) hQ t‖ ≤ R := by | |
| intro ha ha1 hQ | |
| obtain ⟨R, hR, hbound⟩ := radialMatchedCutoff_sampling_bound n z hX hz k hk χ hχ hχzero | |
| refine ⟨R, hR, ?_⟩ | |
| intro L T hL t | |
| exact hbound (expandingRadiusCurve L T hL t).1 (expandingRadiusCurve L T hL t).2 | |
| theorem exists_matchedCutoffLinearized_propagator (n : ℕ) (z : ProfileMatchingBall) | |
| (hX : HasRadialExterior (radialShootingNu (n + radialInnerShootingThreshold) z) | |
| (n + radialInnerShootingThreshold) (radialShootingM z) (Real.log innerBoundaryRadius)) | |
| (hz : radialMatchingMap n z = 0) (k : ℝ) (hk : 8 < k) | |
| (χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport (χ : E → ℂ)) | |
| (hχzero : ∀ y : E, 1 ≤ ‖y‖ → χ y = 0) : | |
| let a := radialShootingA n | |
| let b := radialShootingB (profileMatchingParameter z) | |
| let m := n + radialInnerShootingThreshold | |
| let ha := (radialShootingA_bounds n (profileMatchingParameter z)).1 | |
| let ha1 := (radialShootingA_bounds n (profileMatchingParameter z)).2 | |
| let hQ := radialMatchedCartesian_contDiff n z hX hz | |
| ∃ K : ℝ, 0 ≤ K ∧ ∀ (L T : ℝ) (hL : 1 ≤ L) (hT : 0 ≤ T), | |
| let q := sampledCutoffProfilePath a k L T ha ha1 hk hL χ hχ (radialMatchedCartesian n z) hQ | |
| ∃ U : FourierL2 →L[ℝ] C(Icc (0 : ℝ) T, FourierL2), | |
| (∀ u₀ t, U u₀ t = expandingFreeStep a b k L t ha hk hL t.2.1 u₀ + | |
| expandingDuhamel a b k L ha hk hL t (expandingReactionHistory T hT | |
| (expandingProfileReaction a k T ha ha1 hk m (expandingRadiusCurve L T hL) q 0) | |
| (U u₀))) ∧ | |
| ‖U‖ ≤ (K + 1) * Real.exp ((K + 1) * T) := by | |
| intro a b m ha ha1 hQ | |
| obtain ⟨R, hR, hbound⟩ := exists_matchedCutoffProfile_uniform_bound n z hX hz k hk χ hχ hχzero | |
| obtain ⟨K, hK, hprop⟩ := exists_expandingProfile_propagator a b k ha ha1 hk m R hR | |
| refine ⟨K, hK, ?_⟩ | |
| intro L T hL hT | |
| exact hprop L T hL hT _ (hbound L T hL) | |
| end DefocusingNLS | |
| end OAI | |