openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Detector /CylinderLocalEnergyInput.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Detector.CylinderSolution | |
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.PlanarDerivativeBounds | |
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Detector.ExpandingMovingCutoff | |
| namespace OAI | |
| /-! Published local energy identity, restricted to the exact cylinder and | |
| the expanding compact cutoffs used here. | |
| F. Gancedo and A. Hidalgo-Torné, On the Cauchy problem for 3D Navier–Stokes | |
| helical vortex filament, Advances in Mathematics 471 (2025), 110268, | |
| proof of Theorem 5.6, equation (5.4), printed p.23 (arXiv:2311.15413). | |
| Only its integration-by-parts identity is retained. The helical uniqueness | |
| theorem is not used. The identity below assumes the strong classical class | |
| and does not assert uniqueness, a cutoff limit, or any detector property. | |
| -/ | |
| noncomputable section | |
| namespace ForcedComputation.VelocityDetector | |
| open ShearFlows ExpandingDetector MeasureTheory | |
| def cylinderWeight (n : ℕ) (x : Plane) : ℝ := | |
| concentrationCutoff ((n : ℝ) + 1) 0 x ^ 2 | |
| def cylinderDifference (u v : Velocity) (t : ℝ) (x : Space) : Space := u (t,x) - v (t,x) | |
| def localizedDifferenceEnergy (u v : Velocity) (n : ℕ) (t : ℝ) : ℝ := | |
| ∫ y, cylinderWeight n y.1 * | |
| dot (cylinderDifference u v t (atHeight y.1 y.2)) | |
| (cylinderDifference u v t (atHeight y.1 y.2)) ∂cylinderMeasure | |
| def localizedDifferenceDissipation (u v : Velocity) (n : ℕ) (t : ℝ) : ℝ := | |
| ∫ y, cylinderWeight n y.1 * ∑ j : Fin 3, | |
| dot (derivative (cylinderDifference u v t) j (atHeight y.1 y.2)) | |
| (derivative (cylinderDifference u v t) j (atHeight y.1 y.2)) ∂cylinderMeasure | |
| def localizedDifferenceInteraction (u v : Velocity) (n : ℕ) (t : ℝ) : ℝ := | |
| ∫ y, cylinderWeight n y.1 * | |
| dot (cylinderDifference u v t (atHeight y.1 y.2)) | |
| (fderiv ℝ (fun x => u (t,x)) (atHeight y.1 y.2) | |
| (cylinderDifference u v t (atHeight y.1 y.2))) ∂cylinderMeasure | |
| def localizedDiffusionError (u v : Velocity) (n : ℕ) (t : ℝ) : ℝ := | |
| ∫ y, scalarLaplacian (cylinderWeight n) y.1 * | |
| dot (cylinderDifference u v t (atHeight y.1 y.2)) | |
| (cylinderDifference u v t (atHeight y.1 y.2)) ∂cylinderMeasure | |
| def localizedTransportError (u v : Velocity) (n : ℕ) (t : ℝ) : ℝ := | |
| ∫ y, fderiv ℝ (cylinderWeight n) y.1 | |
| (horizontalLinear (v (t, atHeight y.1 y.2))) * | |
| dot (cylinderDifference u v t (atHeight y.1 y.2)) | |
| (cylinderDifference u v t (atHeight y.1 y.2)) ∂cylinderMeasure | |
| def localizedPressureError (u v : Velocity) (p q : Pressure) (n : ℕ) (t : ℝ) : ℝ := | |
| ∫ y, (p (t, atHeight y.1 y.2) - q (t, atHeight y.1 y.2)) * | |
| fderiv ℝ (cylinderWeight n) y.1 | |
| (horizontalLinear (cylinderDifference u v t (atHeight y.1 y.2))) ∂cylinderMeasure | |
| def CylinderLocalEnergyIdentity : Prop := | |
| ∀ (ν : ℝ), 0 < ν → ∀ (f u v : Velocity) (p q : Pressure), | |
| IsCylinderClassicalSolution ν f u p → IsCylinderClassicalSolution ν f v q → | |
| ∀ (n : ℕ) (t : ℝ), 0 < t → | |
| HasDerivAt (localizedDifferenceEnergy u v n) | |
| (-2 * ν * localizedDifferenceDissipation u v n t | |
| - 2 * localizedDifferenceInteraction u v n t | |
| + ν * localizedDiffusionError u v n t | |
| + localizedTransportError u v n t | |
| + 2 * localizedPressureError u v p q n t) t | |
| end ForcedComputation.VelocityDetector | |
| end | |
| end OAI | |