openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Flow /PlanarFlowSmooth.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.FlowNonautonomous | |
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.PlanarTransition | |
| namespace OAI | |
| /-! The time-augmented smooth-flow theorem applied to the actual finite | |
| Hamiltonian expression used by the planar processor. -/ | |
| noncomputable section | |
| namespace ForcedComputation | |
| open ShearFlows | |
| open scoped ContDiff NNReal | |
| def phaseSpacePoint (p : ℝ × Plane) : Space := atHeight p.2 p.1 | |
| theorem phaseSpacePoint_smooth : ContDiff ℝ ∞ phaseSpacePoint := by | |
| apply contDiff_pi.mpr | |
| intro j | |
| fin_cases j | |
| · change ContDiff ℝ ∞ (fun p : ℝ × Plane => p.2 0) | |
| exact (contDiff_apply ℝ ℝ 0).comp contDiff_snd | |
| · change ContDiff ℝ ∞ (fun p : ℝ × Plane => p.2 1) | |
| exact (contDiff_apply ℝ ℝ 1).comp contDiff_snd | |
| · exact contDiff_fst | |
| theorem phaseSpacePoint_lipschitz : LipschitzWith 1 phaseSpacePoint := by | |
| apply LipschitzWith.of_dist_le_mul | |
| intro p q | |
| rw [NNReal.coe_one, one_mul, dist_eq_norm, dist_eq_norm] | |
| apply (pi_norm_le_iff_of_nonneg (norm_nonneg (p - q))).mpr | |
| intro j | |
| fin_cases j | |
| · exact (norm_le_pi_norm (p.2 - q.2) 0).trans (norm_snd_le (p - q)) | |
| · exact (norm_le_pi_norm (p.2 - q.2) 1).trans (norm_snd_le (p - q)) | |
| · exact norm_fst_le (p - q) | |
| theorem horizontal_smooth : ContDiff ℝ ∞ horizontal := by | |
| apply contDiff_pi.mpr | |
| intro j | |
| fin_cases j <;> exact contDiff_apply ℝ ℝ _ | |
| theorem horizontal_lipschitz : LipschitzWith 1 horizontal := by | |
| apply LipschitzWith.of_dist_le_mul | |
| intro x y | |
| have he : horizontal (x - y) = horizontal x - horizontal y := by | |
| ext j | |
| fin_cases j <;> rfl | |
| simpa only [NNReal.coe_one, one_mul, dist_eq_norm, he] using horizontal_norm_le (x - y) | |
| theorem planarSlice_joint_smooth {H : FieldExpr} (hH : H.Valid) : | |
| ContDiff ℝ ∞ (fun p : ℝ × Plane => planarSlice H p.1 p.2) := by | |
| have he : (fun p : ℝ × Plane => planarSlice H p.1 p.2) = | |
| horizontal ∘ SpatialExpression.suspensionField H ∘ phaseSpacePoint := by | |
| funext p | |
| exact planarSlice_eq_horizontal hH p.1 p.2 | |
| rw [he] | |
| exact horizontal_smooth.comp | |
| ((SpatialExpression.suspensionField_smooth hH).comp phaseSpacePoint_smooth) | |
| theorem planarSlice_joint_lipschitz {H : FieldExpr} (hH : H.Valid) | |
| (hT : SpatialExpression.NoTime H) : | |
| LipschitzWith (suspensionLipschitzBound H : ℝ≥0) | |
| (fun p : ℝ × Plane => planarSlice H p.1 p.2) := by | |
| have h := horizontal_lipschitz.comp | |
| ((suspensionField_lipschitz hH hT).comp phaseSpacePoint_lipschitz) | |
| simpa only [one_mul, mul_one, Function.comp_def, phaseSpacePoint, | |
| ← planarSlice_eq_horizontal hH] using h | |
| theorem planarTransition_joint_smooth {H : FieldExpr} (hH : H.Valid) | |
| (hT : SpatialExpression.NoTime H) {Ω : ℝ → ℝ → Plane → Plane} | |
| (hΩ : IsPlanarTransition (planarSlice H) Ω) : | |
| ContDiff ℝ ∞ (fun p : ℝ × (ℝ × Plane) => Ω p.1 p.2.1 p.2.2) := | |
| Flow.transition_contDiff | |
| ⟨_, (planarSlice_joint_smooth hH).continuous⟩ | |
| (planarSlice_joint_smooth hH) (planarSlice_joint_lipschitz hH hT) | |
| hΩ.ode hΩ.initial | |
| theorem planarTransition_backward_smooth {H : FieldExpr} (hH : H.Valid) | |
| (hT : SpatialExpression.NoTime H) {Ω : ℝ → ℝ → Plane → Plane} | |
| (hΩ : IsPlanarTransition (planarSlice H) Ω) : | |
| ContDiff ℝ ∞ (fun p : ℝ × Plane => Ω p.1 (-p.1) p.2) := | |
| (planarTransition_joint_smooth hH hT hΩ).comp | |
| (contDiff_fst.prodMk (contDiff_fst.neg.prodMk contDiff_snd)) | |
| end ForcedComputation | |
| end | |
| end OAI | |