openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Flow /CompactPlanarVariations.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.CompactPlanarFlow | |
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.PlanarReverseVariations | |
| namespace OAI | |
| /-! Quantitative estimates for the compact planar extension, assuming the | |
| variational equations for the actual periodic transition map. -/ | |
| noncomputable section | |
| namespace ForcedComputation.Recorder.Planar | |
| open ShearFlows PlanarHamiltonian Set Filter | |
| open scoped Topology | |
| theorem compactTransition_germ (I : Alternating.MachineInput) | |
| (hI : Alternating.ValidInput I) {Ψ Ω : ℝ → ℝ → Plane → Plane} | |
| (hΨ : IsPlanarTransition (compactVelocity I hI) Ψ) | |
| (hΩ : IsPlanarTransition (planarSlice (normalizedHamiltonian I hI)) Ω) | |
| {x : Plane} (hx : ∀ j : Fin 2, 0 < x j ∧ x j < 1) (a t : ℝ) : | |
| Ψ a t =ᶠ[𝓝 x] Ω a t := by | |
| have ho : IsOpen {y : Plane | ∀ j : Fin 2, 0 < y j ∧ y j < 1} := by | |
| have he : {y : Plane | ∀ j : Fin 2, 0 < y j ∧ y j < 1} = | |
| ⋂ j : Fin 2, (fun y : Plane => y j) ⁻¹' Ioo 0 1 := by | |
| ext y | |
| simp only [mem_ofPred_eq, mem_iInter, mem_preimage, mem_Ioo] | |
| rw [he] | |
| exact isOpen_iInter_of_finite (fun j => isOpen_Ioo.preimage (continuous_apply j)) | |
| filter_upwards [ho.mem_nhds hx] with y hy | |
| exact compactTransition_eq_slice I hI hΨ hΩ hy a t | |
| theorem compactTransition_germ_id (I : Alternating.MachineInput) | |
| (hI : Alternating.ValidInput I) {Ψ : ℝ → ℝ → Plane → Plane} | |
| (hΨ : IsPlanarTransition (compactVelocity I hI) Ψ) | |
| {x : Plane} (hx : x ∉ commonSupport (normalizedPulse I hI)) (a t : ℝ) : | |
| Ψ a t =ᶠ[𝓝 x] id := by | |
| have ho := (commonSupport_compact (normalizedPulse I hI)).isClosed.isOpen_compl | |
| filter_upwards [ho.mem_nhds hx] with y hy | |
| exact compactTransition_fixed I hI hΨ hy a t | |
| theorem compactTransition_derivative_bounds (I : Alternating.MachineInput) | |
| (hI : Alternating.ValidInput I) {Ψ Ω : ℝ → ℝ → Plane → Plane} | |
| (hΨ : IsPlanarTransition (compactVelocity I hI) Ψ) | |
| (hΩ : IsPlanarTransition (planarSlice (normalizedHamiltonian I hI)) Ω) | |
| (hv : PlanarVariations (planarSlice (normalizedHamiltonian I hI)) Ω) | |
| (a : ℝ) (x : Plane) {t : ℝ} (ht : 0 ≤ t) : | |
| ‖fderiv ℝ (Ψ a t) x‖ ≤ Real.exp ((planarFlowBound (normalizedHamiltonian I hI) : ℝ) * t) ∧ | |
| ‖fderiv ℝ (fderiv ℝ (Ψ a t)) x‖ ≤ | |
| Real.exp (2 * (planarFlowBound (normalizedHamiltonian I hI) : ℝ) * t) - | |
| Real.exp ((planarFlowBound (normalizedHamiltonian I hI) : ℝ) * t) := by | |
| by_cases hx : x ∈ commonSupport (normalizedPulse I hI) | |
| · have hg := compactTransition_germ I hI hΨ hΩ | |
| (commonSupport_in_unit (normalizedPulse_inUnit I hI) hx) a t | |
| rw [hg.fderiv.self_of_nhds, hg.fderiv.fderiv.self_of_nhds] | |
| exact planar_quantitative_variations (normalizedHamiltonian_valid I hI) | |
| (normalizedHamiltonian_noTime I hI) hv a x ht | |
| · have hg := compactTransition_germ_id I hI hΨ hx a t | |
| have hi : fderiv ℝ (id : Plane → Plane) = fun _ => ContinuousLinearMap.id ℝ Plane := | |
| funext fun _ => fderiv_id | |
| have hz : fderiv ℝ (fderiv ℝ (id : Plane → Plane)) x = 0 := by | |
| rw [hi] | |
| exact fderiv_const_apply (ContinuousLinearMap.id ℝ Plane) | |
| have hL : (0 : ℝ) ≤ planarFlowBound (normalizedHamiltonian I hI) := Nat.cast_nonneg _ | |
| constructor | |
| · rw [hg.fderiv.self_of_nhds, fderiv_id] | |
| exact (ContinuousLinearMap.norm_id_le).trans (Real.one_le_exp (mul_nonneg hL ht)) | |
| · rw [hg.fderiv.fderiv.self_of_nhds, hz] | |
| have hn : ‖(0 : Plane →L[ℝ] (Plane →L[ℝ] Plane))‖ = 0 := | |
| ContinuousLinearMap.opNorm_zero | |
| rw [hn] | |
| apply sub_nonneg.mpr | |
| apply Real.exp_le_exp.mpr | |
| nlinarith | |
| end ForcedComputation.Recorder.Planar | |
| end | |
| end OAI | |