openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Flow /CompactPlanarBounds.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.PlanarCompactBridge | |
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.PlanarTransition | |
| namespace OAI | |
| /-! Bounds and actual global transitions for the compact planar version of | |
| the normalized program. The periodic spatial expression agrees on the | |
| chart; outside the common support all derivatives vanish. -/ | |
| noncomputable section | |
| namespace ForcedComputation.Recorder.Planar | |
| open ShearFlows PlanarHamiltonian Set Filter | |
| open scoped Topology ContDiff NNReal | |
| def compactVelocity (I : Alternating.MachineInput) (hI : Alternating.ValidInput I) | |
| (s : ℝ) : Plane → Plane := | |
| PlanarHamiltonian.periodicVelocity | |
| (n := (actions (freshMachine I.1) (freshInput_wellFormed hI)).length) | |
| (normalizedPulse I hI) s | |
| theorem compactVelocity_smooth (I : Alternating.MachineInput) | |
| (hI : Alternating.ValidInput I) : | |
| ContDiff ℝ ∞ (fun y : ℝ × Plane => compactVelocity I hI y.1 y.2) := | |
| periodicVelocity_smooth (normalizedPulse_valid I hI) | |
| theorem compactVelocity_germ (I : Alternating.MachineInput) | |
| (hI : Alternating.ValidInput I) (s : ℝ) {x : Plane} | |
| (hx₀ : x 0 ∈ Ioo (0 : ℝ) 1) (hx₁ : x 1 ∈ Ioo (0 : ℝ) 1) : | |
| compactVelocity I hI s =ᶠ[𝓝 x] planarSlice (normalizedHamiltonian I hI) s := by | |
| have ho : IsOpen {y : Plane | y 0 ∈ Ioo (0 : ℝ) 1 ∧ y 1 ∈ Ioo (0 : ℝ) 1} := | |
| (isOpen_Ioo.preimage (continuous_apply 0)).inter | |
| (isOpen_Ioo.preimage (continuous_apply 1)) | |
| filter_upwards [ho.mem_nhds ⟨hx₀, hx₁⟩] with y hy | |
| exact normalized_planar_field_on_chart I hI s y hy.1 hy.2 | |
| theorem compactVelocity_support (I : Alternating.MachineInput) | |
| (hI : Alternating.ValidInput I) (s : ℝ) : | |
| tsupport (compactVelocity I hI s) ⊆ commonSupport (normalizedPulse I hI) := | |
| (periodicVelocity_properties (normalizedPulse_valid I hI) s).1 | |
| theorem compactVelocity_off_chart (I : Alternating.MachineInput) | |
| (hI : Alternating.ValidInput I) (s : ℝ) {x : Plane} | |
| (hx : ¬ (x 0 ∈ Ioo (0 : ℝ) 1 ∧ x 1 ∈ Ioo (0 : ℝ) 1)) : | |
| x ∉ tsupport (compactVelocity I hI s) := by | |
| intro hm | |
| have h := commonSupport_in_unit (normalizedPulse_inUnit I hI) | |
| (compactVelocity_support I hI s hm) | |
| exact hx ⟨h 0, h 1⟩ | |
| theorem compactVelocity_bound (I : Alternating.MachineInput) | |
| (hI : Alternating.ValidInput I) (s : ℝ) (x : Plane) : | |
| ‖compactVelocity I hI s x‖ ≤ | |
| ((SpatialExpression.suspensionCode (normalizedHamiltonian I hI)).bound [] : ℝ) := by | |
| by_cases hx : x 0 ∈ Ioo (0 : ℝ) 1 ∧ x 1 ∈ Ioo (0 : ℝ) 1 | |
| · rw [(compactVelocity_germ I hI s hx.1 hx.2).eq_of_nhds] | |
| exact planarSlice_bound (normalizedHamiltonian_valid I hI) | |
| (normalizedHamiltonian_noTime I hI) s x | |
| · rw [image_eq_zero_of_notMem_tsupport (compactVelocity_off_chart I hI s hx), norm_zero] | |
| exact Nat.cast_nonneg _ | |
| theorem compactVelocity_derivative_bounds (I : Alternating.MachineInput) | |
| (hI : Alternating.ValidInput I) (s : ℝ) (x : Plane) : | |
| ‖fderiv ℝ (compactVelocity I hI s) x‖ ≤ | |
| (planarVariationCoefficient (normalizedHamiltonian I hI) : ℝ) ∧ | |
| ‖fderiv ℝ (fderiv ℝ (compactVelocity I hI s)) x‖ ≤ | |
| (planarVariationCoefficient (normalizedHamiltonian I hI) : ℝ) := by | |
| by_cases hx : x 0 ∈ Ioo (0 : ℝ) 1 ∧ x 1 ∈ Ioo (0 : ℝ) 1 | |
| · have he := compactVelocity_germ I hI s hx.1 hx.2 | |
| rw [he.fderiv_eq, he.fderiv.fderiv_eq] | |
| exact planar_derivative_bounds (normalizedHamiltonian_valid I hI) | |
| (normalizedHamiltonian_noTime I hI) s x | |
| · have hn := compactVelocity_off_chart I hI s hx | |
| have hn' : x ∉ tsupport (fderiv ℝ (compactVelocity I hI s)) := | |
| fun h => hn (tsupport_fderiv_subset ℝ h) | |
| constructor | |
| · rw [fderiv_of_notMem_tsupport ℝ hn, norm_zero] | |
| exact Nat.cast_nonneg _ | |
| · have hz := fderiv_of_notMem_tsupport ℝ hn' | |
| exact (congrArg norm hz).le.trans ((ContinuousLinearMap.opNorm_zero : | |
| ‖(0 : Plane →L[ℝ] (Plane →L[ℝ] Plane))‖ = 0).le.trans (Nat.cast_nonneg _)) | |
| theorem compactVelocity_lipschitz (I : Alternating.MachineInput) | |
| (hI : Alternating.ValidInput I) (s : ℝ) : | |
| LipschitzWith (planarVariationCoefficient (normalizedHamiltonian I hI) : ℝ≥0) | |
| (compactVelocity I hI s) := by | |
| have hi : ContDiff ℝ ∞ (fun x : Plane => (s, x)) := | |
| contDiff_const.prodMk contDiff_id | |
| have hs := (compactVelocity_smooth I hI).comp hi | |
| apply lipschitzWith_of_nnnorm_fderiv_le (hs.differentiable (by simp)) | |
| intro x | |
| exact_mod_cast (compactVelocity_derivative_bounds I hI s x).1 | |
| theorem compactVelocity_transitions (I : Alternating.MachineInput) | |
| (hI : Alternating.ValidInput I) : | |
| ∃ Ψ, IsPlanarTransition (compactVelocity I hI) Ψ := by | |
| apply exists_planarTransition_of_bounds (B := | |
| ((SpatialExpression.suspensionCode (normalizedHamiltonian I hI)).bound [] : ℝ≥0)) | |
| (compactVelocity_lipschitz I hI) | |
| · intro x | |
| have hi : Continuous (fun t : ℝ => (t, x)) := continuous_id.prodMk continuous_const | |
| have h := (compactVelocity_smooth I hI).continuous.comp hi | |
| exact h | |
| · exact compactVelocity_bound I hI | |
| end ForcedComputation.Recorder.Planar | |
| end | |
| end OAI | |