openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Flow /CompactPlanarFlow.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.CompactPlanarBounds | |
| namespace OAI | |
| /-! The compact planar flow fixes every point off its common support and | |
| preserves that support. In particular, its trajectories cannot leave the | |
| unit-square chart. -/ | |
| noncomputable section | |
| namespace ForcedComputation.Recorder.Planar | |
| open ShearFlows PlanarHamiltonian Set | |
| theorem compactTransition_fixed (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 x = x := by | |
| symm | |
| apply hΨ.unique a x (fun _ => x) rfl | |
| intro s | |
| have hn : x ∉ tsupport (compactVelocity I hI (a + s)) := | |
| fun h => hx (compactVelocity_support I hI (a + s) h) | |
| rw [image_eq_zero_of_notMem_tsupport hn] | |
| exact hasDerivAt_const s x | |
| theorem compactTransition_support (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 x ∈ commonSupport (normalizedPulse I hI) := by | |
| by_contra hn | |
| have he := compactTransition_fixed I hI hΨ hn a t | |
| have hxy : x = Ψ a t x := (hΨ.bijective a t).1 he.symm | |
| exact hn (hxy ▸ hx) | |
| theorem compactTransition_chart (I : Alternating.MachineInput) | |
| (hI : Alternating.ValidInput I) {Ψ : ℝ → ℝ → Plane → Plane} | |
| (hΨ : IsPlanarTransition (compactVelocity I hI) Ψ) {x : Plane} | |
| (hx : ∀ j : Fin 2, 0 < x j ∧ x j < 1) (a t : ℝ) : | |
| ∀ j : Fin 2, 0 < Ψ a t x j ∧ Ψ a t x j < 1 := by | |
| by_cases hm : x ∈ commonSupport (normalizedPulse I hI) | |
| · exact commonSupport_in_unit (normalizedPulse_inUnit I hI) | |
| (compactTransition_support I hI hΨ hm a t) | |
| · rw [compactTransition_fixed I hI hΨ hm a t] | |
| exact hx | |
| theorem compactTransition_eq_slice (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 x = Ω a t x := by | |
| apply hΩ.unique a x (fun s => Ψ a s x) (hΨ.initial a x) | |
| intro s | |
| have hy := compactTransition_chart I hI hΨ hx a s | |
| have he := normalized_planar_field_on_chart I hI (a + s) (Ψ a s x) (hy 0) (hy 1) | |
| exact (hΨ.ode a s x).congr_deriv he | |
| end ForcedComputation.Recorder.Planar | |
| end | |
| end OAI | |