File size: 2,650 Bytes
37bcf52 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 | 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
|