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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 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 | 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
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