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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 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 | 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
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