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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 | import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.BalancedStretch
import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.BalancedPlanarProgram
import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.CompactTimeSupport
namespace OAI
/-! The repeated compact Euclidean field depends only on the machine. Its
fixed vertical cutoff leaves the planar computation on height zero exact. -/
noncomputable section
namespace ForcedComputation.BalancedBody
open ShearFlows Set PlanarHamiltonian Recorder.Planar
open scoped ContDiff
def verticalCutoff : ℝ → ℝ := closedCutoff (-2) (-1) 1 2
theorem verticalCutoff_smooth : ContDiff ℝ ∞ verticalCutoff :=
closedCutoff_smooth _ _ _ _
theorem verticalCutoff_zero : verticalCutoff 0 = 1 :=
closedCutoff_plateau (by norm_num) (by norm_num) (by constructor <;> norm_num)
def raw (M : Alternating.Machine) (hM : M.WellFormed) : Velocity :=
CompactLift.velocity verticalCutoff (BalancedPlanar.bodyVelocity M hM)
def velocity (M : Alternating.Machine) (hM : M.WellFormed) : Velocity :=
BalancedStretch.velocity (raw M hM)
def rawSupport : Set Space := Icc ![0, 0, -2] ![1, 1, 2]
def support : Set Space := BalancedStretch.stretch '' rawSupport
theorem rawSupport_compact : IsCompact rawSupport := isCompact_Icc
theorem support_compact : IsCompact support :=
rawSupport_compact.image BalancedStretch.stretch.continuous
theorem raw_smooth (M : Alternating.Machine) (hM : M.WellFormed) :
ContDiff ℝ ∞ (raw M hM) :=
CompactLift.velocity_smooth verticalCutoff_smooth
(compactVelocity_smooth (BalancedPlanar.referenceInput M) (BalancedPlanar.referenceValid hM))
theorem velocity_smooth (M : Alternating.Machine) (hM : M.WellFormed) :
ContDiff ℝ ∞ (velocity M hM) := BalancedStretch.velocity_smooth (raw_smooth M hM)
theorem raw_support (M : Alternating.Machine) (hM : M.WellFormed) (t : ℝ) :
tsupport (fun x => raw M hM (t, x)) ⊆ rawSupport := by
apply closure_minimal _ rawSupport_compact.isClosed
intro x hx
have hχ : verticalCutoff (x 2) ≠ 0 := by
intro hz
apply hx
simp only [raw, CompactLift.velocity, CompactLift.spatial, hz, zero_smul]
have hV : BalancedPlanar.bodyVelocity M hM t (horizontal x) ≠ 0 := by
intro hz
apply hx
simp only [raw, CompactLift.velocity, CompactLift.spatial, VelocityDetector.triangularLift,
VelocityDetector.horizontalLinear_eq, hz, map_zero, zero_smul, add_zero, smul_zero]
have hp := commonSupport_in_unit
(normalizedPulse_inUnit (BalancedPlanar.referenceInput M) (BalancedPlanar.referenceValid hM))
(compactVelocity_support (BalancedPlanar.referenceInput M) (BalancedPlanar.referenceValid hM)
t (subset_tsupport _ hV))
have hz := closedCutoff_support (by norm_num : (-2 : ℝ) < -1)
(by norm_num : (1 : ℝ) < 2) hχ
constructor <;> intro j <;> fin_cases j
· exact (hp 0).1.le
· exact (hp 1).1.le
· exact hz.1.le
· exact (hp 0).2.le
· exact (hp 1).2.le
· exact hz.2.le
theorem velocity_support (M : Alternating.Machine) (hM : M.WellFormed) (t : ℝ) :
tsupport (fun x => velocity M hM (t, x)) ⊆ support := by
change tsupport (BalancedStretch.push (fun x => raw M hM (t,x))) ⊆
BalancedStretch.stretch '' rawSupport
exact BalancedStretch.push_support (a := fun x => raw M hM (t,x))
(K := rawSupport) rawSupport_compact.isClosed (raw_support M hM t)
theorem raw_divergence (M : Alternating.Machine) (hM : M.WellFormed) (t : ℝ) (x : Space) :
divergence (fun y => raw M hM (t, y)) x = 0 := by
have hs := field_smooth (periodicHamiltonian_slice_smooth
(normalizedPulse (BalancedPlanar.referenceInput M) (BalancedPlanar.referenceValid hM)) t)
change ContDiff ℝ ∞ (BalancedPlanar.bodyVelocity M hM t) at hs
rw [show (fun y => raw M hM (t, y)) =
CompactLift.spatial verticalCutoff (BalancedPlanar.bodyVelocity M hM t) from rfl,
CompactLift.spatial_divergence verticalCutoff_smooth hs]
have hz : PlanarHamiltonian.divergence (BalancedPlanar.bodyVelocity M hM t)
(horizontal x) = 0 :=
(periodicVelocity_properties
(normalizedPulse_valid (BalancedPlanar.referenceInput M) (BalancedPlanar.referenceValid hM)) t).2.1 _
rw [hz, mul_zero]
theorem velocity_divergence (M : Alternating.Machine) (hM : M.WellFormed) :
Solenoidal (velocity M hM) :=
BalancedStretch.velocity_divergence (raw_divergence M hM)
theorem raw_periodic (M : Alternating.Machine) (hM : M.WellFormed) :
TimePeriodic (raw M hM) := by
intro t x
have he : BalancedPlanar.bodyVelocity M hM (t + 1) = BalancedPlanar.bodyVelocity M hM t :=
funext (fun y => periodicVelocity_periodic _ y t)
simp only [raw, CompactLift.velocity, he]
theorem velocity_periodic (M : Alternating.Machine) (hM : M.WellFormed) :
TimePeriodic (velocity M hM) := by
intro t x
change BalancedStretch.stretch (raw M hM (t + 1, BalancedStretch.stretch.symm x)) =
BalancedStretch.stretch (raw M hM (t, BalancedStretch.stretch.symm x))
rw [raw_periodic M hM]
theorem velocity_bounded (M : Alternating.Machine) (hM : M.WellFormed) :
BoundedMixedDerivatives (velocity M hM) :=
boundedMixed_of_compact_periodic (velocity_smooth M hM) support_compact
(velocity_support M hM) (velocity_periodic M hM)
end ForcedComputation.BalancedBody
end
end OAI
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