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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 | import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.CompactPlanarProcessor
import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.NormalizedPlanarProgram
import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Flow.PlanarPerturbations
namespace OAI
/-! Agreement of the compact planar field with the finite periodic scalar
expression on the open unit-square chart, at every phase. -/
noncomputable section
namespace ForcedComputation
open ShearFlows Set
theorem periodic_eq_of_eqOn_unit {E : Type*} {f g : ℝ → E}
(hf : Function.Periodic f 1) (hg : Function.Periodic g 1)
(he : EqOn f g (Icc (0 : ℝ) 1)) : f = g := by
funext t
have hf' : f (Int.fract t) = f t := by
simpa only [Int.fract, mul_one] using hf.sub_int_mul_eq (x := t) ⌊t⌋
have hg' : g (Int.fract t) = g t := by
simpa only [Int.fract, mul_one] using hg.sub_int_mul_eq (x := t) ⌊t⌋
exact hf'.symm.trans ((he ⟨Int.fract_nonneg t, (Int.fract_lt_one t).le⟩).trans hg')
theorem planarSlice_periodic {H : FieldExpr}
(hP : CubePeriodic 1 (SpatialExpression.spatialValue H)) (x : Plane) :
Function.Periodic (fun s => planarSlice H s x) 1 := by
intro s
have he : (fun Y => SpatialExpression.spatialValue H (atHeight Y (s + 1))) =
(fun Y => SpatialExpression.spatialValue H (atHeight Y s)) := by
funext Y
have hv : atHeight Y (s + 1) = atHeight Y s + latticeVector 1 ![0, 0, 1] := by
ext j
fin_cases j <;> simp [atHeight, latticeVector]
rw [hv]
exact hP _ _
change PlanarHamiltonian.field _ x = PlanarHamiltonian.field _ x
rw [he]
namespace Recorder.Planar
open PlanarHamiltonian
theorem normalized_planar_field_on_chart (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) :
PlanarHamiltonian.periodicVelocity (normalizedPulse I hI) s x =
planarSlice (normalizedHamiltonian I hI) s x := by
have he : (fun t => PlanarHamiltonian.periodicVelocity (normalizedPulse I hI) t x) =
(fun t => planarSlice (normalizedHamiltonian I hI) t x) := by
apply periodic_eq_of_eqOn_unit (PlanarHamiltonian.periodicVelocity_periodic _ x)
(planarSlice_periodic (normalizedHamiltonian_periodic I hI) x)
intro t ht
exact (PlanarHamiltonian.periodicVelocity_on_unit (normalizedPulse_valid I hI)
(normalizedPulse_inUnit I hI) ht x).trans
(processorExpression_planar_field (normalizedPulse_valid I hI)
(normalizedPulse_inUnit I hI) ht hx₀ hx₁).symm
exact congrFun he s
end Recorder.Planar
end ForcedComputation
end
end OAI
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