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be44ea5 | 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 | import NavierStokesSpec
/-!
# Time-dependent bridge for the kinematic alignment predicate
This module does not define the Navier–Stokes equations and does not claim that
`vorticityOf` is the curl of `u`. It only gives a typed interface: a
time-dependent velocity is paired with an explicitly supplied time-dependent
vorticity assignment, and alignment is stated at every nonnegative time.
The missing PDE theorem is the construction of a suitable `vorticityOf` from a
periodic divergence-free velocity solution together with a proof that the
resulting alignment predicate persists. That theorem is not asserted here.
-/
namespace NavierStokesSpacetimeBridge
open NavierStokesSpec
abbrev TimeDependentVelocity := ℝ → Point3 → Vector3
abbrev TimeDependentVorticity := ℝ → VorticityField
/-- An explicit interface for assigning a time-dependent vorticity field to a
velocity field. No curl identity, regularity, periodicity, or PDE is built in. -/
def VorticityAssignment :=
TimeDependentVelocity → TimeDependentVorticity
/-- Alignment of the assigned vorticity field for every nonnegative time.
This is an assumption/interface, not a dynamical persistence theorem. -/
def AlignmentAlongVelocity
(K L ρ : ℝ)
(u : TimeDependentVelocity)
(vorticityOf : VorticityAssignment) : Prop :=
∀ t : ℝ, 0 ≤ t →
AlignmentPredicate K L ρ ((vorticityOf u) t)
/-- The direct bridge statement: an assumed time-indexed alignment hypothesis
implies the same alignment proposition at each nonnegative time. -/
theorem alignment_at_time_of_alignment_along_velocity
(K L ρ : ℝ)
(u : TimeDependentVelocity)
(vorticityOf : VorticityAssignment)
(h_alignment : AlignmentAlongVelocity K L ρ u vorticityOf)
(t : ℝ) (ht : 0 ≤ t) :
AlignmentPredicate K L ρ ((vorticityOf u) t) :=
h_alignment t ht
end NavierStokesSpacetimeBridge
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