| import NavierStokesSpec |
|
|
| /-! |
| |
|
|
| 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 |
|
|
| / |
| velocity field. No curl identity, regularity, periodicity, or PDE is built in. -/ |
| def VorticityAssignment := |
| TimeDependentVelocity → TimeDependentVorticity |
|
|
| / |
| 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) |
|
|
| / |
| 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 |
|
|
