OAI/MathematicalPhysics/ContinuumCoulomb/Nuclei/MoserFlux.lean

GitHub source file

9 declarations, 0 contain sorry, 0 declared as axioms

The explicit density homotopy and continuity equation used to transport uniform slab density in the unit-charge construction. The identities describe the explicit field and its density homotopy.

  1. def OAI.ContinuumCoulomb.homotopyDensity line 14
  2. def OAI.ContinuumCoulomb.moserVelocity line 17
  3. theorem OAI.ContinuumCoulomb.homotopyDensity_bounds line 21
  4. theorem OAI.ContinuumCoulomb.homotopyDensity_pos line 34
  5. theorem OAI.ContinuumCoulomb.homotopyDensity_endpoints line 41
  6. theorem OAI.ContinuumCoulomb.homotopyDensity_time_deriv line 46
  7. theorem OAI.ContinuumCoulomb.moserVelocity_zero line 51
  8. theorem OAI.ContinuumCoulomb.moser_flux line 59
  9. theorem OAI.ContinuumCoulomb.moser_continuity_equation line 69

    Source documentation

    The continuity equation holds as an exact identity before any use of flow transport or a numerical approximation to the flow.

    theorem moser_continuity_equation {rho : ℝ} (hrho : 0 < rho)
    (V : Position → ℝ) (hV : ContDiff ℝ 2 V)
    (hbound : ∀ x, |manufacturedCharge V x| ≤ rho / 2)
    {t : ℝ} (ht0 : 0 ≤ t) (ht1 : t ≤ 1) (x : Position) :
    deriv (fun s => homotopyDensity rho V s x) t +
      (∑ a : Fin 3, dirPartial
        (fun y => homotopyDensity rho V t y * moserVelocity rho V t y a) (axis a) x) = 0 := by
…

    Mentions: OAI.ContinuumCoulomb.Position, OAI.ContinuumCoulomb.manufacturedCharge, OAI.ContinuumCoulomb.homotopyDensity, OAI.ContinuumCoulomb.moserVelocity