| import OAI.MathematicalPhysics.AlternatingFlow.EnergyUniqueness |
| import OAI.MathematicalPhysics.AlternatingFlow.DerivativeBounds |
|
|
| namespace OAI |
|
|
| open scoped BigOperators ENNReal NNReal Topology ContDiff |
| open MeasureTheory |
| namespace AlternatingNS |
| namespace Construction |
|
|
| theorem alternating_analytic (ν : ℝ) (hν : 0 < ν) (M : Machine) (w : List ℕ) |
| (hM : M.WellFormed) (hw : M.ValidInput w) : |
| IsCompact Spatial.box ∧ |
| Smooth (force ν M w) ∧ Smooth (velocity M w) ∧ |
| SupportedIn (force ν M w) Spatial.box ∧ SupportedIn (velocity M w) Spatial.box ∧ |
| Rapid (force ν M w) ∧ Rapid (velocity M w) ∧ |
| force ν M w = residual ν (velocity M w) ∧ |
| NavierStokes ν (force ν M w) (velocity M w) (fun _ _ => 0) ∧ |
| EnergyClass (velocity M w) (fun _ _ => 0) ∧ |
| (∀ (v : Velocity) (p : Pressure), |
| NavierStokes ν (force ν M w) v p → CompetitorEnergyClass v p → |
| ∀ t, 0 ≤ t → ∀ x, v t x = velocity M w t x ∧ p t x = 0) ∧ |
| ∃ X : Space → ℝ → Space, |
| (∀ a, IsTrajectory (velocity M w) a (X a)) ∧ |
| (∀ a γ, IsTrajectory (velocity M w) a γ → ∀ t, 0 ≤ t → γ t = X a t) ∧ |
| ((∃ t : ℝ, 0 ≤ t ∧ 0 < X observedParticle t 0) ↔ M.Halts w) := by |
| obtain ⟨hf, hU, hfs, hUs, hNS, hX⟩ := smooth_forced_simulation ν M hM w hw |
| exact ⟨Spatial.box_compact, hf, hU, hfs, hUs, Bounds.force_rapid ν M w, |
| Bounds.velocity_rapid M w, rfl, hNS, velocity_energyClass M w, |
| velocity_unique ν hν M w, hX⟩ |
|
|
| end Construction |
| end AlternatingNS |
|
|
| end OAI |
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|
