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f18e7de | 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 | 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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