rs_implies_global_regularity_2d

formal source

  44At this stage, “regularity” is represented by the existence of a limit Fourier trajectory
  45`u : ℝ → FourierState2D` together with the packaged identification property.
  46-/
  47theorem rs_implies_global_regularity_2d
  48    (H : RSNS2DPipelineHypothesis) :
  49    ∃ u : ℝ → FourierState2D,
  50      (∀ t : ℝ, ∀ k : Mode2,
  51        Filter.Tendsto (fun N : ℕ => (extendByZero (H.Hbounds.uN N t)) k) Filter.atTop
  52          (nhds ((u t) k)))
  53        ∧ H.Hid.IsSolution u
  54        ∧ (∀ t ≥ 0, ∀ k : Mode2, ‖(u t) k‖ ≤ H.Hbounds.B) := by
  55  -- The result is exactly the continuum limit theorem from M5.
  56  simpa using (continuum_limit_exists H.Hbounds H.Hconv H.Hid)
  57
  58/-- Variant of the top-level theorem where the “identification” is the **proved** coefficient bound:
  59we do not need a separate `IdentificationHypothesis` argument for this minimal `IsSolution` notion. -/
  60theorem rs_implies_global_regularity_2d_coeffBound
  61    (Hbounds : UniformBoundsHypothesis)
  62    (Hconv : ConvergenceHypothesis Hbounds) :
  63    ∃ u : ℝ → FourierState2D,
  64      (∀ t : ℝ, ∀ k : Mode2,
  65        Filter.Tendsto (fun N : ℕ => (extendByZero (Hbounds.uN N t)) k) Filter.atTop
  66          (nhds ((u t) k)))
  67        ∧ (∀ t ≥ 0, ∀ k : Mode2, ‖(u t) k‖ ≤ Hbounds.B) := by
  68  -- Take the limit object from convergence, and use the derived bound lemma.
  69  refine ⟨Hconv.u, ?_, ?_⟩
  70  · simpa using Hconv.converges
  71  · intro t ht k
  72    simpa using (ConvergenceHypothesis.coeff_bound_of_uniformBounds (HC := Hconv) t ht k)
  73
  74/-- Variant of the top-level theorem where the derived “identification” is:
  75
  76- the **proved** coefficient bound (from uniform bounds + convergence), and
  77- the **proved** divergence-free constraint (passed to the limit under modewise convergence),