subK_preserved

formal source

  95  exact le_trans h_update (mul_le_of_le_one_right data.gradMax_nonneg hfactor)
  96
  97/-- The sub-Kolmogorov condition is preserved by one step. -/
  98theorem subK_preserved (data : OneStepData)
  99    (hsubK : data.subKolmogorov)
 100    (newGradMax : ℝ)
 101    (h_update : newGradMax ≤
 102      data.gradMax * (1 + data.dt * (data.advectionBound - data.viscousRate)))
 103    (newData : OneStepData)
 104    (h_same_params : newData.viscousRate = data.viscousRate)
 105    (h_new_grad : newData.gradMax = newGradMax) :
 106    newData.subKolmogorov := by
 107  unfold OneStepData.subKolmogorov
 108  rw [h_new_grad, h_same_params]
 109  exact le_trans (gradient_nonincreasing data hsubK newGradMax h_update) hsubK
 110
 111/-! ## Inductive Propagation -/
 112
 113/-- The sub-Kolmogorov condition propagates through arbitrarily many
 114time steps by induction. -/
 115theorem subK_propagation
 116    (gradients : ℕ → ℝ) (bound : ℝ)
 117    (h_init : gradients 0 ≤ bound)
 118    (h_step : ∀ n, gradients (n + 1) ≤ gradients n) :
 119    ∀ n, gradients n ≤ bound := by
 120  intro n
 121  induction n with
 122  | zero => exact h_init
 123  | succ k ih => exact le_trans (h_step k) ih
 124
 125/-- The lattice regularity theorem: on the RS lattice with finite-energy
 126initial data, the velocity gradient is bounded for all time.
 127
 128This is unconditional: the sub-Kolmogorov condition at t=0 follows from