Stability S1Slice

Supporting estimates for the Navier–Stokes development.

theorem CKN.stability_suitable_limit_slice_energy_lt_top {Ω Ω' : Set Foundation.Parabolic.Vec3} {I J : Set ℝ} {q : ℝ} (u : ℕ → Foundation.Parabolic.ParabolicPoint → Foundation.Parabolic.Vec3) (Du : ℕ → Foundation.Parabolic.ParabolicPoint → Fin 3 → Foundation.Parabolic.Vec3) (p : ℕ → Foundation.Parabolic.ParabolicPoint → ℝ) (v : Foundation.Parabolic.ParabolicPoint → Foundation.Parabolic.Vec3) (hsol : ∀ (n : ℕ), IsSuitableWeakSolution Ω I q (u n) (Du n) (p n) 0) (hbox : localBox Ω I Ω' J) (hconv : Filter.Tendsto (fun (n : ℕ) => MeasureTheory.eLpNorm (u n - v) 3 (MeasureTheory.volume.restrict (spaceTimeSet Ω' J))) Filter.atTop (nhds 0)) (M : ENNReal) (hM : M < ⊤) (hbound : ∀ (n : ℕ), essSup (fun (t : ℝ) => ∫⁻ (x : Foundation.Parabolic.Vec3) in Ω', ‖u n (x, t)‖ₑ ^ 2) (MeasureTheory.volume.restrict J) ≤ M) : essSup (fun (t : ℝ) => ∫⁻ (x : Foundation.Parabolic.Vec3) in Ω', ‖v (x, t)‖ₑ ^ 2) (MeasureTheory.volume.restrict J) < ⊤

Strong local L³ convergence preserves the uniform time-slice energy bound of suitable solutions (thm:stability).