Stability Energy Quadratic

Supporting estimates for the Navier–Stokes development.

theorem CKN.stability_energy_quadraticTerm_tendsto {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) (huConv : Filter.Tendsto (fun (n : ℕ) => MeasureTheory.eLpNorm (u n - v) 3 (MeasureTheory.volume.restrict (spaceTimeSet Ω' J))) Filter.atTop (nhds 0)) (ψ : Foundation.Parabolic.Vec3 × ℝ → ℝ) (hψ : ψ ∈ spaceTimeTestFunction Ω I) : Filter.Tendsto (fun (n : ℕ) => ∫ (z : Foundation.Parabolic.ParabolicPoint) in spaceTimeSet Ω' J, ∑ i : Fin 3, u n z i * u n z i * (timePartial ψ z + ∑ j : Fin 3, spatialSecondPartial ψ j j z)) Filter.atTop (nhds (∫ (z : Foundation.Parabolic.ParabolicPoint) in spaceTimeSet Ω' J, ∑ i : Fin 3, v z i * v z i * (timePartial ψ z + ∑ j : Fin 3, spatialSecondPartial ψ j j z)))

The quadratic term on the right side of (S4) passes to a strong local L³ limit.