Stability Momentum Nonlinear

Supporting estimates for the Navier–Stokes development.

theorem CKN.stability_momentum_nonlinearTerm_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)) (φ : Foundation.Parabolic.Vec3 × ℝ → Foundation.Parabolic.Vec3) (hφ : φ ∈ spaceTimeTestFunction Ω I) : Filter.Tendsto (fun (n : ℕ) => ∫ (z : Foundation.Parabolic.ParabolicPoint) in spaceTimeSet Ω' J, ∑ i : Fin 3, ∑ j : Fin 3, u n z i * u n z j * spatialPartial (fun (w : Foundation.Parabolic.ParabolicPoint) => φ w i) j z) Filter.atTop (nhds (∫ (z : Foundation.Parabolic.ParabolicPoint) in spaceTimeSet Ω' J, ∑ i : Fin 3, ∑ j : Fin 3, v z i * v z j * spatialPartial (fun (w : Foundation.Parabolic.ParabolicPoint) => φ w i) j z))

The quadratic velocity term in (S3) passes to a strong local L³ limit.