Stability Momentum Viscous

Supporting estimates for the Navier–Stokes development.

theorem CKN.stability_momentum_viscousTerm_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) (Dv : Foundation.Parabolic.ParabolicPoint → Fin 3 → Foundation.Parabolic.Vec3) (hsol : ∀ (n : ℕ), IsSuitableWeakSolution Ω I q (u n) (Du n) (p n) 0) (hbox : localBox Ω I Ω' J) (hDv : MeasureTheory.MemLp Dv 2 (MeasureTheory.volume.restrict (spaceTimeSet Ω' J))) (hDuWeak : ∀ (i j : Fin 3) (w : Foundation.Parabolic.Vec3 × ℝ → ℝ), MeasureTheory.MemLp w 2 ((MeasureTheory.volume.restrict Ω').prod (MeasureTheory.volume.restrict J)) → Filter.Tendsto (fun (n : ℕ) => ∫ (z : Foundation.Parabolic.Vec3 × ℝ), Du n (z.1, z.2) i j * w z ∂(MeasureTheory.volume.restrict Ω').prod (MeasureTheory.volume.restrict J)) Filter.atTop (nhds (∫ (z : Foundation.Parabolic.Vec3 × ℝ), Dv (z.1, z.2) i j * w z ∂(MeasureTheory.volume.restrict Ω').prod (MeasureTheory.volume.restrict J)))) (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, Du n z i 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, Dv z i j * spatialPartial (fun (w : Foundation.Parabolic.ParabolicPoint) => φ w i) j z))

The viscous term in (S3) passes through weak local L² convergence of the gradient.