Stability S3

Supporting estimates for the Navier–Stokes development.

theorem CKN.stability_suitable_momentum {Ω : Set Foundation.Parabolic.Vec3} {I : 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) (Dv : Foundation.Parabolic.ParabolicPoint → Fin 3 → Foundation.Parabolic.Vec3) (r : Foundation.Parabolic.ParabolicPoint → ℝ) (hsol : ∀ (n : ℕ), IsSuitableWeakSolution Ω I q (u n) (Du n) (p n) 0) (hdata : IsSuitableWeakSolutionData Ω I q v Dv r 0) (huConv : ∀ (Ω' : Set Foundation.Parabolic.Vec3) (J : Set ℝ), localBox Ω I Ω' J → Filter.Tendsto (fun (n : ℕ) => MeasureTheory.eLpNorm (u n - v) 3 (MeasureTheory.volume.restrict (spaceTimeSet Ω' J))) Filter.atTop (nhds 0)) (hDv : ∀ (Ω' : Set Foundation.Parabolic.Vec3) (J : Set ℝ), localBox Ω I Ω' J → MeasureTheory.MemLp Dv 2 (MeasureTheory.volume.restrict (spaceTimeSet Ω' J))) (hDuWeak : ∀ (Ω' : Set Foundation.Parabolic.Vec3) (J : Set ℝ), localBox Ω I Ω' J → ∀ (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)))) (hpConv : ∀ (Ω' : Set Foundation.Parabolic.Vec3) (J : Set ℝ), localBox Ω I Ω' J → Filter.Tendsto (fun (n : ℕ) => MeasureTheory.eLpNorm (p n - r) (3 / 2) (MeasureTheory.volume.restrict (spaceTimeSet Ω' J))) Filter.atTop (nhds 0)) (φ : Foundation.Parabolic.Vec3 × ℝ → Foundation.Parabolic.Vec3) : φ ∈ spaceTimeTestFunction Ω I → ∫ (z : Foundation.Parabolic.ParabolicPoint) in spaceTimeSet Ω I, -∑ i : Fin 3, v z i * timePartial (fun (w : Foundation.Parabolic.ParabolicPoint) => φ w i) z - ∑ i : Fin 3, ∑ j : Fin 3, v z i * v z j * spatialPartial (fun (w : Foundation.Parabolic.ParabolicPoint) => φ w i) j z + ∑ i : Fin 3, ∑ j : Fin 3, Dv z i j * spatialPartial (fun (w : Foundation.Parabolic.ParabolicPoint) => φ w i) j z - r z * ∑ i : Fin 3, spatialPartial (fun (w : Foundation.Parabolic.ParabolicPoint) => φ w i) i z - ∑ i : Fin 3, 0 z i * φ z i = 0

The zero-force weak momentum identity (S3) passes to the limits in thm:stability.