Subcritical convergence in the Leray limit

Strong L² convergence on a time slab together with a uniform L^(10/3) bound gives strong convergence in every exponent 2 ≤ q < 10/3, as in prop:leray-limit. Neither square integrability of the individual fields nor integrability of the limit is assumed: both follow from the two hypotheses.

theorem CKN.Leray.lerayLimit_subcritical_of_strongL2_and_tenThirds {α : Type u_1} {E : Type u_2} [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {U : ℕ → α → E} {u : α → E} {B : ENNReal} (hB : B < ⊤) (hhigh : ∀ (n : ℕ), MeasureTheory.MemLp (U n) (ENNReal.ofReal (10 / 3)) μ ∧ MeasureTheory.eLpNorm (U n) (ENNReal.ofReal (10 / 3)) μ ≤ B) (h2 : Filter.Tendsto (fun (n : ℕ) => MeasureTheory.eLpNorm (U n - u) (ENNReal.ofReal 2) μ) Filter.atTop (nhds 0)) (q : ℝ) : 2 ≤ q → q < 10 / 3 → Filter.Tendsto (fun (n : ℕ) => MeasureTheory.eLpNorm (U n - u) (ENNReal.ofReal q) μ) Filter.atTop (nhds 0)

A strong L² limit of a sequence that is uniformly bounded in L^(10/3) is a strong limit in every intermediate exponent. The limit is not assumed to be measurable or integrable; this follows from the two hypotheses.

theorem CKN.Leray.lerayLimit_regularised_subcritical (ρ : RegMollifierProfile) (uε : (a : Foundation.Parabolic.Vec3 → Foundation.Parabolic.Vec3) → IsInJ a → ℝ → Foundation.Parabolic.ParabolicPoint → Foundation.Parabolic.Vec3) (pε : (a : Foundation.Parabolic.Vec3 → Foundation.Parabolic.Vec3) → IsInJ a → ℝ → Foundation.Parabolic.ParabolicPoint → ℝ) (hregularised : ∀ (a : Foundation.Parabolic.Vec3 → Foundation.Parabolic.Vec3) (ha : IsInJ a) (ε : ℝ) (hε : 0 < ε), have u := uε a ha ε; have p := pε a ha ε; have D := fun (z : Foundation.Parabolic.ParabolicPoint) (i j : Fin 3) => spatialPartial (fun (y : Foundation.Parabolic.ParabolicPoint) => u y i) j z; have DD := fun (z : Foundation.Parabolic.ParabolicPoint) (i j k : Fin 3) => spatialPartial (fun (y : Foundation.Parabolic.ParabolicPoint) => spatialPartial (fun (x : Foundation.Parabolic.ParabolicPoint) => u x i) j y) k z; have Dt := fun (z : Foundation.Parabolic.ParabolicPoint) (i : Fin 3) => timePartial (fun (y : Foundation.Parabolic.ParabolicPoint) => u y i) z; have Dp := fun (z : Foundation.Parabolic.ParabolicPoint) (i : Fin 3) => spatialPartial (fun (y : Foundation.Parabolic.ParabolicPoint) => p y) i z; (∃ (hSlice : ∀ (t : ℝ), 0 ≤ t → MeasureTheory.MemLp (fun (x : Foundation.Parabolic.Vec3) => u (x, t)) 2 MeasureTheory.volume), (Continuous fun (t : ↑(Set.Ici 0)) => realVectorL2OfCoordinateFunction (fun (x : Foundation.Parabolic.Vec3) => u (x, ↑t)) ⋯) ∧ (fun (x : Foundation.Parabolic.Vec3) => u (x, 0)) =ᵐ[MeasureTheory.volume] regUniformMollifiedInitial ρ ε hε a ∧ ∀ (t : ℝ), 0 ≤ t → IsWeakDivFreeL2 fun (x : Foundation.Parabolic.Vec3) => u (x, t)) ∧ (∀ (i : Fin 3), ContinuousOn (fun (z : Foundation.Parabolic.ParabolicPoint) => u z i) (spaceTimeSet Set.univ (Set.Ioi 0))) ∧ (∀ (i j : Fin 3), ContinuousOn (fun (z : Foundation.Parabolic.ParabolicPoint) => D z i j) (spaceTimeSet Set.univ (Set.Ioi 0))) ∧ (∀ (i j k : Fin 3), ContinuousOn (fun (z : Foundation.Parabolic.ParabolicPoint) => DD z i j k) (spaceTimeSet Set.univ (Set.Ioi 0))) ∧ (∀ (i : Fin 3), ContinuousOn (fun (z : Foundation.Parabolic.ParabolicPoint) => Dt z i) (spaceTimeSet Set.univ (Set.Ioi 0))) ∧ ContinuousOn p (spaceTimeSet Set.univ (Set.Ioi 0)) ∧ (∀ (i : Fin 3), ContinuousOn (fun (z : Foundation.Parabolic.ParabolicPoint) => Dp z i) (spaceTimeSet Set.univ (Set.Ioi 0))) ∧ (∀ (i : Fin 3), ContDiffOn ℝ 1 (fun (z : Foundation.Parabolic.ParabolicPoint) => u z i) (spaceTimeSet Set.univ (Set.Ioi 0))) ∧ (∀ z ∈ spaceTimeSet Set.univ (Set.Ioi 0), ∀ (i j : Fin 3), DifferentiableAt ℝ (fun (x : Foundation.Parabolic.Vec3) => D (x, z.2) i j) z.1) ∧ (∀ z ∈ spaceTimeSet Set.univ (Set.Ioi 0), DifferentiableAt ℝ (fun (x : Foundation.Parabolic.Vec3) => p (x, z.2)) z.1) ∧ (∀ (δ T : ℝ), 0 < δ → δ < T → ∃ (C : ℝ), 0 ≤ C ∧ ∀ z ∈ spaceTimeSet Set.univ (Set.Icc δ T), Foundation.Parabolic.vec3EuclideanNorm (u z) ≤ C ∧ |p z| ≤ C ∧ (∀ (i j : Fin 3), |D z i j| ≤ C) ∧ (∀ (i j k : Fin 3), |DD z i j k| ≤ C) ∧ (∀ (i : Fin 3), |Dt z i| ≤ C) ∧ ∀ (i : Fin 3), |Dp z i| ≤ C) ∧ (∀ (δ T : ℝ), 0 < δ → δ < T → (∀ (i : Fin 3), MeasureTheory.MemLp (fun (z : Foundation.Parabolic.ParabolicPoint) => u z i) 2 (MeasureTheory.volume.restrict (spaceTimeSet Set.univ (Set.Ioo δ T)))) ∧ (∀ (i j : Fin 3), MeasureTheory.MemLp (fun (z : Foundation.Parabolic.ParabolicPoint) => D z i j) 2 (MeasureTheory.volume.restrict (spaceTimeSet Set.univ (Set.Ioo δ T)))) ∧ (∀ (i j k : Fin 3), MeasureTheory.MemLp (fun (z : Foundation.Parabolic.ParabolicPoint) => DD z i j k) 2 (MeasureTheory.volume.restrict (spaceTimeSet Set.univ (Set.Ioo δ T)))) ∧ (∀ (i : Fin 3), MeasureTheory.MemLp (fun (z : Foundation.Parabolic.ParabolicPoint) => Dt z i) 2 (MeasureTheory.volume.restrict (spaceTimeSet Set.univ (Set.Ioo δ T)))) ∧ MeasureTheory.MemLp p 2 (MeasureTheory.volume.restrict (spaceTimeSet Set.univ (Set.Ioo δ T)))) ∧ (∀ (z : Foundation.Parabolic.ParabolicPoint), 0 < z.2 → ∀ (i : Fin 3), Dt z i - ∑ j : Fin 3, DD z i j j + ∑ j : Fin 3, regUniformMollifiedVelocity ρ ε hε u z j * D z i j + Dp z i = 0) ∧ (∀ (t : ℝ), 0 < t → ∃ (hF : ∀ (i j : Fin 3), MeasureTheory.MemLp (fun (x : Foundation.Parabolic.Vec3) => regUniformMollifiedVelocity ρ ε hε u (x, t) i * u (x, t) j) (ENNReal.ofReal 2) MeasureTheory.volume), ((fun (x : Foundation.Parabolic.Vec3) => p (x, t)) =ᵐ[MeasureTheory.volume] rieszPressureSliceRepresentative 2 lerayLimit_regularised_subcritical._proof_55 fun (i j : Fin 3) => MeasureTheory.MemLp.toLp (fun (x : Foundation.Parabolic.Vec3) => regUniformMollifiedVelocity ρ ε hε u (x, t) i * u (x, t) j) ⋯) ∧ ∀ (ψ : Foundation.Parabolic.Vec3 → ℝ), ContDiff ℝ (↑⊤) ψ → HasCompactSupport ψ → ∫ (x : Foundation.Parabolic.Vec3), p (x, t) * spatialLaplacian ψ x = -∑ i : Fin 3, ∑ j : Fin 3, ∫ (x : Foundation.Parabolic.Vec3), regUniformMollifiedVelocity ρ ε hε u (x, t) i * u (x, t) j * mixedSecond ψ i j x) ∧ ∀ (t : ℝ), 0 ≤ t → MeasureTheory.eLpNorm (regUniformVelocitySlice u t) 2 MeasureTheory.volume ^ 2 + 2 * regUniformDissipation u D t = MeasureTheory.eLpNorm (regMollifyVector ρ ε hε (regUniformSpatialField a)) 2 MeasureTheory.volume ^ 2) (a : Foundation.Parabolic.Vec3 → Foundation.Parabolic.Vec3) (ha : IsInJ a) (εseq : ℕ → ℝ) (hseq : ∀ (n : ℕ), 0 < εseq n ∧ εseq n ≤ 1) (σ : ℕ → ℕ) (T : ℝ) (u : Foundation.Parabolic.ParabolicPoint → Foundation.Parabolic.Vec3) (h2 : Filter.Tendsto (fun (n : ℕ) => MeasureTheory.eLpNorm (uε a ha (εseq (σ n)) - u) (ENNReal.ofReal 2) (MeasureTheory.volume.restrict (spaceTimeSet Set.univ (Set.Ioo 0 T)))) Filter.atTop (nhds 0)) (q : ℝ) : 2 ≤ q → q < 10 / 3 → Filter.Tendsto (fun (n : ℕ) => MeasureTheory.eLpNorm (uε a ha (εseq (σ n)) - u) (ENNReal.ofReal q) (MeasureTheory.volume.restrict (spaceTimeSet Set.univ (Set.Ioo 0 T)))) Filter.atTop (nhds 0)

The subcritical convergence of prop:leray-limit for a selected sequence of regularized solutions: strong L² convergence on the time slab (0,T) upgrades to strong convergence in every exponent 2 ≤ q < 10/3, the uniform L^(10/3) bound coming from the regularized energy identity.