Local endpoint regularity

The local Escauriaza–Seregin–Šverák theorem thm:ess-local: a local weak solution on the unit parabolic cylinder, with pressure in L^{3/2} and velocity bounded in L³ in space uniformly in time, agrees almost everywhere on the half cylinder with a field that is parabolically Hölder continuous on its closure. The regularity argument of thm:ess-local is combined with the blow-up limit prop:blowup-limit at a point that is not good.

theorem ESS.Main.essLocal (u : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3) (Du : CKN.Foundation.Parabolic.ParabolicPoint → Fin 3 → CKN.Foundation.Parabolic.Vec3) (p : CKN.Foundation.Parabolic.ParabolicPoint → ℝ) : MeasureTheory.AEStronglyMeasurable u (MeasureTheory.volume.restrict (CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0))) → MeasureTheory.AEStronglyMeasurable Du (MeasureTheory.volume.restrict (CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0))) → MeasureTheory.AEStronglyMeasurable p (MeasureTheory.volume.restrict (CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0))) → essSup (fun (t : ℝ) => ∫⁻ (x : CKN.Foundation.Parabolic.Vec3) in CKN.Foundation.Parabolic.vec3Ball 0 1, ‖u (x, t)‖ₑ ^ 2) (MeasureTheory.volume.restrict (Set.Ioo (-1) 0)) < ⊤ → ∫⁻ (z : CKN.Foundation.Parabolic.ParabolicPoint) in CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0), ‖u z‖ₑ ^ 2 + ‖Du z‖ₑ ^ 2 < ⊤ → MeasureTheory.MemLp p (ENNReal.ofReal (3 / 2)) (MeasureTheory.volume.restrict (CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0))) → essSup (fun (t : ℝ) => ∫⁻ (x : CKN.Foundation.Parabolic.Vec3) in CKN.Foundation.Parabolic.vec3Ball 0 1, ENNReal.ofReal (CKN.Foundation.Parabolic.vec3EuclideanNorm (u (x, t))) ^ 3) (MeasureTheory.volume.restrict (Set.Ioo (-1) 0)) < ⊤ → (∀ᵐ (t : ℝ) ∂MeasureTheory.volume.restrict (Set.Ioo (-1) 0), ∀ (i : Fin 3), CKN.HasWeakGradientOn (CKN.Foundation.Parabolic.vec3Ball 0 1) (fun (x : CKN.Vec 3) => u (x, t) i) fun (x : CKN.Vec 3) => Du (x, t) i) → (∀ ψ ∈ CKN.spaceTimeTestFunction (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0), ∫ (z : CKN.Foundation.Parabolic.ParabolicPoint) in CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0), ∑ i : Fin 3, u z i * CKN.spatialPartial ψ i z = 0) → (∀ φ ∈ CKN.spaceTimeTestFunction (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0), ∫ (z : CKN.Foundation.Parabolic.ParabolicPoint) in CKN.spaceTimeSet (CKN.Foundation.Parabolic.vec3Ball 0 1) (Set.Ioo (-1) 0), -∑ i : Fin 3, u z i * CKN.timePartial (fun (y : CKN.Foundation.Parabolic.ParabolicPoint) => φ y i) z - ∑ i : Fin 3, ∑ j : Fin 3, u z i * u z j * CKN.spatialPartial (fun (y : CKN.Foundation.Parabolic.ParabolicPoint) => φ y i) j z + ∑ i : Fin 3, ∑ j : Fin 3, Du z i j * CKN.spatialPartial (fun (y : CKN.Foundation.Parabolic.ParabolicPoint) => φ y i) j z - p z * ∑ i : Fin 3, CKN.spatialPartial (fun (y : CKN.Foundation.Parabolic.ParabolicPoint) => φ y i) i z - ∑ i : Fin 3, 0 z i * φ z i = 0) → ∃ (γ : ℝ), 0 < γ ∧ γ ≤ 1 ∧ ∃ (w : CKN.Foundation.Parabolic.ParabolicPoint → CKN.Foundation.Parabolic.Vec3), w =ᵐ[MeasureTheory.volume.restrict (CKN.Foundation.Parabolic.parabolicCylinder 0 0 (1 / 2))] u ∧ CKN.ParabolicHolderVecOn (closure (CKN.Foundation.Parabolic.parabolicCylinder 0 0 (1 / 2))) w γ

The local Escauriaza–Seregin–Šverák regularity theorem thm:ess-local.