Harmonicity of the pressure remainder

The momentum equation and the Poisson identity for the fixed Riesz pressure give distributional harmonicity of the remainder on almost every time slice.

theorem ESS.pressureSplitRemainder_harmonic_slices {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} (hu : MeasureTheory.AEStronglyMeasurable u (MeasureTheory.volume.restrict pressureSplitDomain)) (hDu : MeasureTheory.AEStronglyMeasurable Du (MeasureTheory.volume.restrict pressureSplitDomain)) (hpmeas : MeasureTheory.AEStronglyMeasurable p (MeasureTheory.volume.restrict pressureSplitDomain)) (hL2 : essSup (fun (t : ℝ) => ∫⁻ (x : CKN.Foundation.Parabolic.Vec3) in pressureSplitBall, ‖u (x, t)‖ₑ ^ 2) (MeasureTheory.volume.restrict pressureSplitTime) < ⊤) (henergy : ∫⁻ (z : CKN.Foundation.Parabolic.ParabolicPoint) in pressureSplitDomain, ‖u z‖ₑ ^ 2 + ‖Du z‖ₑ ^ 2 < ⊤) (hp : MeasureTheory.MemLp p (ENNReal.ofReal (3 / 2)) (MeasureTheory.volume.restrict pressureSplitDomain)) (hL3 : essSup (fun (t : ℝ) => ∫⁻ (x : CKN.Foundation.Parabolic.Vec3) in pressureSplitBall, ENNReal.ofReal (CKN.Foundation.Parabolic.vec3EuclideanNorm (u (x, t))) ^ 3) (MeasureTheory.volume.restrict pressureSplitTime) < ⊤) (hgrad : ∀ᵐ (t : ℝ) ∂MeasureTheory.volume.restrict pressureSplitTime, ∀ (i : Fin 3), CKN.HasWeakGradientOn pressureSplitBall (fun (x : CKN.Vec 3) => u (x, t) i) fun (x : CKN.Vec 3) => Du (x, t) i) (hdiv : ∀ χ ∈ CKN.spaceTimeTestFunction pressureSplitBall pressureSplitTime, ∫ (z : CKN.Foundation.Parabolic.ParabolicPoint) in pressureSplitDomain, ∑ i : Fin 3, u z i * CKN.spatialPartial χ i z = 0) (hMomentum : ∀ φ ∈ CKN.spaceTimeTestFunction pressureSplitBall pressureSplitTime, ∫ (z : CKN.Foundation.Parabolic.ParabolicPoint) in pressureSplitDomain, -∑ 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) : ∀ᵐ (t : ℝ) ∂MeasureTheory.volume.restrict pressureSplitTime, CKN.Foundation.Heat.WeaklyHarmonicOn pressureSplitBall fun (x : CKN.Foundation.Parabolic.Vec3) => pressureSplitRemainder p (pressureSplitRieszPressure (CKN.pressureSplitTensor u) ⋯) (x, t)

The fixed pressure remainder is weakly harmonic in the source ball for almost every time slice.