Distributional equation for space-time pressure slices #
The spatial sign identity holds on almost every time slice of a space-time tensor, for every smooth compactly supported spatial test function.
theorem
CKN.Leray.rieszPressureSpaceTime_slice_laplacian_identity
(r : ℝ)
(hr : 1 < r)
(F : Fin 3 → Fin 3 → Foundation.Parabolic.Vec3 × ℝ → ℝ)
(hF : ∀ (i j : Fin 3), MeasureTheory.MemLp (F i j) (ENNReal.ofReal r) MeasureTheory.volume)
:
∀ᵐ (t : ℝ), ∀ (ψ : Foundation.Parabolic.Vec3 → ℝ),
ContDiff ℝ (↑⊤) ψ →
HasCompactSupport ψ →
∫ (x : Foundation.Parabolic.Vec3), rieszPressureSpaceTime r hr F hF (x, t) * -spatialLaplacian ψ x = ∑ i : Fin 3, ∑ j : Fin 3, ∫ (x : Foundation.Parabolic.Vec3), F i j (x, t) * mixedSecond ψ i j x