Blowup Riesz Exterior Distribution #
Pressure reconstruction and convergence bounds for the rescaled sequence.
theorem
ESS.blowup_rieszPressureSpaceTime_compact_duality
(F : Fin 3 → Fin 3 → CKN.Foundation.Parabolic.Vec3 × ℝ → ℝ)
(hF : ∀ (i j : Fin 3), MeasureTheory.MemLp (F i j) (ENNReal.ofReal (3 / 2)) MeasureTheory.volume)
{ψ : CKN.Foundation.Parabolic.Vec3 × ℝ → ℝ}
(hψ : ContDiff ℝ (↑⊤) ψ)
(hψc : HasCompactSupport ψ)
:
∫ (z : CKN.Foundation.Parabolic.Vec3 × ℝ), CKN.Leray.rieszPressureSpaceTime (3 / 2) blowup_rieszPressureSpaceTime_compact_duality._proof_1 F hF z * CKN.Leray.rieszPressureJointLaplacian ψ z = -∑ i : Fin 3,
∑ j : Fin 3, ∫ (z : CKN.Foundation.Parabolic.Vec3 × ℝ), F i j z * CKN.Leray.rieszPressureJointHessian ψ i j z
The canonical space-time Riesz pressure has its Poisson distribution identity against every smooth compact test.
theorem
ESS.blowup_rieszPressureSpaceTime_exterior_distribution_zero
(F : Fin 3 → Fin 3 → CKN.Foundation.Parabolic.Vec3 × ℝ → ℝ)
(hF : ∀ (i j : Fin 3), MeasureTheory.MemLp (F i j) (ENNReal.ofReal (3 / 2)) MeasureTheory.volume)
(U : Set CKN.Foundation.Parabolic.Vec3)
(hzero : ∀ (i j : Fin 3) (z : CKN.Foundation.Parabolic.Vec3 × ℝ), z.1 ∈ U → F i j z = 0)
{ψ : CKN.Foundation.Parabolic.Vec3 × ℝ → ℝ}
(hψ : ContDiff ℝ (↑⊤) ψ)
(hψc : HasCompactSupport ψ)
(hψU : ∀ z ∈ tsupport ψ, z.1 ∈ U)
:
The pressure of a tensor supported outside a spatial region has zero spatial Laplacian there in space-time distributions.