Spatial decay of the causal heat potential #
The causal heat potential of a continuous compactly supported source decays
like |x|^{-3} in space, uniformly in time. This justifies the whole-space
integrations by parts in the smooth-data energy estimates of lem:pv-stokes.
theorem
ESS.causalHeatConv_abs_le_decay
{h : CKN.Foundation.Parabolic.Vec3 × ℝ → ℝ}
(hh : Continuous h)
(hc : HasCompactSupport h)
:
∃ (M : ℝ),
0 ≤ M ∧ ∀ (v : CKN.Foundation.Parabolic.Vec3 × ℝ),
|causalHeatConv h v| ≤ M / (1 + CKN.Foundation.Parabolic.vec3EuclideanNorm v.1) ^ 3
The causal heat potential of a continuous compactly supported source
decays like |x|^{-3} uniformly in time.