Heat-kernel pairings of a double-divergence-free slice #
At a fixed positive time r, the spatial derivative ∂_j Γ(·, r) of the heat
kernel is paired with the translates y ↦ ∫ g_ij(w) ∂_i ψ(w + y) dw of a
square-integrable tensor g against the derivatives of a test function ψ.
Moving ∂_j from the kernel onto ψ and exchanging the integrals turns the
pairing into -∫ Γ(y, r) ∑_ij ∫ g_ij(w) ∂_i ∂_j ψ(w + y) dw dy, which vanishes
when ∑_ij ∂_i ∂_j g_ij = 0 in the sense of distributions. This is the
fixed-time step in the proof that the forced heat response of a
double-divergence-free tensor is divergence free (prop:pv-local-solution).
For integrable k, square-integrable g and continuous compactly
supported φ, the function (y, w) ↦ k(y) g(w) φ(w + y) is integrable on
Vec3 × Vec3: the inner integral is bounded by (‖g‖₂² + ‖φ‖₂²) / 2
uniformly in y.
At a positive time the spatial derivative of the heat kernel is integrable in space.
Integration by parts in space against the spatial derivative of the heat
kernel at a positive time: ∫ ∂_j Γ(y, r) φ(w + y) dy = -∫ Γ(y, r) ∂_j φ(w + y) dy.
Moving the spatial derivative from the heat kernel onto the test function:
for square-integrable g and a smooth compactly supported ψ,
∫ ∂_j Γ(y, r) ∫ g(w) ∂_i ψ(w + y) dw dy = -∫ Γ(y, r) ∫ g(w) ∂_i ∂_j ψ(w + y) dw dy.
The heat-kernel pairing of a double-divergence-free square-integrable
tensor vanishes: if ∑_ij ∫ g_ij ∂_i ∂_j φ = 0 for every smooth compactly
supported φ, then ∑_ij ∫ ∂_j Γ(y, r) ∫ g_ij(w) ∂_i ψ(w + y) dw dy = 0.