lem:local-heat-gain #
Let B_r ⋐ B_R be concentric balls, a < s < b, and m ∈ {1, 2}. If
z ∈ L²((a, b); H^m(B_R)) solves ∂ₜ z - Δ z = G in distributions on
B_R × (a, b) with G ∈ L²((a, b); H^{m-1}(B_R)), then
z ∈ C([s, b]; H^m(B_r)) ∩ L²((s, b); H^{m+1}(B_r)) with the estimate
‖z‖_{C([s,b];H^m(B_r))} + ‖z‖_{L²((s,b);H^{m+1}(B_r))} ≤ C (‖z‖_{L²((a,b);H^m(B_R))} + ‖G‖_{L²((a,b);H^{m-1}(B_R))}),
where C depends only on m, r, R, s - a and b - a.
The proof multiplies by a cutoff η vanishing near t = a and outside a
compact subset of B_R. Each spatial derivative ∂^β (ηz), |β| ≤ m, solves
the heat equation on ℝ³ × (a, b) with a square-integrable source in source
or divergence form and has zero initial trace, so the energy estimate of
lem:localized-vorticity-energy applies. It gives a curve continuous in L²
and a square-integrable weak gradient. On B_r × [s, b] the cutoff equals
one.
Almost everywhere equality on B × I gives almost everywhere equality of
almost every time slice.
Distributional derivatives are insensitive to modification of the function almost everywhere.
lem:local-heat-gain.