  We use Hodge theory and functional analysis to develop a clean approach to
heat flows and Onsager's conjecture on Riemannian manifolds with boundary,
where the weak solution lies in the trace-critical Besov space
$B_{3,1}^{\frac{1}{3}}$. We also introduce heatable currents as the natural
analogue to tempered distributions and justify their importance in Hodge
theory.
