UCLA

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

In the second half of the thesis, by applying techniques from geometric microlocal analysis to construct the Hodge-Neumann heat kernel, we obtain off-diagonal decay and local Bernstein estimates, and then use them to extend the result to the Besov space $\widehat{B}_{3,V}^{\frac{1}{3}}$, which generalizes both the space $\widehat{B}_{3,c(\mathbb{N})}^{1/3}$ from (Isett-Oh 2014) and the space $\underline{B}_{3,\text{VMO}}^{1/3}$ from (Bardos et al. 2019, Nguyen-Nguyen 2020) --- the best known function space where Onsager's conjecture holds on flat backgrounds.