Hodge-theoretic analysis on manifolds with boundary, heatable currents, and Onsager's conjecture in fluid dynamics

TL;DR

This paper applies Hodge theory and functional analysis to study heat flows and Onsager's conjecture on manifolds with boundary, introducing heatable currents as a new tool for understanding weak solutions in critical Besov spaces.

Contribution

It develops a novel approach combining Hodge theory with heat flows and introduces heatable currents as a new concept for analyzing weak solutions on manifolds with boundary.

Findings

Established a framework for heat flows on manifolds with boundary.

Introduced heatable currents as an analogue to tempered distributions.

Provided insights into Onsager's conjecture in the context of Riemannian manifolds.

Abstract

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.