Construction of the Hodge-Neumann heat kernel, local Bernstein estimates, and Onsager's conjecture in fluid dynamics

TL;DR

This paper constructs the Hodge-Neumann heat kernel on manifolds with boundary, derives local estimates, and extends Onsager's conjecture results to new Besov spaces, advancing understanding in fluid dynamics and geometric analysis.

Contribution

It develops the Hodge-Neumann heat kernel using microlocal analysis and extends Onsager's conjecture to broader Besov spaces on manifolds with boundary.

Findings

Established off-diagonal decay of the heat kernel

Derived local Bernstein estimates

Extended Onsager's conjecture to new Besov spaces

Abstract

Most recently, in arXiv:1907.05360 [math.AP], we introduced the theory of heatable currents and proved Onsager's conjecture on Riemannian manifolds with boundary, where the weak solution has $B_{3,1}^{\frac{1}{3}}$ spatial regularity. In this sequel, 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 arXiv:1310.7947 [math.AP] and the space $\underline{B}_{3,\text{VMO}}^{1/3}$ from arXiv:1902.07120 [math.AP] -- the best known function space where Onsager's conjecture holds on flat backgrounds.