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

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 B133,1 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 Bˆ133,V, which generalizes both the space Bˆ1/33,c(ℕ) from arXiv:1310.7947 [math.AP] and the space B⎯⎯⎯1/33,VMO from arXiv:1902.07120 [math.AP] -- the best known function space where Onsager's conjecture holds on flat backgrounds.