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.
Categories: math.AP math-ph math.DG math.MP