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

2020/10/27 by Khang Manh Huynh, Huynh, Khang Manh
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2010.14762

openalex publication_date 2020/10/27 · openalex created_date 2020/11/09 · openalex updated_date 2026/07/28

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^(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 \widehatB_3,V^(1)/(3), which generalizes both the space \widehatB_3,c(ℕ)^1/3 from arXiv:1310.7947 [math.AP] and the space \underlineB_3,VMO^1/3 from arXiv:1902.07120 [math.AP] -- the best known function space where Onsager's conjecture holds on flat backgrounds.

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