Sharp nonuniqueness for the Navier-Stokes equations

• 成果类型: Article
• 署名作者: Cheskidov, Alexey; Luo, Xiaoyutao
• 署名单位: University of Illinois System; University of Illinois Chicago; University of Illinois Chicago Hospital; Institute for Advanced Study - USA; Duke University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-022-01116-x
• 发表日期: 2022
• 页码: 987-1054
• 关键词: weak solutions; PARTIAL REGULARITY; incompressible euler; energy-conservation; onsagers conjecture; mild solutions; uniqueness; DISSIPATION; INEQUALITY

摘要

In this paper, we prove a sharp nonuniqueness result for the incompressible Navier-Stokes equations in the periodic setting. In any dimension d >= 2 and given any p < 2, we show the nonuniqueness of weak solutions in the class (LtL infinity)-L-p, which is sharp in view of the classical Ladyzhenskaya-Prodi-Serrin criteria. The proof is based on the construction of a class of non-Leray-Hopf weak solutions. More specifically, for any p < 2, q < infinity, and epsilon > 0, we construct non-Leray-Hopf weak solutions u is an element of (LtL infinity)-L-p boolean AND(LtW1,q)-W-1 that are smooth outside a set of singular times with Hausdorff dimension less than epsilon. As a byproduct, examples of anomalous dissipation in the class (Lt3/2-epsilon C1/3) are given in both the viscous and inviscid case.