数学
豪斯多夫维数
无粘流
维数(图论)
班级(哲学)
数学分析
纳维-斯托克斯方程组
压缩性
弱解
数学物理
纯数学
组合数学
经典力学
物理
热力学
人工智能
计算机科学
作者
Alexey Cheskidov,Xiaoyutao Luo
标识
DOI:10.1007/s00222-022-01116-x
摘要
In this paper, we prove a sharp nonuniqueness result for the incompressible Navier-Stokes equations in the periodic setting. In any dimension $d \geq 2$ and given any $ p<2$, we show the nonuniqueness of weak solutions in the class $L^{p}_t L^\infty$, 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$, $q<\infty$, and $\varepsilon>0$, we construct non-Leray-Hopf weak solutions $ u \in L^{p}_t L^\infty \cap L^1_t W^{1,q}$ that are smooth outside a set of singular times with Hausdorff dimension less than $\varepsilon$. As a byproduct, examples of anomalous dissipation in the class $L^{ {3}/{2} - \varepsilon}_t C^{ {1}/{3}} $ are given in both the viscous and inviscid case.
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