论文标题
Navier-Stokes方程的急剧零件
Sharp nonuniqueness for the Navier-Stokes equations
论文作者
论文摘要
在本文中，我们证明了在周期性环境中不可压缩的Navier-Stokes方程的鲜明效果。在任何维度$ d \ geq 2 $中，并且给定任何$ p <2 $，我们在类$ l^{p} _t l^\ infty $中显示了弱解决方案的非唯一性，鉴于经典的Ladyzhenskaya-prodi-serrin-serrin标准，这很敏锐。证明是基于建造一类非Leray-Hopf弱解决方案的。更具体地说，对于任何$ p <2 $，$ q <\ infty $和$ \ varepsilon> 0 $，我们构造了非leray-hopf弱解决方案$ u \ in l^{p} _t l^{p} _t l^\ infty \ infty \ infty \ cap l^cap l^1_t w^{1，q} $，与var $ bats nict of var bats y heausdorn bats nate n of a hausdorn bate hyusdorffff nate hyusdorffff hausdorffffffff in.作为一个副产品，类别$ l^{{3}/{2} - \ varepsilon} _t c^{{1}/{3}} $在类$ l^{{3}/{2} - \ varepsilon} _t均给出了异常耗散的示例。
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.
