Sharp non-uniqueness for the 2D hyper-dissipative Navier-Stokes equations

TL;DR

This paper demonstrates sharp non-uniqueness of weak solutions for 2D hyper-dissipative Navier-Stokes equations in super-critical spaces, extending recent results and highlighting unpredictable solution behavior even with high viscosity.

Contribution

It extends non-uniqueness results to hyper-dissipative 2D Navier-Stokes equations for lpha in [1, 1.5), using intermittency and generalized Ladyzenskaya-Prodi-Serrin conditions, applicable in Lebesgue and Besov spaces.

Findings

Non-uniqueness at endpoints is sharp under specific conditions.

Solution behavior remains unpredictable even with high viscosity.

Results extend to hyper-dissipative cases and various function spaces.

Abstract

In this article, we study the non-uniqueness of weak solutions for the two-dimensional hyper-dissipative Navier-Stokes equations in the super-critical spaces $L_{t}^{\gamma}W_{x}^{s,p}$ when $\alpha\in[1,\frac{3}{2})$, and obtain the conclusion that the non-uniqueness of the weak solutions at the two endpoints is sharp in view of the generalized Lady\v{z}enskaya-Prodi-Serrin condition with the triplet $(s,\gamma,p)=(s,\infty, \frac{2}{2\alpha-1+s})$ and $(s, \frac{2\alpha}{2\alpha-1+s}, \infty)$. As a good observation, we use the intermittency of the temporal concentrated function in an almost optimal way. The research results extend the recent elegant works on 2D Navier-Stokes equations in [Cheskidov and Luo, Invent. Math., 229 (2022), pp. 987--1054; Cheskidov and Luo, Ann. PDE, 9:13 (2023)] to the hyper-dissipative case $\alpha \in(1,\frac{3}{2})$, and are also applicable in Lebesgue and Besov spaces. It is proved that even in the case of high viscosity, the behavior of the solution remains unpredictable and stochastic due to the lack of integrability and regularity.