Navier-Stokes方程在R^3空间弱解的非唯一性
摘要
我们了解到，凸堆积法已广泛应用于周期区域Navier-Stokes方程解的非唯一性研究，但在整个空间或其他区域的方法鲜有文献。本文证明了在具有有限动能的弱解类别中，Navier-Stokes方程的弱解非唯一，这扩展了(Buckmaster和Vicol, Ann. of Math., 189 (2019), pp.101-144)在 torus T^3上的非唯一性结果到R^3。该证明的关键技术包括 developing an iterative scheme where the approximation solution is refined by decomposing it into local and non-local parts. For the non-local part, we introduce the localized corrector which plays a crucial role in balancing the compact support of the Reynolds stress error with the non-compact support of the solution. As applications of this argument, we first prove that there exist infinitely many weak solutions that dissipate the kinetic energy in smooth bounded domain. Moreover, we show the instability of the Navier-Stokes equations near Couette flow in L2(R3).
关键词
引用
@article{arxiv.2412.10404,
title = {Non-uniqueness of weak solutions to the Navier-Stokes equations in R^3},
author = {Changxing Miao and Yao Nie and Weikui Ye},
journal= {arXiv preprint arXiv:2412.10404},
year = {2024}
}