关于三维Navier-Stokes方程的弱解在第一奇性假设下的正则性研究

The project is devoted to study the regularity of weak solutions with Type I singularity for 3D Navier-Stokes equations. As is well-known, the 3D Navier-Stokes equations have a deep Physical background and great practical applications, and whether their weak solutions are regular is one of the seven problems for the Clay Millennium Prize. Even if we assume that weak solutions belong to the class of Type I singularity, whether the weak solutions are regular is still a great open problem. The applicant has made preliminary research on this topic, and published two papers on the journals Commun. Contemp. Math. and J. Differential Equations, two important journals for PDE. In the first paper, we obtained the regularity of weak solutions under the condition Type I singularity for the space (｜u｜<C/｜x｜) and another weaker condition (u belongs to L_t^{infty}(VMO^{-1}(R^3))), and we got an improved result by applying the idea of Navier-Stokes equations to MHD model in the second paper. In this project, we aim to obtain better results by considering the weaker assumptions on the components of the velocity, and attain our final objective.

本项目主要研究三维Navier-Stokes方程的弱解在第一奇性假设下的正则性。众所周知，三维Navier-Stokes方程有很深的物理背景和实际应用，其弱解正则与否是千禧年七大问题之一。即使假设弱解具有第一奇性性质，此弱解是否是正则的，仍然是一个巨大的公开问题。申请人在本课题已经做了初步研究,已有两篇论文被偏微分方程的重要杂志 Commun. Contemp. Math.和 J. Differential Equations 接收发表。 在第一篇里，我们对弱解加了空间第一奇性条件（｜u｜<C/｜x｜）与另一个弱的条件(u属于 L_t^{infty}(VMO^{-1}(R^3)))，从而获得弱解的正则性；第二篇文章，我们把证明Navier-Stokes 方程的反向唯一性方法用到MHD模型，得到一个改进的结果。在本项目中，我们重点放在给速度的分量加上弱的条件,期望逐步改进获得最终的结果。

Navier-Stokes方程的研究是偏微分方程以及流体力学的核心内容。本项目主要研究了：1、三维Navier-Stokes方程的弱解在第一奇性假设下或临界范数下的正则性问题（见Wang, Wendong & Zhang, Zhifei. Sci. China Math. 2017），特别我们改进了Gallagher-Koch-Planchon(CMP, 2016)的结果；2、一般Oseen-Frank能量与一般Leslie 应力下的液晶模型Ericksen-Leslie 系统弱解的全局存在性(Calculus of Variations and Partial Differential Equations, 51 (2014), 915–962. Wang Meng,Wang Wendong)与唯一性（Discrete and Continuous Dynamical System-B, 21(3),919-941, 2016. Wang Meng, Wang Wendong，Zhang Zhifei）；3、逼近调和映射的最佳能量恒等式(Journal of Functional Analysis, 272(2017), 776-803.Wang Wendong, Wei Dongyi,Zhang Zhifie)，由Parker的反例知这是最佳的；4、抛物算子在锥上的反向唯一性(J. Differential Equations, 258(2015), 224–241. Wu Jie,Wang Wendong).