二维流体的整体光滑解和弱解

Navier-Stokes equations are the basic equations for fluid mechanics, which have been one of the scientific focuses over the two centuries. We plan to study the two dimensional Navier-Stokes equations and some coupled systems, with the emphasis on the regularity, uniqueness, blow-up mechanics of the solutions. The first goal of this project is to get some new understanding of the full Navier-Stokes system, in particular, its blow-up mechanics. We wish to find the key factor which plays the most important role in the regularity of solutions. Meanwhile, the second goal is to prove the existence and uniqueness of solutions to 2D density-dependent MHD system. It is a coupled system of Navier-Stokes equations, which involves the change of density. The possible appearance of vacuum can make the system be degenerate and bring new difficulties.

Navier-Stokes方程是描述流体运动的基本方程。两个世纪以来，此方程一直是科学界的热点问题之一。我们计划研究二维Navier-Stokes方程及其耦合问题的解的一些性质，例如光滑性，唯一性，爆破机制等。本项目的第一个目标是，了解二维带温度的可压Navier-Stokes方程解的爆破机制，找出对解的正则性起主导作用的量。第二个目标是，研究二维非齐次不可压磁流体(MHD)方程（Navier-Stokes方程的一类耦合模型）整体强解的存在唯一性或局部解的爆破机制。此模型涉及密度的变化情况，甚至真空有可能出现，导致方程退化，所以不是传统意义上的抛物类方程组，有新的难度。

经过一年的努力，我们实现了项目所提的研究目标，解决了项目中的主要问题。关于二维可压Navier-Stokes方程，我们给出了一个新的爆破准则。此准则只与速度场的散度相关，从而建立了可压与不可压Navier-Stokes方程的某种联系。关于粘性依赖于密度的非齐次不可压Navier-Stokes方程，我们证明了只要粘性系数的梯度相对小，方程有整体强解。