非齐次不可压缩Navier-Stokes方程的整体适定性

Incompressible Navier-Stokes equations are the basic model for fluid dynamics. Earlier researches focus on the homogenenous case, which is devoted to the homogeneous fluid, i.e. the density for the fluid is a constant.In real world, the fluid is usually a combination of several fluids, which is hence nonhomogenous. For this type of fluid, viscosity depends on the density, and the dynamics is described by the nonhomogeneous incompressible Navier-Stokes equations. We will study the strong solution to this model. The first goal of this project is to get some initial data, for which the global strong solution with vacuum exists, unique and stable. Previously, the local strong solution was derived under a strict compatibility condition, we wish to relax it or give another physically reasonable compatibility condition.

不可压缩Navier-Stokes方程是描述流体运动的基本方程。早期的数学研究集中在齐次不可压Navier-Stokes方程上，关注的对象是齐次流体。现实环境中，流体往往是多种流体的混合，故为非齐次流体。对于非齐次流体，粘性系数常常依赖于密度。刻画此类流体的模型是非齐次不可压缩Navier-Stokes方程。我们将研究此方程的强解。本项目的第一个主要目标是，找出合适的初始条件，保证含真空强解的整体存在性，唯一性，稳定性。由于已知局部强解的存在性在一个苛刻的相容性条件下证明存在，因此本项目的第二个主要目标是，适当放松此相容性条件，提出符合物理常识的相容性条件。

我们主要做了以下结果：（1）在初始速度场的临界模小的条件下，我们得到了非齐次不可压缩Navier-Stokes方程的整体强解。相较于之前的结果，我们允许密度含真空或者是常值状态的大扰动。（2）对于三维不可压MHD方程，我们证明了只要初始速度场和磁场相差较小，则方程有整体光滑解。相较于之前的结果，我们并不需要初始速度场与磁场同时小，而是只要求它们的某个线性差比较小。