Qualitative Research On The Incompressible Navier-Stokes Equations

There are two parts in this article. The first part concerns the regularity issue while the second one considers the existence, uniqueness and asymptotic behavior for exterior Navier-Stokes flow. Whether the weak solution to the 3D incompressible Navier-Stokes equations is regular or not is a challenging open problem in this field. At present, there are two main directions in this field. One is to study the size of the singular set (so called partial regularity), the other one is to find sufficient conditions to guarantee no singularity formation (so called regularity criteria). The present paper is the second di-rection. We will present some interesting and different regularity conditions to guarantee the smoothness of solution. The second part is viewed as the main ingredients of this article and contributions in this area. Exterior Navier-Stokes flow is also a meaningful issue. Precisely, when a body moves through a fluid with constant velocity in a regime of Reynolds numbers of less than about fifty, the resulting fluid flow is then laminar. The stokes equations provide a good quantitative description for Reynolds numbers signifi-cantly less than one. For larger Reynolds numbers the Navier-Stokes equations need to be solved in order to obtain precise results. We consider a body which is moving with constant speed in an incompressible fluid with Reynolds numbers less than fifty parallel to a wall in an unbounded domain, the flow around this body can be modeled by the incom-pressible Navier-Stokes equations in an exterior domain with the boundary conditions on the wall, the surface of the body and at infinity. A very important application is that it can model the case of bubbles rising in a liquid which are moving parallel to the wall. In this thesis, we are concerned with the situation that a bubble with fixed shape is moving parallel to the wall in an otherwise unbounded domain filled with fluid. We introduce a smooth cutoff function to simplify the case where the bubble is replaced by a source term with compact support and obtain a stationary Navier-Stokes equations, which actually is a simplified model of the bubble problem, then choose a suitable variable as the "time" variable to show the existence and obtain a uniform bound for stationary solutions by us-ing the dynamical system method and Fourier transform method. Based on this existence result, we analyze the components of each velocity component in the framework of our function space in order to find the leading order term to extract asymptotics from them as the asymptotics of our velocity component in Fourier space. Finally, by taking inverse Fourier transform we can get the asymptotic behavior of strong solutions.