A Class Of Non-homogeneous Navier-stokes Equations Qualitative Research

When the speed of the flow is slower and the viscosity is larger, the convection term p(u ·V)u in the general non-homogeneous incompressible Navier-Stokes equations may be neglectedwhere Ω is an open bounded subset of R3 with a smooth boundary.The unknown function u = u(x, t) is the velocity field of the flow, p = p(x, t) is the density of the flow ,p = p(x,t) is the pressure of the flow,f = f(x,t) is the known external potential , u0 = u0(x)and p0 = p0(x) are the initial velocity and density rcspectively,and constant u > 0 is the viscosity coefficient of the flow .In this paper ,we mainly study the global existence of the weak solutions ,the global existence and uniqueness of the strong solutions and the classical solutions. The contents of the paper include two parts:First , we consider the global existence of the weak solutions of the 3-D non-homogeneous incompressible Navier-Stokes equations. To do so, we employ the "scmi-Galerkin method" to construct the sequence of approximate solutions, and further to get the estimates of the approximate solutions and their derivatives. Then we employ the Sobolev uniform imbedding theorems to obtain the global existence of the weak solutions.Second, we consider the global existence and uniqueness of strong solutions. By improving the regularity estimates of the approximate solutions in Sobolev space, we arc able to prove the global existence and uniqueness of the strong solutions.