The global existence of solutions for the motion of viscous incompressible fluids

本文研究了R^3中有界域上非定常N-S系统的初边值问题。我们找到了一个修正的N-S系统。通过使用修正的系统，我们在下文中证明了Navier-Stokes流转变为非牛顿流的存在性。对于给定的初始速度a(X)，我们找到了一个时间整体强解u(x，t)，当速度梯度低于一个正数(一个物理量)时，它始终满足Navier-Stokes系统，当速度梯度大于一个正数时，它一直满足一个非牛顿系统。此外，我们还证明了对于所有的t∈[T_a，∞)，存在T_a&gt；0使得整体解满足N-S系统，并且映射a(X)→u(x，t)在[T_a，∞)中是一对一的。在物理流体动力学中，假设流体的变形率足够小，从而粘性应力与变形率成线性关系，建立了N-S方程。由于变形速率取决于流体中的速度梯度，因此对于较小的速度梯度，Navier-Stokes方程似乎能很好地描述流体的运动。从这个考虑出发，我们得到了一个修正的方程，它还考虑了大速度梯度的流体运动。

This report is concerned with the initial boundary value problem for the nonstationary Navier-Stokes system in a bounded domain in R^3. We have found a modified Navier-Stokes system. By using the modified system we have shown the existence of Navier-Stokes flows changing to non-Newtonian flows in the following. For a given initial velocity a(x) we find a time-global strong solution u(x, t) which satisfies the Navier-Stokes system for all the time when the velocity gradient is below a positive number (a physical quantity) and satisfies a non-Newtonian system for all the time when the velocity gradient is above the number. Furthermore, we have shown that there exists T_a > 0 such that the global solution satisfies the Navier-Stokes system for all t∈[T_a, ∞), and the mapping a(x) → u(x, t) in [T_a, ∞) is one to one.In the physical fluid dynamics, the Navier-Stokes equation is formulated under the assumption that the rate of deformation of fluids is sufficiently small and therefore the viscous stress is linearly related to the rate of deformation. Since the rate of deformation depends on the velocity gradient in the fluids the Navier-Stokes equation seems to be representing the motion of fluids well for small velocity gradients. From such consideration, we have found out a modified equation by taking the motion of fluids for large velocity gradients also into account.