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

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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.