Abstract: In 1966, O. A. Ladyzhenskaya proposed a kind of modified Navier-Stokes equations to describe the three-dimensional nonstationary flows of viscous incompressible fluids without assuming small gradients of the velocities. This paper considered large time behavior of a solution for the modified Navier-Stokes equations in a bounded domain and showed that decay of the solution is exactly of exponential type when force term is equal to zero.
Moreover the ratio of the enstrophy over the energy has a limit as time tends to infinity, and the limit is an eigenvalue of the Stokes operator.

Key words: large time behavior, decay rate, modified Navier-Stokes equations, large time behavior, decay rate