This study proves the existence of smooth periodic solutions for Navier- Stokes three-dimensional equations under the assumption of a given pe- riodic initial velocity vector field with positive viscosity. The solution proposed solves the equation by utilizing a Fourier series representation of periodic initial velocity vector fields and predicting the velocity vec- tor field at all times. The significance of this finding is that it con- tributes positively towards understanding the behavior of solutions of Navier-Stokes equations and suggests that smooth periodic solutions for the given problem can indeed exist under certain conditions. Addition- ally, the authors suggest that their solution can be used to settle the Clay Mathematics Millennium Prize Problem, which seeks to find a solution for Navier-Stokes equations meeting specific criteria. It is important to note, however, that this study does not provide a complete solution to the problem, but it provides a significant contribution to the understanding of the behavior of solutions of Navier-Stokes equations. Overall, this re- search demonstrates that the smooth periodic solutions for Navier-Stokes equations can exist for a given initial velocity vector field with positive viscosity, and it presents a new approach for the Navier-Stokes equation.