highly sensitive to initial conditions, and it is the main reason why the Millennium Prize conjectures are so challenging. In addition to the mathematical challenges of solving the Navier-Stokes equations, there are also many practical challenges in applying these equations to real-world situations. For example, the Navier-Stokes equations are often used to model fluid flows that are turbulent, which means that the fluid is highly chaotic and unpredictable. Turbulence is a difficult phenomenon to model and understand, and it adds another layer of complexity to the problem of solving the Navier-Stokes equations. To solve the Navier-Stokes equations, we need to find a velocity field and a pressure field that satisfy the equations and the given boundary conditions. This can be done using a variety of numerical techniques, such as finite element methods, spectral methods, or finite difference methods. For example, consider the case of a two-dimensional fluid flow in a rectangular domain, with velocity and pressure fields and a pressure field ,respectively. The Navier-Stokes equations can be written as: where is the density, is the kinematic viscosity, and is an external force. The boundary conditions might specify that the velocity is fixed at the walls of the domain, or that the pressure is fixed at certain points. To solve these equations numerically, we can divide the domain into a series of smaller elements, and solve the equations locally within each element. For example, using a finite element method, we might represent the velocity and pressure fields as: where is the number of elements, and are the shape functions associated with each element. Substituting these expressions into the Navier-Stokes equations and applying the finite element method, we can derive a system of ordinary differential equations Statement of the periodic problem Hypotheses The functions sought now are periodic in the space variables of period 1. More precisely, let be the unitary vector in the i- direction: Then is periodic in the space variables if for any , then: Notice that this is considering the coordinates mod 1. This allows working not on the whole space but on the quotient space , which turns out to be the 3-dimensional torus: Now the hypotheses can be stated properly. The initial condition is assumed to be a smooth and divergence-free function and the external force is assumed to be a smooth function as well. The type of solutions that are physically relevant are those who