**Claude-Louis Navier**and

**George Gabriel Stokes**, describe the motion of a fluid into the space. Given its velocity $\vec{v}$, the pressure $p$, and the kinematic viscosity $\nu$, in presence of an external force $\vec{f}$, the particles' motion in the fluid could be described by the following vector equation: \[\frac{\partial \vec{v}}{\partial t} + ( \vec{v} \cdot \vec \nabla ) \vec{v} = -\vec \nabla p + \nu \Delta \vec{v} +\vec{f}(\vec{x},t)\] The trouble is that, to obtain solutions of this equation, we must introduce approximations that simplify the search of them: for example, a major difficulty is to determine the solutions in the presence of some turbulence. To this problem, that it has a physical nature, we must add another mathematical question: the difficulty in proving, given the initial conditions, the existence of smooth solutions for the equations. Given these difficulties, the Clay Mathematics Institute included it in the list of the seven Millennium Problems:

In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations.