text: (A) Existence and smoothness of the Navier–Stokes solutions in 
  Let . For any initial condition  satisfying the above hypotheses there exist smooth and globally defined solutions to the Navier–Stokes equations, i.e. there is a velocity vector  and a pressure  satisfying conditions 1 and 2 above.
  (B) Breakdown of the Navier–Stokes solutions in 
  There exists an initial condition  and an external force  such that there exists no solutions  and  satisfying conditions 1 and 2 above.
  The Millennium Prize conjectures are two mathematical problems that were chosen by the Clay Mathematics Institute as the most important unsolved problems in mathematics. The first conjecture, which is known as the "smoothness" conjecture, states that there should always exist smooth and globally defined solutions to the Navier–Stokes equations in three-dimensional space. The second conjecture, known as the "breakdown" conjecture, states that there should be at least one set of initial conditions and external forces for which there are no smooth solutions to the Navier–Stokes equations.
  The Navier–Stokes equations are a set of partial differential equations that describe the motion of fluids. They are given by:
  where  is the velocity field of the fluid,  is the pressure,  is the density,  is the kinematic viscosity, and  is an external force. The first equation is known as the momentum equation, and the second equation is known as the continuity equation.
  These equations are typically accompanied by boundary conditions, which describe the behavior of the fluid at the edges of the domain. For example, in the case of a fluid flowing through a pipe, the boundary conditions might specify that the velocity and pressure are fixed at the walls of the pipe.
  The Navier–Stokes equations are nonlinear and highly coupled, making them difficult to solve in general. In particular, the difficulty of solving these equations lies in the term , which represents the nonlinear advection of the velocity field by itself. This term makes the Navier–Stokes equations 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. The last identity occurs because the flow is solenoidal.
  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