satisfies a very analogous inequality (now the multi-index includes time derivatives as well): for all For physically reasonable conditions, the type of solutions expected are smooth functions that do not grow large as . More precisely, the following assumptions are made: There exists a constant such that for all Condition 1 implies that the functions are smooth and globally defined and condition 2 means that the kinetic energy of the solution is globally bounded. The Millennium Prize conjectures in the whole space (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