Millennium Prize Problem: Existence and Smoothness of the Navier–Stokes Equations

Let ( n = 3 ). The incompressible Navier–Stokes equations on ( \mathbb{R}^3 ) are [ \frac{\partial}{\partial t} u_i * \sum_{j=1}^{3} u_j \frac{\partial u_i}{\partial x_j} = \nu \Delta u_i * \frac{\partial p}{\partial x_i} * f_i(x,t), \quad (x \in \mathbb{R}^3,, t \ge 0) ] [ \operatorname{div} u \sum_{i=1}^{3} \frac{\partial u_i}{\partial x_i} = 0 ] with initial condition [\nu(x,0) = u^\circ(x). ] Here, ( u(x,t) \in \mathbb{R}^3 ) is the velocity field, ( p(x,t) \in \mathbb{R} ) the pressure, ( f(x,t) ) an externally applied force, and ( \nu > 0 ) the viscosity. Admissible Data on ( \mathbb{R}^3 ): The initial velocity ( u^\circ(x) ) must be smooth, divergence-free, and satisfy rapid decay; the force ( f(x,t) ) must be smooth and satisfy decay bounds in space and time. Physically Reasonable Solutions on ( \mathbb{R}^3 ): A solution ( (u,p) ) is acceptable only if ( u, p \in C^\infty(\mathbb{R}^3 \times [0,\infty)) ) and the energy is uniformly bounded. Periodic Formulation: Alternatively, consider spatially periodic solutions on ( \mathbb{R}^3/\mathbb{Z}^3 ) with periodic data and solutions in space. Official Statements (exactly one must be proved): (A) Existence and Smoothness on ( \mathbb{R}^3 ); (B) Existence and Smoothness on ( \mathbb{R}^3/\mathbb{Z}^3 ); (C) Breakdown on ( \mathbb{R}^3 ); (D) Breakdown on ( \mathbb{R}^3/\mathbb{Z}^3 ). Context & Limitations: In two dimensions, global smooth solutions are known to exist. In three dimensions, local-in-time smooth solutions exist, and global smooth solutions are known for sufficiently small initial data. Weak solutions always exist, but uniqueness and global regularity remain open. Standard PDE techniques have so far been insufficient to resolve the full problem.