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 ( v ⋅ ⋅ ∇ ∇ ) v , 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.