  We introduce and analyze a space-time least-squares method associated to the
unsteady Navier-Stokes system. Weak solution in the two dimensional case and
regular solution in the three dimensional case are considered. From any initial
guess, we construct a minimizing sequence for the least-squares functional
which converges strongly to a solution of the Navier-Stokes system. After a
finite number of iterates related to the value of the viscosity constant, the
convergence is quadratic. Numerical experiments within the two dimensional case
support our analysis. This globally convergent least-squares approach is
related to the damped Newton method when used to solve the Navier-Stokes system
through a variational formulation.
