  The object of the present paper is to show the existence and the uniqueness
of a reproductive strong solution of the Navier-Stokes equations, i.e. the
solution $\boldsymbol{u} $ belongs to $\text{}\mathbf{L}% ^{\infty}(0,T;V) \cap
\mathbf{L}^{2}(0,T;\mathbf{H}% ^{2}(\Omega))$ and satisfies the property
$\boldsymbol{u}% (\boldsymbol{x,}T) =\boldsymbol{u}% (\boldsymbol{x,}0)
=\boldsymbol{u}_{0}(\boldsymbol{x})$. One considers the case of an
incompressible fluid in two dimensions with nonhomogeneous boundary conditions,
and external forces are neglected.
