On the existence and uniqueness of the solution to the Navier-Stokes equations

In this paper we discuss the mixed boundary problem (1) - (3) of the Navier-Stokes equations for the flow of an incompressible viscous fluid in a bounded domain. We prove that when g is an element of L (infinity) (Q), phi is an element of C-1 (Gamma(1) X [0, t(1)]), and psi is an element of L(1) (Gamma(z) X [0, t(1)]), there exists a weak solution of (1) - (3) t and when u is an element of L(2) (0, t(1) ; H-s (Omega)), T is an element of L(2) (0, t(1) ; H-s (Omega)), the weak solution is unique, if it exists.