  We consider the motion of incompressible viscous fluids bounded above by a
free surface and below by a solid surface in the $N$-dimensional Euclidean
space for $N\geq 2$ when the gravity is not taken into account. The aim of this
paper is to show the global solvability of the Naiver-Stokes equations with a
free surface, describing the above-mentioned motion, in the maximal
$L_p\text{-}L_q$ regularity class. Our approach is based on the maximal
$L_p\text{-}L_q$ regularity with exponential stability for the linearized
equations, and solutions to the original nonlinear problem are also
exponentially stable.
