  We analyze optimal control problems for two-phase Navier-Stokes equations
with surface tension. Based on $L_p$-maximal regularity of the underlying
linear problem and recent well-posedness results of the problem for
sufficiently small data we show the differentiability of the solution with
respect to initial and distributed controls for appropriate spaces resulting
form the $L_p$-maximal regularity setting. We consider first a formulation
where the interface is transformed to a hyperplane. Then we deduce
differentiability results for the solution in the physical coordinates.
Finally, we state an equivalent Volume-of-Fluid type formulation and use the
obtained differentiability results to derive rigorosly the corresponding
sensitivity equations of the Volume-of-Fluid type formulation. For objective
functionals involving the velocity field or the discontinuous pressure or phase
indciator field we derive differentiability results with respect to controls
and state formulas for the derivative. The results of the paper form an
analytical foundation for stating optimality conditions, justifying the
application of derivative based optimization methods and for studying the
convergence of discrete sensitivity schemes based on Volume-of-Fluid
discretizations for optimal control of two-phase Navier-Stokes equations.
