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We analyze optimal control problems for two-phase Navier-Stokes equationswith surface tension. Based on $L_p$-maximal regularity of the underlyinglinear problem and recent well-posedness results of the problem forsufficiently small data we show the differentiability of the solution withrespect to initial and distributed controls for appropriate spaces resultingform the $L_p$-maximal regularity setting. We consider first a formulationwhere the interface is transformed to a hyperplane. Then we deducedifferentiability results for the solution in the physical coordinates.Finally, we state an equivalent Volume-of-Fluid type formulation and use theobtained differentiability results to derive rigorosly the correspondingsensitivity equations of the Volume-of-Fluid type formulation. For objectivefunctionals involving the velocity field or the discontinuous pressure or phaseindciator field we derive differentiability results with respect to controlsand state formulas for the derivative. The results of the paper form ananalytical foundation for stating optimality conditions, justifying theapplication of derivative based optimization methods and for studying theconvergence of discrete sensitivity schemes based on Volume-of-Fluiddiscretizations for optimal control of two-phase Navier-Stokes equations.

Differentiability results and sensitivity calculation for optimal control of incompressible two-phase Navier-Stokes equations with surface tension