Institute of Mathematics of the Academy of Sciences of the Czech Republic

**Abstract**: We study a shape optimization problem for the paper
machine headbox which distributes a mixture of water and wood fibers in the
paper making process. The aim is to find a shape which a priori ensures the
given velocity profile on the outlet part. The mathematical formulation leads
to an optimal control problem in which the control variable is the shape of the
domain representing the header, the state problem is represented by a
generalized stationary Navier-Stokes system with nontrivial mixed boundary
conditions. We prove the existence of solutions both to the generalized
Navier-Stokes system and to the shape optimization problem.