Shape sensitivity analysis for the work functional

The non stationary, compressible Navier-Stokes equations are considered in a bounded hold-all domain. The nonhomegeneous boundary conditions are prescribed on the boundary of hold-all domain. The existence of the so-called weak normalized solutions for the model is established in [2]. In this talk we consider the associated shape optimization problems for the model. In the stationary case the drag of an obstacle is minimized. In the non stationary case the work for a fight scenario of an obstacle is minimized. It means that the boundary of an obstacle flying in a gas is optimized in such a way that for the given trajectory the energy required to attain a given point in the domain is minimized with respect to the shape of the obstacle. The family of admissible obstacles is sufficiently general and includes the standard shapes of an airfoil. We present the new results on the shape sensitivity analysis of the work functional for non stationary compressible Navier-Stokes equations [1]. In particular, the shape gradient is determined for the shape functional. The obtained results can be justified from mathematical point of view for the local solutions, or the global classical solutions of the model.