| Abstract: |
| We consider the steady Navier-Stokes equations with mixed boundary conditions, where, instead of the classical do-nothing (CDN) outflow boundary condition, a regularized directional do-nothing (RDDN) condition is imposed. For computational purposes, a saddle point approach is chosen for the weak formulation of the problem. An auxiliary reference flow is used, which also works as a lifting of the inhomogeneous Dirichlet boundary conditions.
After proving the well-posedness of the Navier-Stokes equations with RDDN condition, we consider the minimization of a quadratic cost functional of velocity tracking type by means of a control localized on a portion of the inflow boundary. We prove the existence of a solution for this optimal control problem and derive a system of first-order optimality conditions. All these results are obtained under suitable assumptions on the size of the data and the controls, which, however, are less restrictive compared with the case of a CDN outflow condition. |
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