Speaker
Description
In this talk, we will present a well posed optimal control problem subject to the Navier-Stokes equations with inhomogeneous do-nothing boundary conditions, as they are frequently used in hemodynamic computations. The simulation of cardiovascular blood flow is an area of research rapidly gaining in significance due to the increased availability of computational resources, and the widespread occurrence of cardiovascular diseases, which these simulations can help diagnose and treat. As a full CFD model of the whole blood circuit however remains infeasible, most blood flow simulations are carried out on truncated domains with different types of boundary conditions: Dirichlet boundary conditions for walls (homogeneous) and inflow boundaries (inhomogeneous), as well as do-nothing boundary conditions on outflow boundaries. For the clinical diagnosis, the spatial distribution of blood pressure is of most importance, which however cannot be measured directly. Due to recent advancements in technology, it is however possible to obtain fully space and time dependent data of the blood velocity (4D-MRI). Pressure then can be recovered by solving an optimal control problem, where the unknown boundary data on the in and outflow boundaries act as controls. Until now, no wellposedness results of such an optimal control problem were available, due to the Navier-Stokes equations with do-nothing boundary conditions only being solvable for small data. As this issue is inherent to the choice of boundary condition, and also present in the 2D Navier-Stokes equations, we present a formulation of an optimal control problem in 2D, that is well posed, despite the issues of the boundary condition. We deduce first and second order optimality conditions and discuss regularity of the optimal state, adjoint state and control. To this end, we present some specially tailored regularity results for the instationary Stokes equations subject to mixed inhomogeneous boundary conditions. Our theoretical results are illustrated with numerical examples.
