Optimal distributed control of two-dimensional nonlocal Cahn--Hilliard--Navier--Stokes systems with degenerate mobility and singular potential
- Frigeri, Sergio
- Grasselli, Maurizio
- Sprekels, Jürgen
2010 Mathematics Subject Classification
- 35Q30 35R09 49J20 49J50 76T99
- Navier-Stokes equations, nonlocal Cahn-Hilliard equations, degenerate mobility, incompressible binary fluids, phase separation, distributed optimal control, first-order necessary optimality conditions
In this paper, we consider a two-dimensional diffuse interface model for the phase separation of an incompressible and isothermal binary fluid mixture with matched densities. This model consists of the Navier-Stokes equations, nonlinearly coupled with a convective nonlocal Cahn-Hilliard equation. The system rules the evolution of the volume-averaged velocity of the mixture and the (relative) concentration difference of the two phases. The aim of this work is to study an optimal control problem for such a system, the control being a time-dependent external force acting on the fluid. We first prove the existence of an optimal control for a given tracking type cost functional. Then we study the differentiability properties of the control-to-state map, and we establish first-order necessary optimality conditions. These results generalize the ones obtained by the first and the third authors jointly with E. Rocca in . There the authors assumed a constant mobility and a regular potential with polynomially controlled growth. Here, we analyze the physically more relevant case of a degenerate mobility and a singular (e.g., logarithmic) potential. This is made possible by the existence of a unique strong solution which was recently proved by the authors and C. G. Gal in .
- Appl. Math. Optim., 81 (2020), pp. 889--931 (published online on 24.09.2018), DOI 10.1007/s00245-018-9524-7 .