2018/01/05 by Frigeri, S., Grasselli, M., Sprekels, J. · 1 citation
#35Q30 #35R09 #49J20 #49J50 #76T99 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1801.02502
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 \uvec of the mixture and the (relative) concentration difference φ 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 \vvec ↦ [\uvec,φ], 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 [19]. 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 [14].