Optimal Distributed Control of Two-Dimensional Nonlocal Cahn–Hilliard–Navier–Stokes Systems with Degenerate Mobility and Singular Potential
Articolo
Data di Pubblicazione:
2018
Citazione:
Optimal Distributed Control of Two-Dimensional Nonlocal Cahn–Hilliard–Navier–Stokes Systems with Degenerate Mobility and Singular Potential / S. Frigeri, M. Grasselli, J. Sprekels. - In: APPLIED MATHEMATICS AND OPTIMIZATION. - ISSN 0095-4616. - (2018). [Epub ahead of print] [10.1007/s00245-018-9524-7]
Abstract:
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 u 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 v 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 v↦ [ u, φ] , and we establish first-order necessary optimality conditions. These results generalize the ones obtained by the first and the third authors jointly with Rocca (SIAM J Control Optim 54:221–250, 2016). 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 Gal (WIAS preprint series No. 2309, Berlin, 2016).
Tipologia IRIS:
01 - Articolo su periodico
Keywords:
Navier–Stokes equations; Nonlocal Cahn–Hilliard equations; Degenerate mobility; Incompressible binary fluids; Phase separation; Distributed optimal control; First-order necessary optimality conditions
Elenco autori:
S. Frigeri, M. Grasselli, J. Sprekels
Link alla scheda completa: