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).
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]
Optimal Distributed Control of Two-Dimensional Nonlocal Cahn–Hilliard–Navier–Stokes Systems with Degenerate Mobility and Singular Potential
S. Frigeri;
2018
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).File | Dimensione | Formato | |
---|---|---|---|
FGS_AMO.pdf
accesso aperto
Tipologia:
Post-print, accepted manuscript ecc. (versione accettata dall'editore)
Dimensione
222.27 kB
Formato
Adobe PDF
|
222.27 kB | Adobe PDF | Visualizza/Apri |
Frigeri2018_Article_OptimalDistributedControlOfTwo.pdf
accesso riservato
Tipologia:
Publisher's version/PDF
Dimensione
653.98 kB
Formato
Adobe PDF
|
653.98 kB | Adobe PDF | Visualizza/Apri Richiedi una copia |
1801.02502.pdf
accesso aperto
Tipologia:
Pre-print (manoscritto inviato all'editore)
Dimensione
378.76 kB
Formato
Adobe PDF
|
378.76 kB | Adobe PDF | Visualizza/Apri |
Pubblicazioni consigliate
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.