Skip to main content
SpringerLink
Log in

Optimal Distributed Control of Two-Dimensional Nonlocal Cahn–Hilliard–Navier–Stokes Systems with Degenerate Mobility and Singular Potential

  • Published:
Applied Mathematics & Optimization Submit manuscript

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 \(\varvec{u}\) of the mixture and the (relative) concentration difference \(\varphi \) 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 \(\varvec{v}\) 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 \(\varvec{v}\mapsto [\varvec{u},\varphi ]\), 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).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Anderson, D.M., McFadden, G.B., Wheeler, A.A.: Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30, 139–165 (1998)

    Article  MathSciNet  Google Scholar 

  2. Agmon, S.: Lectures on Elliptic Boundary Value Problems. Mathematical Studies. Van Nostrand, New York (1965)

    MATH  Google Scholar 

  3. Bedrossian, J., Rodríguez, N., Bertozzi, A.: Local and global well-posedness for an aggregation equation and Patlak–Keller–Segel models with degenerate diffusion. Nonlinearity 24, 1683–1714 (2011)

    Article  MathSciNet  Google Scholar 

  4. Besov, O.V., Il’in, V.P., Nikol’skiĭ, S.M.: Integral Representations of Functions and Embedding Theorems. Scripta Series in Mathematics, vol. II. M.H. Taibleson. V.H. Winston & Sons, Washington, DC; Halsted Press [Wiley], New York (1979)

  5. Bastea, S., Esposito, R., Lebowitz, J.L., Marra, R.: Sharp interface motion of a binary fluid mixture. J. Stat. Phys. 124, 445–483 (2006)

    Article  MathSciNet  Google Scholar 

  6. Boyer, F.: Mathematical study of multi-phase flow under shear through order parameter formulation. Asymptot. Anal. 20, 175–212 (1999)

    MathSciNet  MATH  Google Scholar 

  7. Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)

    Article  Google Scholar 

  8. Colli, P., Frigeri, S., Grasselli, M.: Global existence of weak solutions to a nonlocal Cahn–Hilliard–Navier–Stokes system. J. Math. Anal. Appl. 386, 428–444 (2012)

    Article  MathSciNet  Google Scholar 

  9. Colli, P., Gilardi, G., Sprekels, J.: Optimal velocity control of a viscous Cahn–Hilliard system with convection and dynamic boundary conditions. SIAM J. Control Optim. 56, 1665–1691 (2018)

    Article  MathSciNet  Google Scholar 

  10. Colli, P., Gilardi, G., Sprekels, J.: Optimal velocity control of a convective Cahn–Hilliard system with double obstacles and dynamic boundary conditions: a ‘deep quench’ approach, to appear 2019 in J. Convex Anal. 26 (see also Preprint arXiv:1709.03892 [math. AP] (2017) 1–30)

  11. DiBenedetto, E.: Real Analysis. Advanced Text Series. Birkhäuser, Boston (2002)

    Book  Google Scholar 

  12. Elliott, C.M., Garcke, H.: On the Cahn–Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27, 404–423 (1996)

    Article  MathSciNet  Google Scholar 

  13. Frigeri, S., Gal, C.G., Grasselli, M.: On nonlocal Cahn–Hilliard–Navier–Stokes systems in two dimensions. J. Nonlinear Sci. 26, 847–893 (2016)

    Article  MathSciNet  Google Scholar 

  14. Frigeri, S., Gal, C.G., Grasselli, M., Sprekels, J.: Two-Dimensional Nonlocal Cahn–Hilliard–Navier–Stokes Systems with Variable Viscosity, Degenerate Mobility and Singular Potential. WIAS Preprint Series No. 2309, Berlin (2016). Submitted

  15. Frigeri, S., Grasselli, M.: Global and trajectories attractors for a nonlocal Cahn–Hilliard–Navier–Stokes system. J. Dyn. Differ. Equ. 24, 827–856 (2012)

    Article  Google Scholar 

  16. Frigeri, S., Grasselli, M.: Nonlocal Cahn–Hilliard–Navier–Stokes systems with singular potentials. Dyn. Partial Differ. Equ. 9, 273–304 (2012)

    Article  MathSciNet  Google Scholar 

  17. Frigeri, S., Grasselli, M., Krejčí, P.: Strong solutions for two-dimensional nonlocal Cahn–Hilliard–Navier–Stokes systems. J. Differ. Equ. 255, 2597–2614 (2013)

    Article  MathSciNet  Google Scholar 

  18. Frigeri, S., Grasselli, M., Rocca, E.: A diffuse interface model for two-phase incompressible flows with nonlocal interactions and nonconstant mobility. Nonlinearity 28, 1257–1293 (2015)

    Article  MathSciNet  Google Scholar 

  19. Frigeri, S., Rocca, E., Sprekels, J.: Optimal distributed control of a nonlocal Cahn–Hilliard/Navier–Stokes system in 2D. SIAM J. Control Optim. 54, 221–250 (2016)

    Article  MathSciNet  Google Scholar 

  20. Gal, C.G., Giorgini, A., Grasselli, M.: The nonlocal Cahn–Hilliard equation with singular potential: well-posedness, regularity and strict separation property. J. Differ. Equ. 263, 5253–5297 (2017)

    Article  MathSciNet  Google Scholar 

  21. Giacomin, G., Lebowitz, J.L.: Exact macroscopic description of phase segregation in model alloys with long range interactions. Phys. Rev. Lett. 76, 1094–1097 (1996)

    Article  Google Scholar 

  22. Giacomin, G., Lebowitz, J.L.: Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits. J. Stat. Phys. 87, 37–61 (1997)

    Article  MathSciNet  Google Scholar 

  23. Giacomin, G., Lebowitz, J.L.: Phase segregation dynamics in particle systems with long range interactions. II. Phase motion. SIAM J. Appl. Math. 58, 1707–1729 (1998)

    Article  MathSciNet  Google Scholar 

  24. Gurtin, M.E., Polignone, D., Viñals, J.: Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Methods Appl. Sci. 6, 8–15 (1996)

    Article  MathSciNet  Google Scholar 

  25. Heida, M., Málek, J., Rajagopal, K.R.: On the development and generalizations of Cahn–Hilliard equations within a thermodynamic framework. Z. Angew. Math. Phys. 63, 145–169 (2012)

    Article  MathSciNet  Google Scholar 

  26. Hintermüller, M., Hinze, M., Kahle, C.: An adaptive finite element Moreau–Yosida-based solver for a coupled Cahn–Hilliard/Navier–Stokes system. J. Comput. Phys. 235, 810–827 (2013)

    Article  MathSciNet  Google Scholar 

  27. Hintermüller, M., Hinze, M., Kahle, C., Keil, T.: A goal-oriented dual-weighted adaptive finite element approach for the optimal control of a nonsmooth Cahn–Hilliard–Navier–Stokes system, WIAS Preprint No. 2311, Berlin (2016)

  28. Hintermüller, M., Keil, T., Wegner, D.: Optimal control of a semidiscrete Cahn–Hilliard–Navier–Stokes system with non-matched fluid densities. SIAM J. Control Optim. 55, 1954–1989 (2018)

    Article  Google Scholar 

  29. Hintermüller, M., Wegner, D.: Distributed optimal control of the Cahn–Hilliard system including the case of a double-obstacle homogeneous free energy density. SIAM J. Control Optim. 50, 388–418 (2012)

    Article  MathSciNet  Google Scholar 

  30. Hintermüller, M., Wegner, D.: Optimal control of a semidiscrete Cahn–Hilliard–Navier–Stokes system. SIAM J. Control Optim. 52, 747–772 (2014)

    Article  MathSciNet  Google Scholar 

  31. Hintermüller, M., Wegner, D.: Distributed and boundary control problems for the semidiscrete Cahn–Hilliard/Navier–Stokes system with nonsmooth Ginzburg–Landau energies. Topol. Optim. Optim. Transp. Radon Ser. Comput. Appl. Math. 17, 40–63 (2017)

    MathSciNet  MATH  Google Scholar 

  32. Lions, J.-L.: Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires. Dunod Gauthier-Villars, Paris (1969)

    MATH  Google Scholar 

  33. Rocca, E., Sprekels, J.: Optimal distributed control of a nonlocal convective Cahn–Hilliard equation by the velocity in three dimensions. SIAM J. Control Optim. 53, 1654–1680 (2015)

    Article  MathSciNet  Google Scholar 

  34. Temam, R.: Navier–Stokes Equations and Nonlinear Functional Analysis, Second Edition, CBMS-NSF Regional Conference Series Applied Mathematics, vol. 66. SIAM, Philadelphia, PA (1995)

Download references

Acknowledgements

Sergio Frigeri and Maurizio Grasselli are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Sergio Frigeri is “titolare di un Assegno di Ricerca dell’Istituto Nazionale di Alta Matematica”.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica “N. Tartaglia”, Università Cattolica del Sacro Cuore (Brescia), Via Musei 41, 25121, Brescia, Italy

    Sergio Frigeri

  2. Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi 9, 20133, Milano, Italy

    Maurizio Grasselli

  3. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117, Berlin, Germany

    Jürgen Sprekels

  4. Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Jürgen Sprekels

Authors
  1. Sergio Frigeri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maurizio Grasselli
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jürgen Sprekels
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maurizio Grasselli.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Frigeri, S., Grasselli, M. & Sprekels, J. Optimal Distributed Control of Two-Dimensional Nonlocal Cahn–Hilliard–Navier–Stokes Systems with Degenerate Mobility and Singular Potential. Appl Math Optim 81, 899–931 (2020). https://doi.org/10.1007/s00245-018-9524-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00245-018-9524-7

Keywords

Mathematics Subject Classification

Search

Navigation