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arxiv: 1907.04285 · v1 · pith:KUP46G7Xnew · submitted 2019-07-09 · 🧮 math.OC

Simulation and Control of a Nonsmooth Cahn-Hilliard Navier-Stokes System

Carmen Gr\"a{\ss}le , Michael Hinterm\"uller , Michael Hinze , Tobias Keil This is my paper

Pith reviewed 2026-05-25 00:14 UTC · model grok-4.3

classification 🧮 math.OC
keywords optimal controlCahn-Hilliard Navier-Stokesnonsmooth potentialbilevel optimizationstationarity conditionsmodel order reductionproper orthogonal decompositionadaptive error estimator
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The pith

Optimal solutions exist for controlling a nonsmooth two-phase flow model, with C- and strong stationarity conditions derived for the bilevel problem.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes existence of optimal solutions to a bilevel optimal control problem for a coupled Cahn-Hilliard Navier-Stokes system with nonsmooth energy potential. It derives stationarity conditions in two forms, C-stationarity and strong stationarity, and demonstrates their numerical use through adaptive algorithms based on a goal-oriented error estimator. The work also develops a proper orthogonal decomposition model order reduction that incorporates space-adapted snapshots while preserving the solenoidal property of the velocity field. A sympathetic reader would care because these results make rigorous optimization feasible for two-phase flow simulations where phase separation must be controlled under nonsmooth physics.

Core claim

We establish the existence of optimal solutions and present two distinct approaches to derive suitable stationarity conditions for the bilevel problem, namely C- and strong stationarity. The numerical realization relies on two adaptive solution algorithms that use a specifically developed goal-oriented error estimator. A model order reduction approach using proper orthogonal decomposition replaces high-fidelity models by low-order surrogates, combining POD with space-adapted snapshots to handle different spatial resolutions while conserving the solenoidal property.

What carries the argument

Bilevel optimal control problem for the nonsmooth Cahn-Hilliard Navier-Stokes system, with derivation of C-stationarity and strong stationarity conditions as the core mechanism for characterizing optimality.

If this is right

  • Existence of optimal solutions makes the bilevel control problem well-posed for two-phase flows.
  • C- and strong stationarity conditions supply necessary optimality criteria that support numerical solvers.
  • Goal-oriented adaptive algorithms become applicable for accurate simulation and control.
  • POD-MOR with space-adapted snapshots yields low-order surrogates that maintain the divergence-free velocity constraint.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The stationarity approaches might extend to other coupled nonsmooth fluid models with similar bilevel structure.
  • The handling of varying snapshot resolutions could inform adaptive reduced-order methods in related multiphase problems.
  • Numerical tests could check whether strong stationarity yields tighter control bounds than C-stationarity in practice.

Load-bearing premise

The nonsmooth energy potential and the coupled PDE system are assumed to be sufficiently regular to admit optimal solutions and to permit the derivation of C- and strong stationarity conditions.

What would settle it

A concrete example of a nonsmooth Cahn-Hilliard Navier-Stokes system where no optimal control exists, or where computed solutions violate the derived C- or strong stationarity conditions.

Figures

Figures reproduced from arXiv: 1907.04285 by Carmen Gr\"a{\ss}le, Michael Hinterm\"uller, Michael Hinze, Tobias Keil.

Figure 1
Figure 1. Figure 1: The initial shape ϕ0, the desired shape ϕd, the ansatz for the control u [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The evolution of the phase field ϕ. The optimal solution on the first level and for the initial value for α is found after 26 steepest descent iterations, while the complete algorithm terminates after 419 steepest descent steps. Hereby, the algorithm solves the auxiliary optimization problems 10 times, i.e. line 3 of Algorithm 1 is executed 10 times. After the first two solves the Moreau–Yosida parameter w… view at source ↗
Figure 3
Figure 3. Figure 3: The magnitude of v in grayscale and the isolines ϕ ≡ ±1 (left), and the associated triangulation (right). where the stabilizing term khk 2 L2 ensures the existence of solutions. If a solution h of (29) equals zero, then u ∗ is a B-stationary point, otherwise it is indeed a descent direction, since J 0 [u ∗ ](h) ≤ −khk 2 < 0. In combination with a classical line search procedure, this leads to the following… view at source ↗
Figure 4
Figure 4. Figure 4: The initial shape ϕ0, the desired shape ϕd, the ansatz for the control u [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The evolution of the phase field ϕ, the slack variable a and the magnitude of v at the final time. The parameters for the physical model and the adaptation procedure are adopted from the previous example. In this example, the algorithm terminates at a C￾stationary point after performing the Armijo line search (in line 3) 276 times. The maximum number of cells is exceeded after 6 mesh refinement steps [PIT… view at source ↗
Figure 6
Figure 6. Figure 6: Decay of the normalized eigenvalues for the phase field ϕ considering a Moreau-Yosida relaxation (DOEr) for different re￾laxation parameters r and a polynomial free energy (pDWE) In future research, we plan to apply POD model order reduction for the Cahn￾Hilliard equations using a nonsmooth double-obstacle potential. This involves reduced-order modeling for variational inequalities, see e.g. [16] for a red… view at source ↗
Figure 7
Figure 7. Figure 7: Finite element snapshots of the phase field at t = 0, t = T /2 and t = T (top) with the associated adapted finite element meshes (bottom) of node points varies between 16779 and 19808 and the finite element simulation time is 1674 sec. In order to construct a POD reduced-order model, we utilize the adapted finite element solutions for the phase field as snapshots in (31), where we choose X = L 2 (Ω) for th… view at source ↗
Figure 8
Figure 8. Figure 8: POD reduced-order approximation of the phase field at t = 0, t = T /2 and t = T using ` = 10 POD modes (top) and ` = 20 POD modes (bottom) The relative L 2 (0, T; Ω)-error between the adaptive finite element solution and the POD reduced-order solution using ` = 20 POD modes is 2.793 · 10−4 . The solution time for the reduced-order simulation is 88 sec, which leads to a speed up factor of 19 compared to the… view at source ↗
read the original abstract

We are concerned with the simulation and control of a two phase flow model governed by a coupled Cahn-Hilliard Navier-Stokes system involving a nonsmooth energy potential. We establish the existence of optimal solutions and present two distinct approaches to derive suitable stationarity conditions for the bilevel problem, namely C- and strong stationarity. Moreover, we demonstrate the numerical realization of these concepts at the hands of two adaptive solution algorithms relying on a specifically developed goal-oriented error estimator. In addition, we present a model order reduction approach using proper orthogonal decomposition (POD-MOR) in order to replace high-fidelity models by low order surrogates. In particular, we combine POD with space-adapted snapshots and address the challenges which are the consideration of snapshots with different spatial resolutions and the conservation of a solenoidal property.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript studies the simulation and optimal control of a coupled Cahn-Hilliard Navier-Stokes system with a nonsmooth energy potential modeling two-phase flows. It proves existence of optimal solutions to the associated bilevel control problem and derives C-stationarity and strong stationarity conditions via two distinct approaches. Numerically, the work develops two adaptive solution algorithms that employ a goal-oriented error estimator and introduces a POD-based model-order-reduction technique that accommodates snapshots of differing spatial resolutions while preserving the solenoidal constraint.

Significance. If the existence and stationarity results are valid, the paper supplies rigorous first-order conditions for a class of nonsmooth, coupled PDE-constrained optimization problems that are otherwise difficult to treat. The combination of limiting subdifferential calculus with regularization arguments and the practical demonstration of adaptive and reduced-order methods constitute a concrete advance for control of two-phase flow models.

minor comments (2)
  1. [Introduction] The abstract states that two distinct approaches are used to obtain C- and strong stationarity, yet the introduction does not explicitly contrast the two routes (e.g., which assumptions each route relaxes). Adding a short comparative paragraph would improve readability.
  2. [Section on POD-MOR] In the POD-MOR section the preservation of the divergence-free property after projection onto the reduced basis is asserted, but the precise construction of the solenoidal reduced space (e.g., via a discrete Helmholtz decomposition or constrained POD) is not detailed; a short algorithmic box or equation would clarify the implementation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report correctly identifies the core contributions on existence of optimal solutions, C- and strong stationarity conditions for the bilevel problem, goal-oriented adaptive algorithms, and the POD-MOR approach handling space-adapted snapshots while preserving the divergence-free constraint.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes existence of optimal solutions for the bilevel control problem and derives C- and strong stationarity conditions via standard limiting subdifferential arguments, regularization, and function-space compactness results for the nonsmooth Cahn-Hilliard-Navier-Stokes system. No load-bearing step reduces by definition or self-citation to a fitted input or prior result whose validity depends on the present work; the stationarity systems are obtained from external PDE theory rather than internal renaming or ansatz smuggling. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the well-posedness of the nonsmooth Cahn-Hilliard Navier-Stokes system and the applicability of bilevel optimal control theory; no free parameters, invented entities, or additional ad-hoc axioms are stated in the abstract.

axioms (1)
  • domain assumption The nonsmooth Cahn-Hilliard Navier-Stokes system admits solutions that allow existence of optimal controls and derivation of stationarity conditions.
    Invoked to establish existence of optimal solutions and stationarity for the bilevel problem.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. An optimal control problem for Stokes-Cahn-Hilliard-Oono equations with regular potential

    math.OC 2026-05 unverdicted novelty 4.0

    Existence of an optimal control is established for the Stokes-Cahn-Hilliard-Oono equations, and first-order optimality conditions are derived via the corresponding adjoint system.

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Works this paper leans on

68 extracted references · 68 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    Aland, S

    S. Aland, S. Boden, A. Hahn, F. Klingbeil, M. Weismann, and S. Weller, Quantitative comparison of Taylor flow simulations based on sharp-interface and diffuse-interface models, International Journal for Numerical Methods in Fluids, 73 (2013), pp. 344–361

    work page 2013

  2. [2]

    Aland, J

    S. Aland, J. Lowengrub, and A. Voigt , Particles at fluid-fluid interfaces: A new Navier-Stokes-Cahn-Hilliard surface-phase-field-crystal model, Physical Review E, 86 (2012), p. 046321

    work page 2012

  3. [3]

    and Voigt, A

    Aland, S. and Voigt, A. (2012). Benchmark computations of diffuse interface models for two-dimensional bubble dynamics. International Journal for Numerical Methods in Fluids , 69:747–761

    work page 2012

  4. [4]

    J. O. Alff, Modellordnungsreduktion f¨ ur das Cahn-Hilliard System, Bachelorarbeit, Univer- sit¨ at Hamburg (2015)

    work page 2015

  5. [5]

    Arian, M

    E. Arian, M. Fahl, E. W. Sachs, Trust-region proper orthogonal decomposition for flow con- trol, Technical Report 2000-25 (2000), ICASE

    work page 2000

  6. [6]

    Abels, H

    H. Abels, H. Garcke, G. Gr¨ un, Thermodynamically consistent, frame indifferent diffuse in- terface models for incompressible two-phase flows with different densities , Math. Models Methods Appl. Sci. 22(3) (2012)

    work page 2012

  7. [7]

    A. Alla, C. Gr¨ aßle, M. Hinze, A-posteriori snapshot location for POD in optimal control of linear parabolic equations, ESAIM: M2AN 52(5) (2018), 1847–1873

    work page 2018

  8. [8]

    A. Alla, N. Kutz, Nonlinear model order reduction via dynamic mode decomposition , SIAM J. Sci. Comput., 39(5) (2017), B778–B796

    work page 2017

  9. [9]

    J. O. Alff, Trust Region POD for Optimal Control of Cahn-Hilliard Systems, Master’s Thesis, Universit¨ at Hamburg (2018)

    work page 2018

  10. [10]

    Barbu, Optimal control of variational inequalities , vol

    V. Barbu, Optimal control of variational inequalities , vol. 100 of Research Notes in Mathe- matics, Pitman (Advanced Publishing Program), Boston, MA, 1984

    work page 1984

  11. [11]

    Borrvall and J

    T. Borrvall and J. Petersson. Topology optimization of fluids in Stokes flow . Internat. J. Numer. Methods Fluids 41(1) (2003), 77–107

    work page 2003

  12. [12]

    Boyer, A theoretical and numerical model for the study of incompressible mixture flows , Computers & fluids, 31 (2002), pp

    F. Boyer, A theoretical and numerical model for the study of incompressible mixture flows , Computers & fluids, 31 (2002), pp. 41–68

    work page 2002

  13. [13]

    Boyer, L

    F. Boyer, L. Chupin, and P. Fabrie , Numerical study of viscoelastic mixtures through a Cahn-Hilliard flow model , Eur. J. Mech. B Fluids, 23 (2004), pp. 759–780

    work page 2004

  14. [14]

    Interfaces Free Bound

    Brett, C., Elliott, C.M., Hinterm¨ uller, M., L¨ obhard, C.: Mesh adaptivity in optimal control of elliptic variational inequalities with point-tracking of the state. Interfaces Free Bound. 17(1), 21–53 (2015)

    work page 2015

  15. [15]

    Blowey, C

    J. Blowey, C. Elliot, The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy. Part I: Mathematical analysis , Eur. J. Appl. Math., 2 (1991), 233–280

    work page 1991

  16. [16]

    Burkovska, B

    O. Burkovska, B. Haasdonk, J. Salomon, B. Wohlmuth, Reduced Basis Methods for Pricing Options with the Black-Scholes and Heston Models , SIAM J. Financial Math., 6(1) (2015), 685–712

    work page 2015

  17. [17]

    empirical interpolation

    M. Barrault, Y. Maday, N. C. Nguyen, A. T. Patera, An “empirical interpolation” method: application to efficient reduced-basis discretization of partial differential equations , C. R. Acad. Sci. Paris, 339(9) (2004), 667–672

    work page 2004

  18. [18]

    Ballarin, A

    F. Ballarin, A. Manzoni, A. Quarteroni, G. Rozza, Supremizer stabilization of POD-Galerkin approximation of parametrized steady incompressible Navier-Stokes equations, Int. J. Numer. Methods Eng. 102(5) (2015), 1136–1161

    work page 2015

  19. [19]

    Colli, M

    P. Colli, M. H. Farshbaf-Shaker, G. Gilardi, and J. Sprekels, Optimal boundary control of a viscous Cahn-Hilliard system with dynamic boundary condition and double obstacle potentials, SIAM J. Control Optim., 53 (2015), pp. 2696–2721. SIMULATION AND CONTROL OF A NONSMOOTH CHNS SYSTEM 23

    work page 2015

  20. [20]

    Cagniart, Y

    N. Cagniart, Y. Maday, B. Stamm, Model Order Reduction with large Convection Effects , Contributions to Partial Differential Equations (2018), 131–150

    work page 2018

  21. [21]

    Chaturantabut, D

    S. Chaturantabut, D. C. Sorensen, Nonlinear model order reduction via discrete empirical interpolation, SIAM J. Sci. Comput., 32(5) (2010), 2737–2764

    work page 2010

  22. [22]

    H. Ding, P. DM Spelt, and C. Shu , Diffuse interface model for incompressible two-phase flows with large density ratios, Journal of Computational Physics, 226 (2007), pp. 2078–2095

    work page 2007

  23. [23]

    Eckert, P

    S. Eckert, P. A. Nikrityuk, B. Willers, D. R ¨abiger, N. Shevchenko, H. Neumann- Heyme, V. Travnikov, S. Odenbach, A. Voigt, and K. Eckert , Electromagnetic melt flow control during solidification of metallic alloys , The European Physical Journal Special Topics, 220 (2013), pp. 123–137

    work page 2013

  24. [24]

    C. M. Elliott, A. Stuart, The global dynamics of discrete semilinear parabolic euqations , SIAM J. Numer. Anal., 30(6) (1993), 1622–1663

    work page 1993

  25. [25]

    D. J. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation , MRS Online Proceedings Library Archive 529 (1998)

    work page 1998

  26. [26]

    C. M. Elliott and Z. Songmu, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986), pp. 339–357

    work page 1986

  27. [27]

    Frigeri, E

    S. Frigeri, E. Rocca, and J. Sprekels , Optimal distributed control of a nonlocal Cahn- Hilliard/Navier-Stokes system in two dimensions , SIAM J. Control Optim., 54 (2016), pp. 221–250

    work page 2016

  28. [28]

    C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire, 27 (2010), pp. 401–436

    work page 2010

  29. [29]

    Gr¨ aßle, M

    C. Gr¨ aßle, M. Hinze,POD reduced-order modeling for evolution equations utilizing arbitrary finite element discretizations , Adv. Comput. Math. 44(6) (2018), 1941–1978

    work page 2018

  30. [30]

    Garcke, M

    H. Garcke, M. Hinze, C. Kahle, A stable and linear time discretization for a thermodynam- ically consistent model for two-phase incompressible flow , Appl. Numer. Math., 99 (2016), 151–171

    work page 2016

  31. [31]

    Garcke, M

    H. Garcke, M. Hinze, C. Kahle, Optimal control of time-discrete two-phase flow driven by a diffuse-interface model, ESAIM Control Optim. Calc. Var., 25 (2019), 2018006, 31

    work page 2019

  32. [32]

    Gr¨ aßle, M

    C. Gr¨ aßle, M. Hinze,The combination of POD model reduction with adaptive finite element methods in the context of phase field models , PAMM, 17(1) (2017), 47–50

    work page 2017

  33. [33]

    Gr¨ aßle, M

    C. Gr¨ aßle, M. Hinze, J. Lang, S. Ullmann, POD model order reduction with space-adapted snapshots for incompressible flows , accepted for publication in Adv. Comput. Math. (2018), preprint available https://arxiv.org/abs/1810.03892

    work page arXiv 2018

  34. [34]

    Gr¨ aßle, M

    C. Gr¨ aßle, M. Hinze, N. Scharmacher,POD for optimal control of the Cahn-Hilliard system using spatially adapted snapshots , in Numerical Mathematics and Advanced Applications ENUMATH 2017 (2019), 703–711

    work page 2017

  35. [35]

    Gr¨ un, F

    G. Gr¨ un, F. Klingbeil, Two-phase flow with mass density contrast: stable schemes for a thermodynamic consistent and frame-indifferent diffuse interface model , J. Comput. Phys., 257 (2014), 708–725

    work page 2014

  36. [36]

    Hinterm¨ uller, M

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

    work page 2013

  37. [37]

    Hinterm¨ uller, M

    M. Hinterm¨ uller, M. Hinze, C. Kahle, T. Keil, A goal-oriented dual-weighted adaptive fi- nite element approach for the optimal control of a nonsmooth Cahn-Hilliard Navier-Stokes system, Optim. Eng. 19(3) (2018), 629–662

    work page 2018

  38. [38]

    Hinterm¨ uller, M

    M. Hinterm¨ uller, M. Hinze, M. Tber,An adaptive finite element Moreau-Yosida-based solver for a nonsmooth Cahn-Hilliard problem , Optim. Method. Softw., 25 (2011), 777-811

    work page 2011

  39. [39]

    Mathematical Programming 160(1), 271–305 (2016)

    Hinterm¨ uller, M., Surowiec, T.: A bundle-free implicit programming approach for a class of elliptic MPECs in function space. Mathematical Programming 160(1), 271–305 (2016)

    work page 2016

  40. [40]

    Hinterm¨ uller, M., Keil, T., Wegner, D.: Optimal control of a semidiscrete Cahn-Hilliard- Navier-Stokes system with nonmatched fluid densities. SIAM J. Control Optim. 55(3), 1954– 1989 (2017)

    work page 1954

  41. [41]

    Hinterm ¨uller and I

    M. Hinterm ¨uller and I. Kopacka , Mathematical programs with complementarity con- straints in function space: C- and strong stationarity and a path-following algorithm , SIAM J. Optim., 20 (2009), pp. 868–902

    work page 2009

  42. [42]

    Hinterm¨uller, B

    M. Hinterm¨uller, B. S. Mordukhovich, and T. M. Surowiec, Several approaches for the derivation of stationarity conditions for elliptic MPECs with upper-level control constraints , Math. Program., 146 (2014), pp. 555–582. 24 GR ¨ASSLE, HINTERM ¨ULLER, HINZE, AND KEIL

    work page 2014

  43. [43]

    Hinterm¨uller and D

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

    work page 2012

  44. [44]

    Hinterm ¨uller and D

    M. Hinterm ¨uller and D. Wegner , Distributed and boundary control problems for the semidiscrete Cahn-Hilliard/Navier-Stokes system with nonsmooth Ginzburg-Landau ener- gies, Isaac Newton Institute preprint:NI14042-FRB, (2014)

    work page 2014

  45. [45]

    Hinterm¨uller and D

    M. Hinterm¨uller and D. Wegner, Optimal control of a semidiscrete Cahn-Hilliard-Navier- Stokes system, SIAM J. Control Optim., 52 (2014), pp. 747–772

    work page 2014

  46. [46]

    P. C. Hohenberg and B. I. Halperin , Theory of dynamic critical phenomena , Reviews of Modern Physics, 49 (1977), p. 435

    work page 1977

  47. [47]

    Hysing, S., Turek, S., Kuzmin, D., Parolini, N., Burman, E., Ganesan, S., and Tobiska, L. (2009). Quantitative benchmark computations of two-dimensional bubble dynamics. Inter- national Journal for Numerical Methods in Fluids , 60(11):1259–1288

    work page 2009

  48. [48]

    Jaruˇsek, M

    J. Jaruˇsek, M. Krbec, M. Rao, and J. Soko lowski, Conical differentiability for evolution variational inequalities, J. Differential Equations, 193 (2003), pp. 131–146

    work page 2003

  49. [49]

    J. Kim, K. Kang, and J. Lowengrub , Conservative multigrid methods for Cahn-Hilliard fluids, J. Comput. Phys., 193 (2004), pp. 511–543

    work page 2004

  50. [50]

    Kim and J

    J. Kim and J. Lowengrub , Interfaces and multicomponent fluids , Encyclopedia of Mathe- matical Physics, (2004), pp. 135–144

    work page 2004

  51. [51]

    Kunisch, S

    K. Kunisch, S. Volkwein, Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics , SIAM J. Numer. Anal., 40(2) (2002), 492–515

    work page 2002

  52. [52]

    Lowengrub and L

    J. Lowengrub and L. Truskinovsky, Quasi-incompressible Cahn-Hilliard fluids and topo- logical transitions, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454 (1998), pp. 2617– 2654

    work page 1998

  53. [53]

    Luo, J.-S

    Z.-Q. Luo, J.-S. Pang, and D. Ralph, Mathematical programs with equilibrium constraints, Cambridge University Press, Cambridge, 1996

    work page 1996

  54. [54]

    Mordukhovich, Variational analysis and generalized differentiation

    Boris S. Mordukhovich, Variational analysis and generalized differentiation. II, vol. 331 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2006. Applications

    work page 2006

  55. [55]

    Outrata, M

    J. Outrata, M. Ko ˇcvara, and J. Zowe , Nonsmooth approach to optimization problems with equilibrium constraints, vol. 28 of Nonconvex Optimization and its Applications, Kluwer Academic Publishers, Dordrecht, 1998. Theory, applications and numerical results

    work page 1998

  56. [56]

    A Navier Stokes Phase Field Crystal Model for Colloidal Suspensions

    S. Praetorius and A. Voigt , A phase field crystal model for colloidal suspensions with hydrodynamic interactions, arXiv preprint arXiv:1310.5495, (2013)

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  57. [57]

    Reiss, P

    J. Reiss, P. Schulze, J. Sesterhenn, V. Mehrmann, The shifted proper orthogonal decomposi- tion: a mode decomposition for multiple transport phenomena , SIAM J. Sci. Comput., 40(3) (2018), A1322–A1344

    work page 2018

  58. [58]

    Rozza, K

    G. Rozza, K. Veroy, On the stability of the reduced basis method for Stokes equations in parametrized domains, Comput. Method Appl. M., 196(7) (2007), 1244–1260

    work page 2007

  59. [59]

    Sirovich, Turbulence and the dynamics of coherent structures I-III , Q

    L. Sirovich, Turbulence and the dynamics of coherent structures I-III , Q. Appl. Math., 45(3) (1987), 561–590

    work page 1987

  60. [60]

    Sieber, C

    M. Sieber, C. O. Paschereit, K. Oberleithner, Spectral proper orthogonal decomposition, J. Fluid Mech., 792 (2016), 798–828

    work page 2016

  61. [61]

    Scheel and S

    H. Scheel and S. Scholtes, Mathematical programs with complementarity constraints: sta- tionarity, optimality, and sensitivity , Math. Oper. Res., 25 (2000), pp. 1–22

    work page 2000

  62. [62]

    Tachim Medjo, T.: Optimal control of a Cahn-Hilliard-Navier-Stokes model with state con- straints. J. Convex Anal. 22(4), 1135–1172 (2015)

    work page 2015

  63. [63]

    Uzunca, B

    M. Uzunca, B. Karas¨ ozen,Energy stable model order reduction for the Allen-Cahn equation , in Model Reduction of Parametrized Systems (2017), 403–419, Springer

    work page 2017

  64. [64]

    Ullmann, M

    S. Ullmann, M. Rotkvic, J. Lang, POD-Galerkin reduced-order modeling with adaptive finite element snapshots, J. Comput. Phys. 325 (2016), 244–258

    work page 2016

  65. [65]

    Wells, Z

    D. Wells, Z. Wang, X. Xie, T. Iliescu, An evolve-then-filter regularized reduced order model for convection-dominated flows, Int. J. Numer. Meth. Fl., 84 (10) (2017), 598–615

    work page 2017

  66. [66]

    J. M. Yong and S. M. Zheng , Feedback stabilization and optimal control for the Cahn- Hilliard equation, Nonlinear Anal., 17 (1991), pp. 431–444

    work page 1991

  67. [67]

    Zhou et al

    B. Zhou et al. , Simulations of polymeric membrane formation in 2D and 3D , PhD thesis, Massachusetts Institute of Technology, 2006. SIMULATION AND CONTROL OF A NONSMOOTH CHNS SYSTEM 25

    work page 2006

  68. [68]

    Zowe and S

    J. Zowe and S. Kurcyusz, Regularity and stability for the mathematical programming prob- lem in Banach spaces , Appl. Math. Optim., 5 (1979), pp. 49–62. University of Hamburg, Bundesstr. 55, 20146 Hamburg, Germany E-mail address: carmen.graessle@uni-hamburg.de Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, Germany E-mail address: michael.hintermueller@...

    work page 1979