{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,20]],"date-time":"2025-09-20T21:00:00Z","timestamp":1758402000687,"version":"3.37.3"},"reference-count":99,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2021,4,1]],"date-time":"2021-04-01T00:00:00Z","timestamp":1617235200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,4,3]],"date-time":"2021-04-03T00:00:00Z","timestamp":1617408000000},"content-version":"vor","delay-in-days":2,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["M 2876"],"award-info":[{"award-number":["M 2876"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2021,4]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A Cahn\u2013Hilliard equation with stochastic multiplicative noise and a random convection term is considered. The model describes isothermal phase-separation occurring in a moving fluid, and accounts for the randomness appearing at the microscopic level both in the phase-separation itself and in the flow-inducing process. The call for a random component in the convection term stems naturally from applications, as the fluid\u2019s stirring procedure is usually caused by mechanical or magnetic devices. Well-posedness of the state system is addressed, and optimisation of a standard tracking type cost with respect to the velocity control is then studied. Existence of optimal controls is proved, and the G\u00e2teaux\u2013Fr\u00e9chet differentiability of the control-to-state map is shown. Lastly, the corresponding adjoint backward problem is analysed, and the first-order necessary conditions for optimality are derived in terms of a variational inequality involving the intrinsic adjoint variables.<\/jats:p>","DOI":"10.1007\/s00332-021-09702-8","type":"journal-article","created":{"date-parts":[[2021,4,3]],"date-time":"2021-04-03T15:02:42Z","timestamp":1617462162000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Analysis and Optimal Velocity Control of a Stochastic Convective Cahn\u2013Hilliard Equation"],"prefix":"10.1007","volume":"31","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6928-8944","authenticated-orcid":false,"given":"Luca","family":"Scarpa","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,4,3]]},"reference":[{"issue":"2","key":"9702_CR1","doi-asserted-by":"publisher","first-page":"463","DOI":"10.1007\/s00205-008-0160-2","volume":"194","author":"H Abels","year":"2009","unstructured":"Abels, H.: On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities. Arch. Ration. Mech. Anal. 194(2), 463\u2013506 (2009)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"3","key":"9702_CR2","doi-asserted-by":"publisher","first-page":"2383","DOI":"10.1016\/j.jde.2015.10.004","volume":"260","author":"DC Antonopoulou","year":"2016","unstructured":"Antonopoulou, D.C., Karali, G., Millet, A.: Existence and regularity of solution for a stochastic Cahn\u2013Hilliard\/Allen\u2013Cahn equation with unbounded noise diffusion. J. Differ. Equ. 260(3), 2383\u20132417 (2016)","journal-title":"J. Differ. Equ."},{"key":"9702_CR3","doi-asserted-by":"crossref","unstructured":"Barbu, V.: Nonlinear differential equations of monotone types in Banach spaces. Springer Monographs in Mathematics. Springer, New York (2010)","DOI":"10.1007\/978-1-4419-5542-5"},{"issue":"4","key":"9702_CR4","doi-asserted-by":"publisher","first-page":"1957","DOI":"10.1214\/17-AOP1217","volume":"46","author":"V Barbu","year":"2018","unstructured":"Barbu, V., R\u00f6ckner, M., Zhang, D.: Optimal bilinear control of nonlinear stochastic Schr\u00f6dinger equations driven by linear multiplicative noise. Ann. Probab. 46(4), 1957\u20131999 (2018)","journal-title":"Ann. Probab."},{"issue":"14","key":"9702_CR5","doi-asserted-by":"publisher","first-page":"5241","DOI":"10.1002\/mma.4383","volume":"40","author":"C Bauzet","year":"2017","unstructured":"Bauzet, C., Bonetti, E., Bonfanti, G., Lebon, F., Vallet, G.: A global existence and uniqueness result for a stochastic Allen\u2013Cahn equation with constraint. Math. Methods Appl. Sci. 40(14), 5241\u20135261 (2017)","journal-title":"Math. Methods Appl. Sci."},{"key":"9702_CR6","doi-asserted-by":"publisher","unstructured":"Bertacco, F.: Stochastic Allen-Cahn equation with logarithmic potential. Nonlinear Anal. (2021). https:\/\/doi.org\/10.1016\/j.na.2020.112122","DOI":"10.1016\/j.na.2020.112122"},{"key":"9702_CR7","doi-asserted-by":"publisher","first-page":"62","DOI":"10.1007\/978-3-642-68038-0_7","volume-title":"Stochastic Nonlinear Systems in Physics, Chemistry, and Biology","author":"K Binder","year":"1981","unstructured":"Binder, K.: Kinetics of phase separation. In: Arnold, L., Lefever, R. (eds.) Stochastic Nonlinear Systems in Physics, Chemistry, and Biology, pp. 62\u201371. Springer, Berlin (1981)"},{"issue":"3","key":"9702_CR8","doi-asserted-by":"publisher","first-page":"553","DOI":"10.1007\/PL00005585","volume":"223","author":"D Bl\u00f6mker","year":"2001","unstructured":"Bl\u00f6mker, D., Maier-Paape, S., Wanner, T.: Spinodal decomposition for the Cahn\u2013Hilliard\u2013Cook equation. Commun. Math. Phys. 223(3), 553\u2013582 (2001)","journal-title":"Commun. Math. Phys."},{"issue":"1","key":"9702_CR9","doi-asserted-by":"publisher","first-page":"449","DOI":"10.1090\/S0002-9947-07-04387-5","volume":"360","author":"D Bl\u00f6mker","year":"2008","unstructured":"Bl\u00f6mker, D., Maier-Paape, S., Wanner, T.: Second phase spinodal decomposition for the Cahn\u2013Hilliard\u2013Cook equation. Trans. Am. Math. Soc. 360(1), 449\u2013489 (2008)","journal-title":"Trans. Am. Math. Soc."},{"issue":"1","key":"9702_CR10","doi-asserted-by":"publisher","first-page":"459","DOI":"10.1137\/15M1028844","volume":"15","author":"D Bl\u00f6mker","year":"2016","unstructured":"Bl\u00f6mker, D., Sander, E., Wanner, T.: Degenerate nucleation in the Cahn\u2013Hilliard\u2013Cook model. SIAM J. Appl. Dyn. Syst. 15(1), 459\u2013494 (2016)","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"9702_CR11","doi-asserted-by":"crossref","unstructured":"Bonetti, E., Colli, P., Scarpa, L., Tomassetti, G.: Bounded solutions and their asymptotics for a doubly nonlinear Cahn\u2013Hilliard system. Calc. Var. Partial Differ. Equ. 59(2), Paper No. 88 (2020)","DOI":"10.1007\/s00526-020-1715-9"},{"issue":"4","key":"9702_CR12","doi-asserted-by":"publisher","first-page":"641","DOI":"10.1142\/S0218202517500129","volume":"27","author":"E Bonetti","year":"2017","unstructured":"Bonetti, E., Colli, P., Tomassetti, G.: A non-smooth regularization of a forward-backward parabolic equation. Math. Models Methods Appl. Sci. 27(4), 641\u2013661 (2017)","journal-title":"Math. Models Methods Appl. Sci."},{"issue":"3","key":"9702_CR13","doi-asserted-by":"publisher","first-page":"1001","DOI":"10.3934\/cpaa.2018049","volume":"17","author":"E Bonetti","year":"2018","unstructured":"Bonetti, E., Colli, P., Scarpa, L., Tomassetti, G.: A doubly nonlinear Cahn\u2013Hilliard system with nonlinear viscosity. Commun. Pure Appl. Anal. 17(3), 1001\u20131022 (2018)","journal-title":"Commun. Pure Appl. Anal."},{"key":"9702_CR14","doi-asserted-by":"crossref","unstructured":"Brezis, H.: Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York (2011)","DOI":"10.1007\/978-0-387-70914-7"},{"issue":"3","key":"9702_CR15","doi-asserted-by":"publisher","first-page":"2664","DOI":"10.1137\/100788574","volume":"51","author":"Za Brze\u017aniak","year":"2013","unstructured":"Brze\u017aniak, Za, Serrano, R.: Optimal relaxed control of dissipative stochastic partial differential equations in Banach spaces. SIAM J. Control Optim. 51(3), 2664\u20132703 (2013)","journal-title":"SIAM J. Control Optim."},{"issue":"2","key":"9702_CR16","first-page":"258","volume":"28","author":"JW Cahn","year":"1958","unstructured":"Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. J. Chem. Phys. I. Interfacial Free Energy 28(2), 258\u2013267 (1958)","journal-title":"J. Chem. Phys. I. Interfacial Free Energy"},{"issue":"2","key":"9702_CR17","doi-asserted-by":"publisher","first-page":"561","DOI":"10.1007\/s00032-011-0165-4","volume":"79","author":"L Cherfils","year":"2011","unstructured":"Cherfils, L., Miranville, A., Zelik, S.: The Cahn\u2013Hilliard equation with logarithmic potentials. Milan J. Math. 79(2), 561\u2013596 (2011)","journal-title":"Milan J. Math."},{"issue":"3\u20134","key":"9702_CR18","first-page":"183","volume":"99","author":"P Colli","year":"2016","unstructured":"Colli, P., Scarpa, L.: From the viscous Cahn\u2013Hilliard equation to a regularized forward-backward parabolic equation. Asymptot. Anal. 99(3\u20134), 183\u2013205 (2016)","journal-title":"Asymptot. Anal."},{"issue":"1","key":"9702_CR19","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1137\/120902422","volume":"53","author":"P Colli","year":"2015","unstructured":"Colli, P., Sprekels, J.: Optimal control of an Allen\u2013Cahn equation with singular potentials and dynamic boundary condition. SIAM J. Control Optim. 53(1), 213\u2013234 (2015)","journal-title":"SIAM J. Control Optim."},{"issue":"2","key":"9702_CR20","doi-asserted-by":"publisher","first-page":"972","DOI":"10.1016\/j.jmaa.2014.05.008","volume":"419","author":"P Colli","year":"2014","unstructured":"Colli, P., Gilardi, G., Sprekels, J.: On the Cahn\u2013Hilliard equation with dynamic boundary conditions and a dominating boundary potential. J. Math. Anal. Appl. 419(2), 972\u2013994 (2014)","journal-title":"J. Math. Anal. Appl."},{"issue":"4","key":"9702_CR21","doi-asserted-by":"publisher","first-page":"2696","DOI":"10.1137\/140984749","volume":"53","author":"P Colli","year":"2015","unstructured":"Colli, P., Farshbaf-Shaker, M.H., Gilardi, G., Sprekels, J.: Optimal boundary control of a viscous Cahn\u2013Hilliard system with dynamic boundary condition and double obstacle potentials. SIAM J. Control Optim. 53(4), 2696\u20132721 (2015)","journal-title":"SIAM J. Control Optim."},{"issue":"1","key":"9702_CR22","first-page":"41","volume":"7","author":"P Colli","year":"2015","unstructured":"Colli, P., Farshbaf-Shaker, M.H., Gilardi, G., Sprekels, J.: Second-order analysis of a boundary control problem for the viscous Cahn\u2013Hilliard equation with dynamic boundary condition. Ann. Acad. Rom. Sci. Ser. Math. Appl. 7(1), 41\u201366 (2015)","journal-title":"Ann. Acad. Rom. Sci. Ser. Math. Appl."},{"issue":"4","key":"9702_CR23","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1515\/anona-2015-0035","volume":"4","author":"P Colli","year":"2015","unstructured":"Colli, P., Gilardi, G., Sprekels, J.: A boundary control problem for the pure Cahn\u2013Hilliard equation with dynamic boundary conditions. Adv. Nonlinear Anal. 4(4), 311\u2013325 (2015)","journal-title":"Adv. Nonlinear Anal."},{"issue":"2","key":"9702_CR24","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1007\/s00245-015-9299-z","volume":"73","author":"P Colli","year":"2016","unstructured":"Colli, P., Gilardi, G., Sprekels, J.: A boundary control problem for the viscous Cahn\u2013Hilliard equation with dynamic boundary conditions. Appl. Math. Optim. 73(2), 195\u2013225 (2016)","journal-title":"Appl. Math. Optim."},{"issue":"5","key":"9702_CR25","doi-asserted-by":"publisher","first-page":"1445","DOI":"10.1007\/s10231-018-0732-1","volume":"197","author":"P Colli","year":"2018","unstructured":"Colli, P., Gilardi, G., Sprekels, J.: On a Cahn\u2013Hilliard system with convection and dynamic boundary conditions. Ann. Mat. Pura Appl. (4) 197(5), 1445\u20131475 (2018)","journal-title":"Ann. Mat. Pura Appl. (4)"},{"issue":"3","key":"9702_CR26","doi-asserted-by":"publisher","first-page":"1665","DOI":"10.1137\/17M1146786","volume":"56","author":"P Colli","year":"2018","unstructured":"Colli, P., Gilardi, G., Sprekels, J.: Optimal velocity control of a viscous Cahn\u2013Hilliard system with convection and dynamic boundary conditions. SIAM J. Control Optim. 56(3), 1665\u20131691 (2018)","journal-title":"SIAM J. Control Optim."},{"issue":"2","key":"9702_CR27","first-page":"485","volume":"26","author":"P Colli","year":"2019","unstructured":"Colli, P., Gilardi, G., Sprekels, J.: Optimal velocity control of a convective Cahn\u2013Hilliard system with double obstacles and dynamic boundary conditions: a \u2018deep quench\u2019 approach. J. Convex Anal. 26(2), 485\u2013514 (2019)","journal-title":"J. Convex Anal."},{"issue":"3","key":"9702_CR28","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1016\/0001-6160(70)90144-6","volume":"18","author":"H Cook","year":"1970","unstructured":"Cook, H.: Brownian motion in spinodal decomposition. Acta Metall. 18(3), 297\u2013306 (1970)","journal-title":"Acta Metall."},{"key":"9702_CR29","doi-asserted-by":"publisher","first-page":"38","DOI":"10.1016\/j.na.2016.03.009","volume":"140","author":"F Cornalba","year":"2016","unstructured":"Cornalba, F.: A nonlocal stochastic Cahn\u2013Hilliard equation. Nonlinear Anal. 140, 38\u201360 (2016)","journal-title":"Nonlinear Anal."},{"issue":"2","key":"9702_CR30","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1016\/0362-546X(94)00277-O","volume":"26","author":"G Da Prato","year":"1996","unstructured":"Da Prato, G., Debussche, A.: Stochastic Cahn\u2013Hilliard equation. Nonlinear Anal. 26(2), 241\u2013263 (1996)","journal-title":"Nonlinear Anal."},{"key":"9702_CR31","doi-asserted-by":"crossref","unstructured":"Da\u00a0Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions, vol. 152 of Encyclopedia of Mathematics and its Applications, second edition. Cambridge University Press, Cambridge (2014)","DOI":"10.1017\/CBO9781107295513"},{"issue":"3","key":"9702_CR32","doi-asserted-by":"publisher","first-page":"1473","DOI":"10.1137\/090769636","volume":"43","author":"A Debussche","year":"2011","unstructured":"Debussche, A., Gouden\u00e8ge, L.: Stochastic Cahn\u2013Hilliard equation with double singular nonlinearities and two reflections. SIAM J. Math. Anal. 43(3), 1473\u20131494 (2011)","journal-title":"SIAM J. Math. Anal."},{"issue":"5","key":"9702_CR33","doi-asserted-by":"publisher","first-page":"1706","DOI":"10.1214\/009117906000000773","volume":"35","author":"A Debussche","year":"2007","unstructured":"Debussche, A., Zambotti, L.: Conservative stochastic Cahn\u2013Hilliard equation with reflection. Ann. Probab. 35(5), 1706\u20131739 (2007)","journal-title":"Ann. Probab."},{"issue":"5","key":"9702_CR34","doi-asserted-by":"publisher","first-page":"1529","DOI":"10.3934\/dcdsb.2015.20.1529","volume":"20","author":"F Della Porta","year":"2015","unstructured":"Della Porta, F., Grasselli, M.: Convective nonlocal Cahn\u2013Hilliard equations with reaction terms. Discrete Contin. Dyn. Syst. Ser. B 20(5), 1529\u20131553 (2015)","journal-title":"Discrete Contin. Dyn. Syst. Ser. B"},{"issue":"2","key":"9702_CR35","doi-asserted-by":"publisher","first-page":"545","DOI":"10.1016\/j.jde.2018.03.002","volume":"265","author":"G Deugou\u00e9","year":"2018","unstructured":"Deugou\u00e9, G., Medjo, T\u00a0Tachim: Convergence of the solution of the stochastic 3D globally modified Cahn\u2013Hilliard\u2013Navier\u2013Stokes equations. J. Differ. Equ. 265(2), 545\u2013592 (2018)","journal-title":"J. Differ. Equ."},{"issue":"1","key":"9702_CR36","doi-asserted-by":"publisher","first-page":"140","DOI":"10.1016\/j.jmaa.2017.11.050","volume":"460","author":"G Deugou\u00e9","year":"2018","unstructured":"Deugou\u00e9, G., Medjo, T\u00a0Tachim: The exponential behavior of a stochastic globally modified Cahn\u2013Hilliard\u2013Navier\u2013Stokes model with multiplicative noise. J. Math. Anal. Appl. 460(1), 140\u2013163 (2018)","journal-title":"J. Math. Anal. Appl."},{"issue":"10","key":"9702_CR37","doi-asserted-by":"publisher","first-page":"1996","DOI":"10.1016\/j.spa.2010.06.001","volume":"120","author":"K Du","year":"2010","unstructured":"Du, K., Meng, Q.: A revisit to $$W^n_2$$-theory of super-parabolic backward stochastic partial differential equations in $${\\mathbb{R}}^d$$. Stochastic Process. Appl. 120(10), 1996\u20132015 (2010)","journal-title":"Stochastic Process. Appl."},{"issue":"6","key":"9702_CR38","doi-asserted-by":"publisher","first-page":"4343","DOI":"10.1137\/120882433","volume":"51","author":"K Du","year":"2013","unstructured":"Du, K., Meng, Q.: A maximum principle for optimal control of stochastic evolution equations. SIAM J. Control Optim. 51(6), 4343\u20134362 (2013)","journal-title":"SIAM J. Control Optim."},{"key":"9702_CR39","unstructured":"Edwards, R.E.: Functional analysis. Theory and applications. Holt, Rinehart and Winston, New York (1965)"},{"key":"9702_CR40","doi-asserted-by":"publisher","first-page":"4725","DOI":"10.1103\/PhysRevB.38.4725","volume":"38","author":"KR Elder","year":"1988","unstructured":"Elder, K.R., Rogers, T.M., Desai, R.C.: Early stages of spinodal decomposition for the Cahn\u2013Hilliard\u2013Cook model of phase separation. Phys. Rev. B 38, 4725\u20134739 (1988)","journal-title":"Phys. Rev. B"},{"issue":"12","key":"9702_CR41","doi-asserted-by":"publisher","first-page":"1169","DOI":"10.1016\/0362-546X(91)90204-E","volume":"16","author":"N Elezovi\u0107","year":"1991","unstructured":"Elezovi\u0107, N., Mikeli\u0107, A.: On the stochastic Cahn\u2013Hilliard equation. Nonlinear Anal. 16(12), 1169\u20131200 (1991)","journal-title":"Nonlinear Anal."},{"issue":"4","key":"9702_CR42","doi-asserted-by":"publisher","first-page":"339","DOI":"10.1007\/BF00251803","volume":"96","author":"CM Elliott","year":"1986","unstructured":"Elliott, C.M., Songmu, Z.: On the Cahn\u2013Hilliard equation. Arch. Rational Mech. Anal. 96(4), 339\u2013357 (1986)","journal-title":"Arch. Rational Mech. Anal."},{"issue":"2","key":"9702_CR43","doi-asserted-by":"publisher","first-page":"387","DOI":"10.1006\/jdeq.1996.0101","volume":"128","author":"CM Elliott","year":"1996","unstructured":"Elliott, C.M., Stuart, A.M.: Viscous Cahn\u2013Hilliard equation. II. Analysis. J. Differ. Equ. 128(2), 387\u2013414 (1996)","journal-title":"J. Differ. Equ."},{"key":"9702_CR44","doi-asserted-by":"crossref","unstructured":"Feireisl, E., Petcu, M.: A diffuse interface model of a two-phase flow with thermal fluctuations. Appl. Math. Optim. (2019)","DOI":"10.1007\/s00245-019-09557-2"},{"issue":"3","key":"9702_CR45","doi-asserted-by":"publisher","first-page":"1836","DOI":"10.1016\/j.jde.2019.03.006","volume":"267","author":"E Feireisl","year":"2019","unstructured":"Feireisl, E., Petcu, M.: Stability of strong solutions for a model of incompressible two-phase flow under thermal fluctuations. J. Differ. Equ. 267(3), 1836\u20131858 (2019)","journal-title":"J. Differ. Equ."},{"issue":"6","key":"9702_CR46","doi-asserted-by":"publisher","first-page":"2821","DOI":"10.1007\/s00332-020-09637-6","volume":"30","author":"Y Feng","year":"2020","unstructured":"Feng, Y., Feng, Y., Iyer, G., Thiffeault, J.-L.: Phase separation in the advective Cahn\u2013Hilliard equation. J. Nonlinear Sci. 30(6), 2821\u20132845 (2020)","journal-title":"J. Nonlinear Sci."},{"key":"9702_CR47","doi-asserted-by":"publisher","first-page":"893","DOI":"10.1103\/PhysRevLett.79.893","volume":"79","author":"HP Fischer","year":"1997","unstructured":"Fischer, H.P., Maass, P., Dieterich, W.: Novel surface modes in spinodal decomposition. Phys. Rev. Lett. 79, 893\u2013896 (1997)","journal-title":"Phys. Rev. Lett."},{"issue":"3","key":"9702_CR48","doi-asserted-by":"publisher","first-page":"367","DOI":"10.1007\/BF01192467","volume":"102","author":"F Flandoli","year":"1995","unstructured":"Flandoli, F., Gatarek, D.: Martingale and stationary solutions for stochastic Navier\u2013Stokes equations. Probab. Theory Related Fields 102(3), 367\u2013391 (1995)","journal-title":"Probab. Theory Related Fields"},{"issue":"1","key":"9702_CR49","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1137\/140994800","volume":"54","author":"S Frigeri","year":"2016","unstructured":"Frigeri, S., Rocca, E., Sprekels, J.: Optimal distributed control of a nonlocal Cahn\u2013Hilliard\/Navier\u2013Stokes system in two dimensions. SIAM J. Control Optim. 54(1), 221\u2013250 (2016)","journal-title":"SIAM J. Control Optim."},{"issue":"2","key":"9702_CR50","doi-asserted-by":"publisher","first-page":"678","DOI":"10.1088\/1361-6544\/aaedd0","volume":"32","author":"S Frigeri","year":"2019","unstructured":"Frigeri, S., Gal, C.G., Grasselli, M., Sprekels, J.: Two-dimensional nonlocal Cahn\u2013Hilliard\u2013Navier\u2013Stokes systems with variable viscosity, degenerate mobility and singular potential. Nonlinearity 32(2), 678\u2013727 (2019)","journal-title":"Nonlinearity"},{"issue":"3","key":"9702_CR51","doi-asserted-by":"publisher","first-page":"899","DOI":"10.1007\/s00245-018-9524-7","volume":"81","author":"S Frigeri","year":"2020","unstructured":"Frigeri, S., Grasselli, M., Sprekels, J.: Optimal distributed control of two-dimensional nonlocal Cahn\u2013Hilliard\u2013Navier\u2013Stokes systems with degenerate mobility and singular potential. Appl. Math. Optim. 81(3), 899\u2013931 (2020)","journal-title":"Appl. Math. Optim."},{"issue":"1","key":"9702_CR52","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1137\/15M1012888","volume":"54","author":"M Fuhrman","year":"2016","unstructured":"Fuhrman, M., Orrieri, C.: Stochastic maximum principle for optimal control of a class of nonlinear SPDEs with dissipative drift. SIAM J. Control Optim. 54(1), 341\u2013371 (2016)","journal-title":"SIAM J. Control Optim."},{"issue":"13\u201314","key":"9702_CR53","doi-asserted-by":"publisher","first-page":"683","DOI":"10.1016\/j.crma.2012.07.009","volume":"350","author":"M Fuhrman","year":"2012","unstructured":"Fuhrman, M., Hu, Y., Tessitore, G.: Stochastic maximum principle for optimal control of SPDEs. C. R. Math. Acad. Sci. Paris 350(13\u201314), 683\u2013688 (2012)","journal-title":"C. R. Math. Acad. Sci. Paris"},{"issue":"2","key":"9702_CR54","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1007\/s00245-013-9203-7","volume":"68","author":"M Fuhrman","year":"2013","unstructured":"Fuhrman, M., Hu, Y., Tessitore, G.: Stochastic maximum principle for optimal control of SPDEs. Appl. Math. Optim. 68(2), 181\u2013217 (2013)","journal-title":"Appl. Math. Optim."},{"issue":"2","key":"9702_CR55","first-page":"255","volume":"6","author":"M Fuhrman","year":"2018","unstructured":"Fuhrman, M., Hu, Y., Tessitore, G.: Stochastic maximum principle for optimal control of partial differential equations driven by white noise. Stoch. Partial Differ. Equ. Anal. Comput. 6(2), 255\u2013285 (2018)","journal-title":"Stoch. Partial Differ. Equ. Anal. Comput."},{"issue":"1","key":"9702_CR56","doi-asserted-by":"publisher","first-page":"126","DOI":"10.1016\/j.jde.2012.02.010","volume":"253","author":"CG Gal","year":"2012","unstructured":"Gal, C.G.: On a class of degenerate parabolic equations with dynamic boundary conditions. J. Differ. Equ. 253(1), 126\u2013166 (2012)","journal-title":"J. Differ. Equ."},{"issue":"3","key":"9702_CR57","doi-asserted-by":"publisher","first-page":"881","DOI":"10.3934\/cpaa.2009.8.881","volume":"8","author":"G Gilardi","year":"2009","unstructured":"Gilardi, G., Miranville, A., Schimperna, G.: On the Cahn\u2013Hilliard equation with irregular potentials and dynamic boundary conditions. Commun. Pure Appl. Anal. 8(3), 881\u2013912 (2009)","journal-title":"Commun. Pure Appl. Anal."},{"issue":"10","key":"9702_CR58","doi-asserted-by":"publisher","first-page":"3516","DOI":"10.1016\/j.spa.2009.06.008","volume":"119","author":"L Gouden\u00e8ge","year":"2009","unstructured":"Gouden\u00e8ge, L.: Stochastic Cahn\u2013Hilliard equation with singular nonlinearity and reflection. Stoch. Process. Appl. 119(10), 3516\u20133548 (2009)","journal-title":"Stoch. Process. Appl."},{"key":"9702_CR59","doi-asserted-by":"publisher","first-page":"3027","DOI":"10.1103\/PhysRevB.31.3027","volume":"31","author":"M Grant","year":"1985","unstructured":"Grant, M., Miguel, M\u00a0San, Vials, J., Gunton, J.D.: Theory for the early stages of phase separation: the long-range-force limit. Phys. Rev. B 31, 3027\u20133039 (1985)","journal-title":"Phys. Rev. B"},{"issue":"7","key":"9702_CR60","doi-asserted-by":"publisher","first-page":"2396","DOI":"10.1016\/j.spa.2016.11.007","volume":"127","author":"G Guatteri","year":"2017","unstructured":"Guatteri, G., Masiero, F., Orrieri, C.: Stochastic maximum principle for SPDEs with delay. Stoch. Process. Appl. 127(7), 2396\u20132427 (2017)","journal-title":"Stoch. Process. Appl."},{"issue":"3\u20134","key":"9702_CR61","doi-asserted-by":"publisher","first-page":"178","DOI":"10.1016\/0167-2789(95)00173-5","volume":"92","author":"ME Gurtin","year":"1996","unstructured":"Gurtin, M.E.: Generalized Ginzburg\u2013Landau and Cahn\u2013Hilliard equations based on a microforce balance. Phys. D 92(3\u20134), 178\u2013192 (1996)","journal-title":"Phys. D"},{"issue":"2","key":"9702_CR62","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1007\/BF01203833","volume":"105","author":"I Gy\u00f6ngy","year":"1996","unstructured":"Gy\u00f6ngy, I., Krylov, N.: Existence of strong solutions for It\u00f4\u2019s stochastic equations via approximations. Probab. Theory Related Fields 105(2), 143\u2013158 (1996)","journal-title":"Probab. Theory Related Fields"},{"key":"9702_CR63","unstructured":"Hawick, K., Playne, D.: Modelling and visualizing the Cahn\u2013Hilliard\u2013Cook equation, pp. 149\u2013155 (2008). cited By 6"},{"key":"9702_CR64","doi-asserted-by":"crossref","unstructured":"Hawick, K.: Numerical simulation and role of noise in the Cahn\u2013Hilliard\u2013Cook equation below the critical dimension, pp. 33\u201340 (2010)","DOI":"10.2316\/P.2010.702-020"},{"issue":"1","key":"9702_CR65","doi-asserted-by":"publisher","first-page":"78","DOI":"10.1504\/IJCAET.2010.029598","volume":"2","author":"K Hawick","year":"2010","unstructured":"Hawick, K., Playne, D.: Modelling, simulating and visualising the Cahn\u2013Hilliard\u2013Cook field equation. Int. J. Comput. Aided Eng. Technol. 2(1), 78\u201393 (2010)","journal-title":"Int. J. Comput. Aided Eng. Technol."},{"issue":"1","key":"9702_CR66","doi-asserted-by":"publisher","first-page":"388","DOI":"10.1137\/110824152","volume":"50","author":"M Hinterm\u00fcller","year":"2012","unstructured":"Hinterm\u00fcller, M., Wegner, D.: Distributed optimal control of the Cahn\u2013Hilliard system including the case of a double-obstacle homogeneous free energy density. SIAM J. Control Optim. 50(1), 388\u2013418 (2012)","journal-title":"SIAM J. Control Optim."},{"key":"9702_CR67","doi-asserted-by":"crossref","unstructured":"Hun, O., Kim, M.-C., Pak, C.-K.: A stochastic Gronwall inequality in random time horizon and its application to BSDE. J. Inequal. Appl. (2020)","DOI":"10.1186\/s13660-020-2304-3"},{"key":"9702_CR68","unstructured":"Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes, volume\u00a024 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, second edition (1989)"},{"issue":"2","key":"9702_CR69","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1016\/S0010-4655(00)00159-4","volume":"133","author":"R Kenzler","year":"2001","unstructured":"Kenzler, R., Eurich, F., Maass, P., Rinn, B., Schropp, J., Bohl, E., Dieterich, W.: Phase separation in confined geometries: solving the Cahn\u2013Hilliard equation with generic boundary conditions. Comput. Phys. Commun. 133(2), 139\u2013157 (2001)","journal-title":"Comput. Phys. Commun."},{"key":"9702_CR70","unstructured":"Krylov, N.V., Rozovski\u012d, B.L.: Stochastic evolution equations. In: Current Problems in Mathematics, Vol. 14 (Russian), pp. 71\u2013147, 256. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow (1979)"},{"key":"9702_CR71","doi-asserted-by":"publisher","first-page":"54","DOI":"10.1016\/j.jcrysgro.2012.11.049","volume":"365","author":"C Kudla","year":"2013","unstructured":"Kudla, C., Blumenau, A.T., B\u00fcllesfeld, F., Dropka, N., Frank-Rotsch, C., Kiessling, F.-M., Klein, O., Lange, P., Miller, W., Rehse, U., Sahr, U., Schellhorn, M., Weidemann, G., Ziem, M., Bethin, G., Fornari, R., M\u00fcller, M.N.O., Sprekels, J., Trautmann, V., Rudolph, P.: Crystallization of 640 kg mc-silicon ingots under traveling magnetic field by using a heater-magnet module. J. Cryst. Growth 365, 54\u201358 (2013)","journal-title":"J. Cryst. Growth"},{"key":"9702_CR72","doi-asserted-by":"publisher","first-page":"1417","DOI":"10.1103\/PhysRevA.11.1417","volume":"11","author":"JS Langer","year":"1975","unstructured":"Langer, J.S., Bar-on, M., Miller, H.D.: New computational method in the theory of spinodal decomposition. Phys. Rev. A 11, 1417\u20131429 (1975)","journal-title":"Phys. Rev. A"},{"key":"9702_CR73","doi-asserted-by":"publisher","first-page":"216","DOI":"10.1016\/j.commatsci.2013.08.027","volume":"81","author":"D Lee","year":"2014","unstructured":"Lee, D., Huh, J.-Y., Jeong, D., Shin, J., Yun, A., Kim, J.: Physical, mathematical, and numerical derivations of the Cahn\u2013Hilliard equation. Comput. Mater. Sci. 81, 216\u2013225 (2014)","journal-title":"Comput. Mater. Sci."},{"key":"9702_CR74","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-22354-4","volume-title":"Stochastic Partial Differential Equations: An Introduction","author":"W Liu","year":"2015","unstructured":"Liu, W., R\u00f6ckner, M.: Stochastic Partial Differential Equations: An Introduction. Springer, Cham (2015)"},{"key":"9702_CR75","doi-asserted-by":"crossref","unstructured":"L\u00fc, Q., Zhang, X.: General Pontryagin-type stochastic maximum principle and backward stochastic evolution equations in infinite dimensions. SpringerBriefs in Mathematics. Springer, Cham (2014)","DOI":"10.1007\/978-3-319-06632-5"},{"key":"9702_CR76","doi-asserted-by":"crossref","unstructured":"Marinelli, C., Scarpa, L.: Refined existence and regularity results for a class of semilinear dissipative SPDEs. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 23(2), 2050014, 34 (2020)","DOI":"10.1142\/S0219025720500149"},{"issue":"3","key":"9702_CR77","doi-asserted-by":"publisher","first-page":"1455","DOI":"10.1214\/17-AOP1207","volume":"46","author":"C Marinelli","year":"2018","unstructured":"Marinelli, C., Scarpa, L.: A variational approach to dissipative SPDEs with singular drift. Ann. Probab. 46(3), 1455\u20131497 (2018)","journal-title":"Ann. Probab."},{"issue":"2","key":"9702_CR78","doi-asserted-by":"publisher","first-page":"1028","DOI":"10.1016\/j.jde.2017.03.008","volume":"263","author":"TT Medjo","year":"2017","unstructured":"Medjo, T.T.: On the existence and uniqueness of solution to a stochastic 2D Cahn\u2013Hilliard\u2013Navier\u2013Stokes model. J. Differ. Equ. 263(2), 1028\u20131054 (2017)","journal-title":"J. Differ. Equ."},{"issue":"2","key":"9702_CR79","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1016\/0001-6160(88)90013-2","volume":"36","author":"A Milchev","year":"1988","unstructured":"Milchev, A., Heermann, D., Binder, K.: Monte-Carlo simulation of the Cahn\u2013Hillard model of spinodal decomposition. Acta Metall. 36(2), 377\u2013383 (1988)","journal-title":"Acta Metall."},{"key":"9702_CR80","doi-asserted-by":"crossref","unstructured":"Miranville, A.: The Cahn\u2013Hilliard Equation: Recent Advances and Applications. Society for Industrial and Applied Mathematics, Philadelphia, PA (2019)","DOI":"10.1137\/1.9781611975925"},{"key":"9702_CR81","unstructured":"Novick-Cohen, A.: On the viscous Cahn\u2013Hilliard equation. In: Material Instabilities in Continuum Mechanics (Edinburgh, 1985\u20131986), Oxford Sci. Publ., pp. 329\u2013342. Oxford Univ. Press, New York (1988)"},{"key":"9702_CR82","doi-asserted-by":"crossref","unstructured":"Orrieri, C., Rocca, E., Scarpa, L.: Optimal control of stochastic phase-field models related to tumor growth. ESAIM Control Optim. Calc. Var. 26, Paper No. 104, 46 (2020)","DOI":"10.1051\/cocv\/2020022"},{"issue":"8","key":"9702_CR83","doi-asserted-by":"publisher","first-page":"4624","DOI":"10.1016\/j.jde.2018.10.007","volume":"266","author":"C Orrieri","year":"2019","unstructured":"Orrieri, C., Scarpa, L.: Singular stochastic Allen\u2013Cahn equations with dynamic boundary conditions. J. Differ. Equ. 266(8), 4624\u20134667 (2019)","journal-title":"J. Differ. Equ."},{"key":"9702_CR84","unstructured":"Pardoux, E.: Equations aux deriv\u00e9es partielles stochastiques nonlin\u00e9aires monotones. Ph.D. thesis, Universit\u00e9 Paris XI (1975)"},{"issue":"1863","key":"9702_CR85","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1098\/rspa.1989.0027","volume":"422","author":"RL Pego","year":"1989","unstructured":"Pego, R.L.: Front migration in the nonlinear Cahn\u2013Hilliard equation. Proc. R. Soc. Lond. Ser. A 422(1863), 261\u2013278 (1989)","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"issue":"3","key":"9702_CR86","doi-asserted-by":"publisher","first-page":"1654","DOI":"10.1137\/140964308","volume":"53","author":"E Rocca","year":"2015","unstructured":"Rocca, E., Sprekels, J.: Optimal distributed control of a nonlocal convective Cahn\u2013Hilliard equation by the velocity in three dimensions. SIAM J. Control Optim. 53(3), 1654\u20131680 (2015)","journal-title":"SIAM J. Control Optim."},{"key":"9702_CR87","doi-asserted-by":"publisher","first-page":"9638","DOI":"10.1103\/PhysRevB.37.9638","volume":"37","author":"TM Rogers","year":"1988","unstructured":"Rogers, T.M., Elder, K.R., Desai, R.C.: Numerical study of the late stages of spinodal decomposition. Phys. Rev. B 37, 9638\u20139649 (1988)","journal-title":"Phys. Rev. B"},{"key":"9702_CR88","doi-asserted-by":"crossref","unstructured":"Scarpa, L.: The stochastic Cahn-Hilliard equation with degenerate mobility and logarithmic potential. Nonlinearity (to appear, 2021)","DOI":"10.1088\/1361-6544\/abf338"},{"key":"9702_CR89","doi-asserted-by":"publisher","unstructured":"Scarpa, L.: The stochastic viscous Cahn\u2013Hilliard equation: well-posedness, regularity and vanishing viscosity limit. Appl. Math. Optim. (2020). https:\/\/doi.org\/10.1007\/s00245-020-09652-9","DOI":"10.1007\/s00245-020-09652-9"},{"key":"9702_CR90","doi-asserted-by":"publisher","first-page":"102","DOI":"10.1016\/j.na.2018.01.016","volume":"171","author":"L Scarpa","year":"2018","unstructured":"Scarpa, L.: On the stochastic Cahn-Hilliard equation with a singular double-well potential. Nonlinear Anal. 171, 102\u2013133 (2018)","journal-title":"Nonlinear Anal."},{"issue":"2","key":"9702_CR91","doi-asserted-by":"publisher","first-page":"730","DOI":"10.1016\/j.jmaa.2018.09.034","volume":"469","author":"L Scarpa","year":"2019","unstructured":"Scarpa, L.: Existence and uniqueness of solutions to singular Cahn\u2013Hilliard equations with nonlinear viscosity terms and dynamic boundary conditions. J. Math. Anal. Appl. 469(2), 730\u2013764 (2019)","journal-title":"J. Math. Anal. Appl."},{"issue":"5","key":"9702_CR92","doi-asserted-by":"publisher","first-page":"3571","DOI":"10.1137\/18M1222223","volume":"57","author":"L Scarpa","year":"2019","unstructured":"Scarpa, L.: Optimal distributed control of a stochastic Cahn\u2013Hilliard equation. SIAM J. Control Optim. 57(5), 3571\u20133602 (2019)","journal-title":"SIAM J. Control Optim."},{"issue":"146","key":"9702_CR93","first-page":"65","volume":"4","author":"J Simon","year":"1987","unstructured":"Simon, J.: Compact sets in the space $$L^p(0, T;B)$$. Ann. Mat. Pura Appl. 4(146), 65\u201396 (1987)","journal-title":"Ann. Mat. Pura Appl."},{"key":"9702_CR94","unstructured":"Stannat, W., Wessels, L.: Deterministic Control of Stochastic Reaction-Diffusion Equations. arXiv e-prints, arXiv:1905.09074, (May 2019)"},{"key":"9702_CR95","doi-asserted-by":"crossref","unstructured":"van\u00a0der Vaart, A.W., Wellner, J.A.: Weak convergence and empirical processes. Springer Series in Statistics. Springer, New York (1996). With applications to statistics","DOI":"10.1007\/978-1-4757-2545-2"},{"key":"9702_CR96","doi-asserted-by":"crossref","unstructured":"Wang, X., Fan, S.: A class of stochastic Gronwall\u2019s inequality and its application. J. Inequal. Appl., Paper No. 336, 10 (2018)","DOI":"10.1186\/s13660-018-1932-3"},{"key":"9702_CR97","doi-asserted-by":"crossref","unstructured":"Yong, J., Zhou, X.Y.: Stochastic controls, volume\u00a043 of Applications of Mathematics (New York). Springer, New York, (1999). Hamiltonian systems and HJB equations","DOI":"10.1007\/978-1-4612-1466-3"},{"issue":"5","key":"9702_CR98","doi-asserted-by":"publisher","first-page":"1028","DOI":"10.1080\/00036811.2011.643786","volume":"92","author":"X Zhao","year":"2013","unstructured":"Zhao, X., Liu, C.: Optimal control of the convective Cahn\u2013Hilliard equation. Appl. Anal. 92(5), 1028\u20131045 (2013)","journal-title":"Appl. Anal."},{"issue":"1","key":"9702_CR99","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1007\/s00245-013-9234-0","volume":"70","author":"X Zhao","year":"2014","unstructured":"Zhao, X., Liu, C.: Optimal control for the convective Cahn\u2013Hilliard equation in 2D case. Appl. Math. Optim. 70(1), 61\u201382 (2014)","journal-title":"Appl. Math. Optim."}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-021-09702-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00332-021-09702-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-021-09702-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,31]],"date-time":"2023-01-31T06:20:56Z","timestamp":1675146056000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00332-021-09702-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4]]},"references-count":99,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2021,4]]}},"alternative-id":["9702"],"URL":"https:\/\/doi.org\/10.1007\/s00332-021-09702-8","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"type":"print","value":"0938-8974"},{"type":"electronic","value":"1432-1467"}],"subject":[],"published":{"date-parts":[[2021,4]]},"assertion":[{"value":"29 July 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 March 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 April 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"45"}}