Degenerate nonlocal Cahn-Hilliard equations: Well-posedness, regularity and local asymptotics
Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 3, pp. 627-651.
Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

Existence and uniqueness of solutions for nonlocal Cahn-Hilliard equations with degenerate potential is shown. The nonlocality is described by means of a symmetric singular kernel not falling within the framework of any previous existence theory. A convection term is also taken into account. Building upon this novel existence result, we prove convergence of solutions for this class of nonlocal Cahn-Hilliard equations to their local counterparts, as the nonlocal convolution kernels approximate a Dirac delta. Eventually, we show that, under suitable assumptions on the data, the solutions to the nonlocal Cahn-Hilliard equations exhibit further regularity, and the nonlocal-to-local convergence is verified in a stronger topology.

DOI : 10.1016/j.anihpc.2019.10.002
Classification : 45K05, 35K25, 35K55, 35B40, 76R05
Mots-clés : Nonlocal Cahn-Hilliard equation, Degenerate potential, Singular kernel, Well-posedness, Nonlocal-to-local convergence, Convection 
@article{AIHPC_2020__37_3_627_0,
     author = {Davoli, Elisa and Ranetbauer, Helene and Scarpa, Luca and Trussardi, Lara},
     title = {Degenerate nonlocal {Cahn-Hilliard} equations: {Well-posedness,} regularity and local asymptotics},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {627--651},
     publisher = {Elsevier},
     volume = {37},
     number = {3},
     year = {2020},
     doi = {10.1016/j.anihpc.2019.10.002},
     mrnumber = {4093616},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2019.10.002/}
}
TY  - JOUR
AU  - Davoli, Elisa
AU  - Ranetbauer, Helene
AU  - Scarpa, Luca
AU  - Trussardi, Lara
TI  - Degenerate nonlocal Cahn-Hilliard equations: Well-posedness, regularity and local asymptotics
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2020
SP  - 627
EP  - 651
VL  - 37
IS  - 3
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2019.10.002/
DO  - 10.1016/j.anihpc.2019.10.002
LA  - en
ID  - AIHPC_2020__37_3_627_0
ER  - 
%0 Journal Article
%A Davoli, Elisa
%A Ranetbauer, Helene
%A Scarpa, Luca
%A Trussardi, Lara
%T Degenerate nonlocal Cahn-Hilliard equations: Well-posedness, regularity and local asymptotics
%J Annales de l'I.H.P. Analyse non linéaire
%D 2020
%P 627-651
%V 37
%N 3
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2019.10.002/
%R 10.1016/j.anihpc.2019.10.002
%G en
%F AIHPC_2020__37_3_627_0
Davoli, Elisa; Ranetbauer, Helene; Scarpa, Luca; Trussardi, Lara. Degenerate nonlocal Cahn-Hilliard equations: Well-posedness, regularity and local asymptotics. Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 3, pp. 627-651. doi : 10.1016/j.anihpc.2019.10.002. http://archive.numdam.org/articles/10.1016/j.anihpc.2019.10.002/

[1] Abels, H.; Bosia, S.; Grasselli, M. Cahn-Hilliard equation with nonlocal singular free energies, Ann. Mat. Pura Appl. (4), Volume 194 (2015) no. 4, pp. 1071–1106 | DOI | MR

[2] Abels, H.; Depner, D.; Garcke, H. On an incompressible Navier-Stokes/Cahn-Hilliard system with degenerate mobility, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 30 (2013) no. 6, pp. 1175–1190 | DOI | Numdam | MR | Zbl

[3] Abels, H.; Röger, M. Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous, incompressible fluids, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 26 (2009) no. 6, pp. 2403–2424 | DOI | Numdam | MR | Zbl

[4] Barbu, V. Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics, Springer, New York, 2010 | DOI | MR | Zbl

[5] Bates, P.W.; Han, J. The Neumann boundary problem for a nonlocal Cahn-Hilliard equation, J. Differ. Equ., Volume 212 (2005) no. 2, pp. 235–277 | DOI | MR | Zbl

[6] Blodgett, K.B. Films built by depositing successive monomolecular layers on a solid surface, J. Am. Chem. Soc., Volume 57 (1935) no. 6, pp. 1007–1022 | DOI

[7] Blowey, J.F.; Elliott, C.M. The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy. I. Mathematical analysis, Eur. J. Appl. Math., Volume 2 (1991) no. 3, pp. 233–280 | DOI | MR | Zbl

[8] Bonacini, M.; Davoli, E.; Morandotti, M. Analysis of a perturbed Cahn-Hilliard model for Langmuir-Blodgett films, 2018 (preprint) | arXiv | MR

[9] Bonetti, E.; Colli, P.; Scarpa, L.; Tomassetti, G. A doubly nonlinear Cahn-Hilliard system with nonlinear viscosity, Commun. Pure Appl. Anal., Volume 17 (2018) no. 3, pp. 1001–1022 | MR

[10] Bourgain, J.; Brezis, H.; Mironescu, P. Another look at Sobolev spaces, Optimal Control and Partial Differential Equations, IOS, Amsterdam, 2001, pp. 439–455 | MR | Zbl

[11] Bourgain, J.; Brezis, H.; Mironescu, P. Limiting embedding theorems for Ws,p when s↑1 and applications, J. Anal. Math., Volume 87 (2002), pp. 77–101 (Dedicated to the memory of Thomas H. Wolff) | DOI | MR | Zbl

[12] Boyer, F. Nonhomogeneous Cahn-Hilliard fluids, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 18 (2001) no. 2, pp. 225–259 | DOI | Numdam | MR | Zbl

[13] Cahn, J.; Hilliard, J. Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., Volume 28 (1958) no. 2, pp. 258–267 | DOI

[14] Cherfils, L.; Gatti, S.; Miranville, A. A variational approach to a Cahn-Hilliard model in a domain with nonpermeable walls, J. Math. Sci. (N.Y.), Volume 189 (2013) no. 4, pp. 604–636 (Problems in Mathematical Analysis, No. 69) | DOI | MR | Zbl

[15] Cherfils, L.; Miranville, A.; Zelik, S. The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., Volume 79 (2011) no. 2, pp. 561–596 | DOI | MR | Zbl

[16] Cherfils, L.; Petcu, M. A numerical analysis of the Cahn-Hilliard equation with non-permeable walls, Numer. Math., Volume 128 (2014) no. 3, pp. 517–549 | DOI | MR

[17] Colli, P.; Farshbaf-Shaker, M.H.; Gilardi, G.; Sprekels, J. Optimal boundary control of a viscous Cahn-Hilliard system with dynamic boundary condition and double obstacle potentials, SIAM J. Control Optim., Volume 53 (2015) no. 4, pp. 2696–2721 | DOI | MR

[18] Colli, P.; Fukao, T. Cahn-Hilliard equation with dynamic boundary conditions and mass constraint on the boundary, J. Math. Anal. Appl., Volume 429 (2015) no. 2, pp. 1190–1213 | DOI | MR

[19] Colli, P.; Fukao, T. Equation and dynamic boundary condition of Cahn-Hilliard type with singular potentials, Nonlinear Anal., Volume 127 (2015), pp. 413–433 | DOI | MR

[20] Colli, P.; Fukao, T. Nonlinear diffusion equations as asymptotic limits of Cahn-Hilliard systems, J. Differ. Equ., Volume 260 (2016) no. 9, pp. 6930–6959 | DOI | MR

[21] Colli, P.; Gilardi, G.; Sprekels, J. On the Cahn-Hilliard equation with dynamic boundary conditions and a dominating boundary potential, J. Math. Anal. Appl., Volume 419 (2014) no. 2, pp. 972–994 | DOI | MR | Zbl

[22] Colli, P.; Gilardi, G.; Sprekels, J. A boundary control problem for the pure Cahn-Hilliard equation with dynamic boundary conditions, Adv. Nonlinear Anal., Volume 4 (2015) no. 4, pp. 311–325 | MR

[23] Colli, P.; Gilardi, G.; Sprekels, J. A boundary control problem for the viscous Cahn-Hilliard equation with dynamic boundary conditions, Appl. Math. Optim., Volume 73 (2016) no. 2, pp. 195–225 | DOI | MR

[24] Colli, P.; Gilardi, G.; Sprekels, J. On an application of Tikhonov's fixed point theorem to a nonlocal Cahn-Hilliard type system modeling phase separation, J. Differ. Equ., Volume 260 (2016) no. 11, pp. 7940–7964 | DOI | MR

[25] Colli, P.; Gilardi, G.; Sprekels, J. Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential, Evol. Equ. Control Theory, Volume 6 (2017) no. 1, pp. 35–58 | MR

[26] Colli, P.; Gilardi, G.; Sprekels, J. On a Cahn-Hilliard system with convection and dynamic boundary conditions, Ann. Mat. Pura Appl. (4), Volume 197 (2018) no. 5, pp. 1445–1475 | DOI | MR

[27] Colli, P.; Scarpa, L. From the viscous Cahn-Hilliard equation to a regularized forward-backward parabolic equation, Asymptot. Anal., Volume 99 (2016) no. 3–4, pp. 183–205 | MR

[28] Colli, P.; Sprekels, J. Optimal boundary control of a nonstandard Cahn-Hilliard system with dynamic boundary condition and double obstacle inclusions, Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, Springer INdAM Ser., vol. 22, Springer, Cham, 2017, pp. 151–182 | MR

[29] Della Porta, F.; Grasselli, M. Convective nonlocal Cahn-Hilliard equations with reaction terms, Discrete Contin. Dyn. Syst., Ser. B, Volume 20 (2015) no. 5, pp. 1529–1553 | MR

[30] Della Porta, F.; Grasselli, M. On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems, Commun. Pure Appl. Anal., Volume 15 (2016) no. 2, pp. 299–317 | MR

[31] Eden, A.; Kalantarov, V.K.; Zelik, S.V. Global solvability and blow up for the convective Cahn-Hilliard equations with concave potentials, J. Math. Phys., Volume 54 (2013) no. 4 | DOI | MR | Zbl

[32] Ei, S.-I. The effect of nonlocal convection on reaction-diffusion equations, Hiroshima Math. J., Volume 17 (1987) no. 2, pp. 281–307 | MR | Zbl

[33] 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., Volume 263 (2017) no. 9, pp. 5253–5297 | MR

[34] Gal, C.G.; Grasselli, M. Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 27 (2010) no. 1, pp. 401–436 | Numdam | MR | Zbl

[35] Gal, C.G.; Grasselli, M. Longtime behavior of nonlocal Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., Volume 34 (2014) no. 1, pp. 145–179 | MR | Zbl

[36] Giacomin, G.; Lebowitz, J. Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits, J. Stat. Phys., Volume 87 (1997) no. 1, pp. 37–61 | MR | Zbl

[37] Gilardi, G.; Miranville, A.; Schimperna, G. On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., Volume 8 (2009) no. 3, pp. 881–912 | MR | Zbl

[38] Gilardi, G.; Miranville, A.; Schimperna, G. Long time behavior of the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Chin. Ann. Math., Ser. B, Volume 31 (2010) no. 5, pp. 679–712 | DOI | MR | Zbl

[39] Han, J. The Cauchy problem and steady state solutions for a nonlocal Cahn-Hilliard equation, Electron. J. Differ. Equ., Volume 113 (2004), pp. 9 | MR | Zbl

[40] 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., Volume 50 (2012) no. 1, pp. 388–418 | DOI | MR | Zbl

[41] Ignat, L.I.; Rossi, J.D. A nonlocal convection-diffusion equation, J. Funct. Anal., Volume 251 (2007) no. 2, pp. 399–437 | DOI | MR | Zbl

[42] Langmuir, I. The constitution and fundamental properties of solids and liquids, J. Am. Chem. Soc., Volume 39 (1917) no. 9, pp. 1848–1906 | DOI

[43] Maz′ya, V.; Shaposhnikova, T. On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., Volume 195 (2002) no. 2, pp. 230–238 | MR | Zbl

[44] Maz′ya, V.; Shaposhnikova, T. Erratum to: “On the Bourgain, Brezis and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces”, J. Funct. Anal., Volume 195 (2002) no. 2, pp. 230–238 MR1940355 (2003j:46051) J. Funct. Anal., 201, 1, 2003, 298–300 | MR | Zbl

[45] Melchionna, S.; Ranetbauer, H.; Scarpa, L.; Trussardi, L. From nonlocal to local Cahn-Hilliard equation, Adv. Math. Sci. Appl., Volume 28 (2019) no. 1, pp. 197–211 | MR

[46] Miranville, A.; Schimperna, G. On a doubly nonlinear Cahn-Hilliard-Gurtin system, Discrete Contin. Dyn. Syst., Ser. B, Volume 14 (2010) no. 2, pp. 675–697 | MR | Zbl

[47] Oono, Y.; Puri, S. Study of phase-separation dynamics by use of cell dynamical systems, Phys. Rev. A, Volume 38 (Jul 1988), pp. 434–453

[48] Ponce, A.C. An estimate in the spirit of Poincaré's inequality, J. Eur. Math. Soc., Volume 6 (2004) no. 1, pp. 1–15 | DOI | MR | Zbl

[49] Ponce, A.C. A new approach to Sobolev spaces and connections to Γ-convergence, Calc. Var. Partial Differ. Equ., Volume 19 (2004) no. 3, pp. 229–255 | DOI | MR

[50] Rocca, E.; Sprekels, J. Optimal distributed control of a nonlocal convective Cahn-Hilliard equation by the velocity in three dimensions, SIAM J. Control Optim., Volume 53 (2015) no. 3, pp. 1654–1680 | DOI | MR | Zbl

[51] Scarpa, L. Existence and uniqueness of solutions to singular Cahn-Hilliard equations with nonlinear viscosity terms and dynamic boundary conditions, J. Math. Anal. Appl., Volume 469 (2019) no. 2, pp. 730–764 | DOI | MR | Zbl

[52] Watson, S.J.; Otto, F.; Rubinstein, B.Y.; Davis, S.H. Coarsening dynamics of the convective Cahn-Hilliard equation, Physica D, Volume 178 (2003) no. 3–4, pp. 127–148 | MR | Zbl

Cité par Sources :