August  2020, 13(8): 2211-2229. doi: 10.3934/dcdss.2020186

Global-in-time Gevrey regularity solutions for the functionalized Cahn-Hilliard equation

1. 

School of Science, Southwest University of Science and Technology, Mianyang, Sichuan 621010, China

2. 

Mathematics Department, The University of Massachusetts, North Dartmouth, MA 02747, USA

3. 

Mathematics Department, The University of Tennessee, Knoxville, TN 37996, USA

4. 

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China

* Corresponding author: swise1@utk.edu

Received  April 2018 Published  August 2020 Early access  November 2019

Fund Project: C. Wang was supported by NSF grant DMS-1418689. S.M. Wise was supported by NSF grants DMS-1418692 and DMS-1719854

The existence and uniqueness of Gevrey regularity solutions for the functionalized Cahn-Hilliard (FCH) and Cahn-Hilliard-Willmore (CHW) equations are established. The energy dissipation law yields a uniform-in-time $ H^2 $ bound of the solution, and the polynomial patterns of the nonlinear terms enable one to derive a local-in-time solution with Gevrey regularity. A careful calculation reveals that the existence time interval length depends on the $ H^3 $ norm of the initial data. A further detailed estimate for the original PDE system indicates a uniform-in-time $ H^3 $ bound. Consequently, a global-in-time solution becomes available with Gevrey regularity.

Citation: Kelong Cheng, Cheng Wang, Steven M. Wise, Zixia Yuan. Global-in-time Gevrey regularity solutions for the functionalized Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2211-2229. doi: 10.3934/dcdss.2020186
References:
[1] R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, 65, Academic Press, New York-London, 1975. 
[2]

S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coursening, Acta. Metall., 27 (1979), 1085-1095.  doi: 10.1016/0001-6160(79)90196-2.

[3]

A. Biswas and D. Swanson, Existence and generalized Gevrey regularity of solutions to the Kuramoto-Sivashinsky equation in $R^n$, J. Differential Equations, 240 (2007), 145-163.  doi: 10.1016/j.jde.2007.05.022.

[4]

A. Biswas and D. Swanson, Gevrey regularity of solutions to the 3-D Navier-Stokes equations with weighted $\ell_p$ initial data, Indiana Univ. Math. J., 56 (2007), 1157-1188.  doi: 10.1512/iumj.2007.56.2891.

[5]

Z. BradshawZ. Grujic and I. Kukavica, Local analyticity radii of solutions to the 3D Navier-Stokes equations with locally analytic forcing, J. Differential Equations, 259 (2015), 3955-3975.  doi: 10.1016/j.jde.2015.05.009.

[6]

J. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801.  doi: 10.1016/0001-6160(61)90182-1.

[7]

J. Cahn and J. Hilliard, Free energy of a nonuniform system. Ⅰ: Interfacial free energy, J. Chem. Phys., 28 (1958). doi: 10.1063/1.1744102.

[8]

C. CaoM. Rammaha and E. Titi, Gevrey regularity for nonlinear analytic parabolic equations on the sphere, J. Dynam. Differential Equations, 12 (2000), 411-433.  doi: 10.1023/A:1009072526324.

[9]

F. Chen and J. Shen, Efficient spectral-Galerkin methods for systems of coupled second-order equations and their applications, J. Comput. Phys., 231 (2012), 5016-5028.  doi: 10.1016/j.jcp.2012.03.001.

[10]

N. ChenC. Wang and S. Wise, Global-in-time Gevrey regularity solution for a class of bistable gradient flows, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1689-1711.  doi: 10.3934/dcdsb.2016018.

[11]

Y. ChenJ. LowengrubJ. ShenC. Wang and S. Wise, Efficient energy stable schemes for isotropic and strongly anisotropic Cahn-Hilliard systems with the Willmore regularization, J. Comput. Phys., 365 (2018), 56-73.  doi: 10.1016/j.jcp.2018.03.024.

[12]

A. ChristliebJ. JonesK. PromislowB. Wetton and M. Willoughby, High accuracy solutions to energy gradient flows from material science models, J. Comput. Phys., 257 (2014), 193-215.  doi: 10.1016/j.jcp.2013.09.049.

[13]

S. Dai and K. Promislow, Geometric evolution of bilayers under the functionalized Cahn–Hilliard equation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 469 (2013), 20pp. doi: 10.1098/rspa.2012.0505.

[14]

A. DoelmanG. HayrapetyanK. Promislow and B. Wetton, Meander and pearling of single-curvature bilayer interfaces in the functionalized Cahn-Hilliard equation, SIAM J. Math. Anal., 46 (2014), 3640-3677.  doi: 10.1137/13092705X.

[15]

A. Eden and V. Kalantarov, The convective Cahn-Hilliard equation, Appl. Math. Lett., 20 (2007), 455-461.  doi: 10.1016/j.aml.2006.05.014.

[16]

W. FengZ. GuanJ. LowengrubC. WangS. Wise and Y. Chen, A uniquely solvable, energy stable numerical scheme for the functionalized Cahn-Hilliard equation and its convergence analysis, J. Sci. Comput., 76 (2018), 1938-1967.  doi: 10.1007/s10915-018-0690-1.

[17]

W. FengZ. GuoJ. Lowengrub and S. Wise, A mass-conservative adaptive FAS multigrid solver for cell-centered finite difference methods on block-structured, locally-cartesian grids, J. Comput. Phys., 352 (2018), 463-497.  doi: 10.1016/j.jcp.2017.09.065.

[18]

A. Ferrari and E. Titi, Gevrey regularity for nonlinear analytic parabolic equations, Comm. Partial Differential Equations, 23 (1998), 1-16.  doi: 10.1080/03605309808821336.

[19]

C. Foias and R. Temam, Gevrey class regularity for the solution of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369.  doi: 10.1016/0022-1236(89)90015-3.

[20]

N. GavishG. HayrapetyanK. Promislow and L. Yang, Curvature driven flow of bi-layer interfaces, Physica D: Nonlinear Phenomena, 240 (2011), 675-693.  doi: 10.1016/j.physd.2010.11.016.

[21]

N. GavishJ. JonesZ. XuA. Christlieb and K. Promislow, Variational models of network formation and ion transport: Applications to perfluorosulfonate ionomer membranes, Polymers, 4 (2012), 630-655.  doi: 10.3390/polym4010630.

[22]

G. Gompper and M. Schick, Correlation between structural and interfacial properties of amphiphilic systems, Phys. Rev. Lett., 65 (1990), 1116-1119.  doi: 10.1103/PhysRevLett.65.1116.

[23]

Z. Grujic and I. Kukavica, Space analyticity for the Navier-Stokes and related equations with initial data in $L^p$, J. Funct. Anal., 152 (1998), 447-466.  doi: 10.1006/jfan.1997.3167.

[24]

R. GuoY. Xu and Z. Xu, Local discontinuous Galerkin methods for the functionalized Cahn-Hilliard equation, J. Sci. Comput., 63 (2015), 913-937.  doi: 10.1007/s10915-014-9920-3.

[25]

W. Hsu and T. Gierke, Ion transport and clustering in nafion perfluorinated membranes, J. Membr. Sci., 13 (1983), 307-326.  doi: 10.1016/S0376-7388(00)81563-X.

[26]

V. KalantarovB. Levant and E. Titi, Gevrey regularity for the attractor of the 3D Navier-Stokes-Voight equations, J. Nonlinear Sci., 19 (2009), 133-152.  doi: 10.1007/s00332-008-9029-7.

[27]

I. KukavicaR. TemamV. Vlad and M. Ziane, On the time analyticity radius of the solutions of the two-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 3 (1991), 611-618.  doi: 10.1007/BF01049102.

[28]

I. KukavicaR. TemamV. Vlad and M. Ziane, Existence and uniqueness of solutions for the hydrostatic Euler equations on a bounded domain with analytic data, C. R. Math. Acad. Sci. Paris, 348 (2010), 639-645.  doi: 10.1016/j.crma.2010.03.023.

[29]

I. Kukavica and V. Vlad, On the radius of analyticity of solutions to the three-dimensional Euler equations, Proc. Amer. Math. Soc., 137 (2009), 669-677.  doi: 10.1090/S0002-9939-08-09693-7.

[30]

I. Kukavica and V. Vlad, The domain of analyticity of solutions to the three-dimensional Euler equations in a half space, Discrete Contin. Dyn. Syst., 29 (2011), 285-303.  doi: 10.3934/dcds.2011.29.285.

[31]

I. Kukavica and V. Vlad, On the analyticity and Gevrey-class regularity up to the boundary for the Euler equations, Nonlinearity, 24 (2011), 765-796.  doi: 10.1088/0951-7715/24/3/004.

[32]

I. Kukavica and V. Vlad, On the local existence of analytic solutions to the Prandtl boundary layer equations, Commun. Math. Sci., 11 (2013), 269-292.  doi: 10.4310/CMS.2013.v11.n1.a8.

[33]

J. LowengrubE. Titi and K. Zhao, Analysis of a mixture model of tumor growth, European J. Appl. Math., 24 (2013), 691-734.  doi: 10.1017/S0956792513000144.

[34]

H. Ly and E. Titi, Global Gevrey regularity for the Bénard convection in a porous medium with zero Darcy-Prandtl number, J. Nonlinear Sci., 9 (1999), 333-362.  doi: 10.1007/s003329900073.

[35]

K. Promislow, Time analyticity and Gevrey regularity for solutions of a class of dissipative partial differential equations, Nonlinear Anal., 16 (1991), 959-980.  doi: 10.1016/0362-546X(91)90100-F.

[36]

K. Promislow and B. Wetton, PEM fuel cells: A mathematical overview, SIAM J. Appl. Math., 70 (2009), 369-409.  doi: 10.1137/080720802.

[37]

K. Promislow and Q. Wu, Existence of pearled patterns in the planar functionalized Cahn-Hilliard equation, J. Differential Equations, 259 (2015), 3298-3343.  doi: 10.1016/j.jde.2015.04.022.

[38] J. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001. 
[39]

R. RyhamF. S. Cohen and R. Eisenberg, A dynamic model of open vesicles in fluids, Commun. Math. Sci., 10 (2012), 1273-1285.  doi: 10.4310/CMS.2012.v10.n4.a12.

[40]

D. Swanson, Gevrey regularity of certain solutions to the Cahn-Hilliard equation with rough initial data, Methods Appl. Anal., 18 (2011), 417-426.  doi: 10.4310/MAA.2011.v18.n4.a4.

[41]

S. TorabiJ. LowengrubA. Voigt and S. Wise, A new phase-field model for strongly anisotropic systems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2009), 1337-1359.  doi: 10.1098/rspa.2008.0385.

[42]

S. Torabi, S. Wise, J. Lowengrub, A. Ratz and A. Voigt, A new method for simulating strongly anisotropic Cahn-Hilliard equations, MST 2007 Conference Proceedings, 3, 2007.

[43]

X. WangL. Ju and Q. Du, Efficient and stable exponential time differencing Runge-Kutta methods for phase field elastic bending energy models, J. Comput. Phys., 316 (2016), 21-38.  doi: 10.1016/j.jcp.2016.04.004.

[44]

S. WiseJ. Kim and J. Lowengrub, Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method, J. Comput. Phys., 226 (2007), 414-446.  doi: 10.1016/j.jcp.2007.04.020.

show all references

References:
[1] R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, 65, Academic Press, New York-London, 1975. 
[2]

S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coursening, Acta. Metall., 27 (1979), 1085-1095.  doi: 10.1016/0001-6160(79)90196-2.

[3]

A. Biswas and D. Swanson, Existence and generalized Gevrey regularity of solutions to the Kuramoto-Sivashinsky equation in $R^n$, J. Differential Equations, 240 (2007), 145-163.  doi: 10.1016/j.jde.2007.05.022.

[4]

A. Biswas and D. Swanson, Gevrey regularity of solutions to the 3-D Navier-Stokes equations with weighted $\ell_p$ initial data, Indiana Univ. Math. J., 56 (2007), 1157-1188.  doi: 10.1512/iumj.2007.56.2891.

[5]

Z. BradshawZ. Grujic and I. Kukavica, Local analyticity radii of solutions to the 3D Navier-Stokes equations with locally analytic forcing, J. Differential Equations, 259 (2015), 3955-3975.  doi: 10.1016/j.jde.2015.05.009.

[6]

J. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801.  doi: 10.1016/0001-6160(61)90182-1.

[7]

J. Cahn and J. Hilliard, Free energy of a nonuniform system. Ⅰ: Interfacial free energy, J. Chem. Phys., 28 (1958). doi: 10.1063/1.1744102.

[8]

C. CaoM. Rammaha and E. Titi, Gevrey regularity for nonlinear analytic parabolic equations on the sphere, J. Dynam. Differential Equations, 12 (2000), 411-433.  doi: 10.1023/A:1009072526324.

[9]

F. Chen and J. Shen, Efficient spectral-Galerkin methods for systems of coupled second-order equations and their applications, J. Comput. Phys., 231 (2012), 5016-5028.  doi: 10.1016/j.jcp.2012.03.001.

[10]

N. ChenC. Wang and S. Wise, Global-in-time Gevrey regularity solution for a class of bistable gradient flows, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1689-1711.  doi: 10.3934/dcdsb.2016018.

[11]

Y. ChenJ. LowengrubJ. ShenC. Wang and S. Wise, Efficient energy stable schemes for isotropic and strongly anisotropic Cahn-Hilliard systems with the Willmore regularization, J. Comput. Phys., 365 (2018), 56-73.  doi: 10.1016/j.jcp.2018.03.024.

[12]

A. ChristliebJ. JonesK. PromislowB. Wetton and M. Willoughby, High accuracy solutions to energy gradient flows from material science models, J. Comput. Phys., 257 (2014), 193-215.  doi: 10.1016/j.jcp.2013.09.049.

[13]

S. Dai and K. Promislow, Geometric evolution of bilayers under the functionalized Cahn–Hilliard equation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 469 (2013), 20pp. doi: 10.1098/rspa.2012.0505.

[14]

A. DoelmanG. HayrapetyanK. Promislow and B. Wetton, Meander and pearling of single-curvature bilayer interfaces in the functionalized Cahn-Hilliard equation, SIAM J. Math. Anal., 46 (2014), 3640-3677.  doi: 10.1137/13092705X.

[15]

A. Eden and V. Kalantarov, The convective Cahn-Hilliard equation, Appl. Math. Lett., 20 (2007), 455-461.  doi: 10.1016/j.aml.2006.05.014.

[16]

W. FengZ. GuanJ. LowengrubC. WangS. Wise and Y. Chen, A uniquely solvable, energy stable numerical scheme for the functionalized Cahn-Hilliard equation and its convergence analysis, J. Sci. Comput., 76 (2018), 1938-1967.  doi: 10.1007/s10915-018-0690-1.

[17]

W. FengZ. GuoJ. Lowengrub and S. Wise, A mass-conservative adaptive FAS multigrid solver for cell-centered finite difference methods on block-structured, locally-cartesian grids, J. Comput. Phys., 352 (2018), 463-497.  doi: 10.1016/j.jcp.2017.09.065.

[18]

A. Ferrari and E. Titi, Gevrey regularity for nonlinear analytic parabolic equations, Comm. Partial Differential Equations, 23 (1998), 1-16.  doi: 10.1080/03605309808821336.

[19]

C. Foias and R. Temam, Gevrey class regularity for the solution of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369.  doi: 10.1016/0022-1236(89)90015-3.

[20]

N. GavishG. HayrapetyanK. Promislow and L. Yang, Curvature driven flow of bi-layer interfaces, Physica D: Nonlinear Phenomena, 240 (2011), 675-693.  doi: 10.1016/j.physd.2010.11.016.

[21]

N. GavishJ. JonesZ. XuA. Christlieb and K. Promislow, Variational models of network formation and ion transport: Applications to perfluorosulfonate ionomer membranes, Polymers, 4 (2012), 630-655.  doi: 10.3390/polym4010630.

[22]

G. Gompper and M. Schick, Correlation between structural and interfacial properties of amphiphilic systems, Phys. Rev. Lett., 65 (1990), 1116-1119.  doi: 10.1103/PhysRevLett.65.1116.

[23]

Z. Grujic and I. Kukavica, Space analyticity for the Navier-Stokes and related equations with initial data in $L^p$, J. Funct. Anal., 152 (1998), 447-466.  doi: 10.1006/jfan.1997.3167.

[24]

R. GuoY. Xu and Z. Xu, Local discontinuous Galerkin methods for the functionalized Cahn-Hilliard equation, J. Sci. Comput., 63 (2015), 913-937.  doi: 10.1007/s10915-014-9920-3.

[25]

W. Hsu and T. Gierke, Ion transport and clustering in nafion perfluorinated membranes, J. Membr. Sci., 13 (1983), 307-326.  doi: 10.1016/S0376-7388(00)81563-X.

[26]

V. KalantarovB. Levant and E. Titi, Gevrey regularity for the attractor of the 3D Navier-Stokes-Voight equations, J. Nonlinear Sci., 19 (2009), 133-152.  doi: 10.1007/s00332-008-9029-7.

[27]

I. KukavicaR. TemamV. Vlad and M. Ziane, On the time analyticity radius of the solutions of the two-dimensional Navier-Stokes equations, J. Dynam. Differential Equations, 3 (1991), 611-618.  doi: 10.1007/BF01049102.

[28]

I. KukavicaR. TemamV. Vlad and M. Ziane, Existence and uniqueness of solutions for the hydrostatic Euler equations on a bounded domain with analytic data, C. R. Math. Acad. Sci. Paris, 348 (2010), 639-645.  doi: 10.1016/j.crma.2010.03.023.

[29]

I. Kukavica and V. Vlad, On the radius of analyticity of solutions to the three-dimensional Euler equations, Proc. Amer. Math. Soc., 137 (2009), 669-677.  doi: 10.1090/S0002-9939-08-09693-7.

[30]

I. Kukavica and V. Vlad, The domain of analyticity of solutions to the three-dimensional Euler equations in a half space, Discrete Contin. Dyn. Syst., 29 (2011), 285-303.  doi: 10.3934/dcds.2011.29.285.

[31]

I. Kukavica and V. Vlad, On the analyticity and Gevrey-class regularity up to the boundary for the Euler equations, Nonlinearity, 24 (2011), 765-796.  doi: 10.1088/0951-7715/24/3/004.

[32]

I. Kukavica and V. Vlad, On the local existence of analytic solutions to the Prandtl boundary layer equations, Commun. Math. Sci., 11 (2013), 269-292.  doi: 10.4310/CMS.2013.v11.n1.a8.

[33]

J. LowengrubE. Titi and K. Zhao, Analysis of a mixture model of tumor growth, European J. Appl. Math., 24 (2013), 691-734.  doi: 10.1017/S0956792513000144.

[34]

H. Ly and E. Titi, Global Gevrey regularity for the Bénard convection in a porous medium with zero Darcy-Prandtl number, J. Nonlinear Sci., 9 (1999), 333-362.  doi: 10.1007/s003329900073.

[35]

K. Promislow, Time analyticity and Gevrey regularity for solutions of a class of dissipative partial differential equations, Nonlinear Anal., 16 (1991), 959-980.  doi: 10.1016/0362-546X(91)90100-F.

[36]

K. Promislow and B. Wetton, PEM fuel cells: A mathematical overview, SIAM J. Appl. Math., 70 (2009), 369-409.  doi: 10.1137/080720802.

[37]

K. Promislow and Q. Wu, Existence of pearled patterns in the planar functionalized Cahn-Hilliard equation, J. Differential Equations, 259 (2015), 3298-3343.  doi: 10.1016/j.jde.2015.04.022.

[38] J. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001. 
[39]

R. RyhamF. S. Cohen and R. Eisenberg, A dynamic model of open vesicles in fluids, Commun. Math. Sci., 10 (2012), 1273-1285.  doi: 10.4310/CMS.2012.v10.n4.a12.

[40]

D. Swanson, Gevrey regularity of certain solutions to the Cahn-Hilliard equation with rough initial data, Methods Appl. Anal., 18 (2011), 417-426.  doi: 10.4310/MAA.2011.v18.n4.a4.

[41]

S. TorabiJ. LowengrubA. Voigt and S. Wise, A new phase-field model for strongly anisotropic systems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2009), 1337-1359.  doi: 10.1098/rspa.2008.0385.

[42]

S. Torabi, S. Wise, J. Lowengrub, A. Ratz and A. Voigt, A new method for simulating strongly anisotropic Cahn-Hilliard equations, MST 2007 Conference Proceedings, 3, 2007.

[43]

X. WangL. Ju and Q. Du, Efficient and stable exponential time differencing Runge-Kutta methods for phase field elastic bending energy models, J. Comput. Phys., 316 (2016), 21-38.  doi: 10.1016/j.jcp.2016.04.004.

[44]

S. WiseJ. Kim and J. Lowengrub, Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method, J. Comput. Phys., 226 (2007), 414-446.  doi: 10.1016/j.jcp.2007.04.020.

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