# American Institute of Mathematical Sciences

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. Bradshaw, Z. 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. Cao, M. 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. Chen, C. 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. Chen, J. Lowengrub, J. Shen, C. 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. Christlieb, J. Jones, K. Promislow, B. 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. Doelman, G. Hayrapetyan, K. 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. Feng, Z. Guan, J. Lowengrub, C. Wang, S. 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. Feng, Z. Guo, J. 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. Gavish, G. Hayrapetyan, K. 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. Gavish, J. Jones, Z. Xu, A. 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. Guo, Y. 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. Kalantarov, B. 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. Kukavica, R. Temam, V. 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. Kukavica, R. Temam, V. 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. Lowengrub, E. 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. Ryham, F. 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. Torabi, J. Lowengrub, A. 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. Wang, L. 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. Wise, J. 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. Bradshaw, Z. 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. Cao, M. 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. Chen, C. 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. Chen, J. Lowengrub, J. Shen, C. 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. Christlieb, J. Jones, K. Promislow, B. 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. Doelman, G. Hayrapetyan, K. 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. Feng, Z. Guan, J. Lowengrub, C. Wang, S. 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. Feng, Z. Guo, J. 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. Gavish, G. Hayrapetyan, K. 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. Gavish, J. Jones, Z. Xu, A. 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. Guo, Y. 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. Kalantarov, B. 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. Kukavica, R. Temam, V. 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. Kukavica, R. Temam, V. 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. Lowengrub, E. 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. Ryham, F. 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. Torabi, J. Lowengrub, A. 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. Wang, L. 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. Wise, J. 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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