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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 |
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.
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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