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Equilibrium validation in models for pattern formation based on Sobolev embeddings
Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA |
In the study of equilibrium solutions for partial differential equations there are so many equilibria that one cannot hope to find them all. Therefore one usually concentrates on finding individual branches of equilibrium solutions. On the one hand, a rigorous theoretical understanding of these branches is ideal but not generally tractable. On the other hand, numerical bifurcation searches are useful but not guaranteed to give an accurate structure, in that they could miss a portion of a branch or find a spurious branch where none exists. In a series of recent papers, we have aimed for a third option. Namely, we have developed a method of computer-assisted proofs to prove both existence and isolation of branches of equilibrium solutions. In the current paper, we extend these techniques to the Ohta-Kawasaki model for the dynamics of diblock copolymers in dimensions one, two, and three, by giving a detailed description of the analytical underpinnings of the method. Although the paper concentrates on applying the method to the Ohta-Kawasaki model, the functional analytic approach and techniques can be generalized to other parabolic partial differential equations.
References:
[1] |
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Elsevier/Academic Press, Amsterdam, 2003.
![]() |
[2] |
G. Arioli and H. Koch,
Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation, Archive for Rational Mechanics and Analysis, 197 (2010), 1033-1051.
doi: 10.1007/s00205-010-0309-7. |
[3] |
S. Cai and Y. Watanabe, A computer-assisted method for the diblock copolymer model, Zeitschrift für Angewandte Mathematik und Mechanik, 99 (2019), e201800125, 14pp.
doi: 10.1002/zamm.201800125. |
[4] |
L. Chierchia, KAM lectures, in Dynamical Systems. Part I, Scuola Normale Superiore, Pisa, Italy, 2003, 1–55. Google Scholar |
[5] |
R. Choksi, M. Maras and J. F. Williams,
2D phase diagram for minimizers of a Cahn-Hilliard functional with long-range interactions, SIAM Journal on Applied Dynamical Systems, 10 (2011), 1344-1362.
doi: 10.1137/100784497. |
[6] |
R. Choksi, M. A. Peletier and J. F. Williams,
On the phase diagram for microphase separation of diblock copolymers: An approach via a nonlocal Cahn-Hilliard functional, SIAM Journal on Applied Mathematics, 69 (2009), 1712-1738.
doi: 10.1137/080728809. |
[7] |
R. Choksi and X. Ren,
On the derivation of a density functional theory for microphase separation of diblock copolymers, Journal of Statistical Physics, 113 (2003), 151-176.
doi: 10.1023/A:1025722804873. |
[8] |
R. Choksi and X. Ren,
Diblock copolymer/homopolymer blends: Derivation of a density functional theory, Physica D, 203 (2005), 100-119.
doi: 10.1016/j.physd.2005.03.006. |
[9] |
J. Cyranka and T. Wanner,
Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model, SIAM Journal on Applied Dynamical Systems, 17 (2018), 694-731.
doi: 10.1137/17M111938X. |
[10] |
S. Day, J.-P. Lessard and K. Mischaikow,
Validated continuation for equilibria of PDEs, SIAM Journal on Numerical Analysis, 45 (2007), 1398-1424.
doi: 10.1137/050645968. |
[11] |
J. P. Desi, H. Edrees, J. Price, E. Sander and T. Wanner,
The dynamics of nucleation in stochastic Cahn-Morral systems, SIAM Journal on Applied Dynamical Systems, 10 (2011), 707-743.
doi: 10.1137/100801378. |
[12] |
A. Dhooge, W. Govaerts and Y. A. Kuznetsov,
MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs, Association for Computing Machinery. Transactions on Mathematical Software, 29 (2003), 141-164.
doi: 10.1145/779359.779362. |
[13] |
E. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, in Proceedings of the Tenth Manitoba Conference on Numerical Mathematics and Computing, Vol. I (Winnipeg, Man., 1980), 30 (1981), 265–284. |
[14] |
M. Gameiro, J.-P. Lessard and K. Mischaikow,
Validated continuation over large parameter ranges for equilibria of PDEs, Mathematics and Computers in Simulation, 79 (2008), 1368-1382.
doi: 10.1016/j.matcom.2008.03.014. |
[15] |
Z. G. Huseynov and A. M. Shykhammedov,
On bases of sines and cosines in Sobolev spaces, Applied Mathematics Letters, 25 (2012), 275-278.
doi: 10.1016/j.aml.2011.08.026. |
[16] |
I. Johnson, E. Sander and T. Wanner,
Branch interactions and long-term dynamics for the diblock copolymer model in one dimension, Discrete and Continuous Dynamical Systems. Series A, 33 (2013), 3671-3705.
doi: 10.3934/dcds.2013.33.3671. |
[17] |
T. Kinoshita, Y. Watanabe and M. T. Nakao,
An alternative approach to norm bound computation for inverses of linear operators in Hilbert spaces, Journal of Differential Equations, 266 (2019), 5431-5447.
doi: 10.1016/j.jde.2018.10.027. |
[18] |
J.-P. Lessard, E. Sander and T. Wanner,
Rigorous continuation of bifurcation points in the diblock copolymer equation, Journal of Computational Dynamics, 4 (2017), 71-118.
doi: 10.3934/jcd.2017003. |
[19] |
S. Maier-Paape, U. Miller, K. Mischaikow and T. Wanner,
Rigorous numerics for the Cahn-Hilliard equation on the unit square, Revista Matematica Complutense, 21 (2008), 351-426.
doi: 10.5209/rev_REMA.2008.v21.n2.16380. |
[20] |
S. Maier-Paape, K. Mischaikow and T. Wanner,
Structure of the attractor of the Cahn-Hilliard equation on a square, International Journal of Bifurcation and Chaos, 17 (2007), 1221-1263.
doi: 10.1142/S0218127407017781. |
[21] |
S. Maier-Paape and T. Wanner,
Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: Nonlinear dynamics, Archive for Rational Mechanics and Analysis, 151 (2000), 187-219.
doi: 10.1007/s002050050196. |
[22] |
T. R. Muradov and V. F. Salmanov,
On the basis property of trigonometric systems with linear phase in a weighted Sobolev space, Dokl. Math., 90 (2014), 611-612.
doi: 10.1134/s1064562414060301. |
[23] |
M. T. Nakao, M. Plum and Y. Watanabe, Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations, Springer-Verlag, Berlin, 2019.
doi: 10.1007/978-981-13-7669-6. |
[24] |
T. Ohta and K. Kawasaki,
Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632.
doi: 10.1021/ma00164a028. |
[25] |
M. Plum,
Existence and enclosure results for continua of solutions of parameter-dependent nonlinear boundary value problems, Journal of Computational and Applied Mathematics, 60 (1995), 187-200.
doi: 10.1016/0377-0427(94)00091-E. |
[26] |
M. Plum, Enclosures for two-point boundary value problems near bifurcation points, in Scientific Computing and Validated Numerics (Wuppertal, 1995), vol. 90 of Mathematical Research, Akademie Verlag, Berlin, 1996,265–279. |
[27] |
M. Plum,
Computer-assisted proofs for semilinear elliptic boundary value problems, Japan Journal of Industrial and Applied Mathematics, 26 (2009), 419-442.
doi: 10.1007/BF03186542. |
[28] |
S. M. Rump, INTLAB - INTerval LABoratory, in Developments in Reliable Computing (ed. T. Csendes), Kluwer Academic Publishers, Dordrecht, 1999, 77–104, http://www.ti3.tuhh.de/rump/. Google Scholar |
[29] |
S. M. Rump,
Verification methods: Rigorous results using floating-point arithmetic, Acta Numerica, 19 (2010), 287-449.
doi: 10.1017/S096249291000005X. |
[30] |
E. Sander and T. Wanner,
Validated saddle-node bifurcations and applications to lattice dynamical systems, SIAM Journal on Applied Dynamical Systems, 15 (2016), 1690-1733.
doi: 10.1137/16M1061011. |
[31] |
L. N. Trefethen and M. Embree, Spectra and Pseudospectra, Princeton University Press, Princeton, NJ, 2005.
![]() |
[32] |
J. B. van den Berg and J. F. Williams,
Validation of the bifurcation diagram in the 2D Ohta-Kawasaki problem, Nonlinearity, 30 (2017), 1584-1638.
doi: 10.1088/1361-6544/aa60e8. |
[33] |
J. B. van den Berg and J. F. Williams, Optimal periodic structures with general space group symmetries in the Ohta-Kawasaki problem, arXiv: 1912.00059. Google Scholar |
[34] |
J. B. van den Berg and J. F. Williams,
Rigorously computing symmetric stationary states of the Ohta-Kawasaki problem in three dimensions, SIAM Journal on Mathematical Analysis, 51 (2019), 131-158.
doi: 10.1137/17M1155624. |
[35] |
T. Wanner, Topological analysis of the diblock copolymer equation, in Mathematical Challenges in a New Phase of Materials Science (eds. Y. Nishiura and M. Kotani), vol. 166 of Springer Proceedings in Mathematics & Statistics, Springer-Verlag, 2016, 27–51.
doi: 10.1007/978-4-431-56104-0_2. |
[36] |
T. Wanner,
Computer-assisted equilibrium validation for the diblock copolymer model, Discrete and Continuous Dynamical Systems, Series A, 37 (2017), 1075-1107.
doi: 10.3934/dcds.2017045. |
[37] |
T. Wanner,
Computer-assisted bifurcation diagram validation and applications in materials science, Proceedings of Symposia in Applied Mathematics, 74 (2018), 123-174.
|
[38] |
T. Wanner,
Validated bounds on embedding constants for Sobolev space Banach algebras, Mathematical Methods in the Applied Sciences, 41 (2018), 9361-9376.
doi: 10.1002/mma.5294. |
[39] |
Y. Watanabe, T. Kinoshita and M. T. Nakao,
An improved method for verifying the existence and bounds of the inverse of second-order linear elliptic operators mapping to dual space, Japan Journal of Industrial and Applied Mathematics, 36 (2019), 407-420.
doi: 10.1007/s13160-019-00344-8. |
[40] |
Y. Watanabe, K. Nagatou, M. Plum and M. T. Nakao,
Norm bound computation for inverses of linear operators in Hilbert spaces, Journal of Differential Equations, 260 (2016), 6363-6374.
doi: 10.1016/j.jde.2015.12.041. |
[41] |
N. Yamamoto,
A numerical verification method for solutions of boundary value problems with local uniqueness by Banach's fixed-point theorem, SIAM Journal on Numerical Analysis, 35 (1998), 2004-2013.
doi: 10.1137/S0036142996304498. |
[42] |
N. Yamamoto, M. T. Nakao and Y. Watanabe,
A theorem for numerical verification on local uniqueness of solutions to fixed-point equations, Numerical Functional Analysis and Optimization, 32 (2011), 1190-1204.
doi: 10.1080/01630563.2011.594348. |
show all references
References:
[1] |
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Elsevier/Academic Press, Amsterdam, 2003.
![]() |
[2] |
G. Arioli and H. Koch,
Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation, Archive for Rational Mechanics and Analysis, 197 (2010), 1033-1051.
doi: 10.1007/s00205-010-0309-7. |
[3] |
S. Cai and Y. Watanabe, A computer-assisted method for the diblock copolymer model, Zeitschrift für Angewandte Mathematik und Mechanik, 99 (2019), e201800125, 14pp.
doi: 10.1002/zamm.201800125. |
[4] |
L. Chierchia, KAM lectures, in Dynamical Systems. Part I, Scuola Normale Superiore, Pisa, Italy, 2003, 1–55. Google Scholar |
[5] |
R. Choksi, M. Maras and J. F. Williams,
2D phase diagram for minimizers of a Cahn-Hilliard functional with long-range interactions, SIAM Journal on Applied Dynamical Systems, 10 (2011), 1344-1362.
doi: 10.1137/100784497. |
[6] |
R. Choksi, M. A. Peletier and J. F. Williams,
On the phase diagram for microphase separation of diblock copolymers: An approach via a nonlocal Cahn-Hilliard functional, SIAM Journal on Applied Mathematics, 69 (2009), 1712-1738.
doi: 10.1137/080728809. |
[7] |
R. Choksi and X. Ren,
On the derivation of a density functional theory for microphase separation of diblock copolymers, Journal of Statistical Physics, 113 (2003), 151-176.
doi: 10.1023/A:1025722804873. |
[8] |
R. Choksi and X. Ren,
Diblock copolymer/homopolymer blends: Derivation of a density functional theory, Physica D, 203 (2005), 100-119.
doi: 10.1016/j.physd.2005.03.006. |
[9] |
J. Cyranka and T. Wanner,
Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model, SIAM Journal on Applied Dynamical Systems, 17 (2018), 694-731.
doi: 10.1137/17M111938X. |
[10] |
S. Day, J.-P. Lessard and K. Mischaikow,
Validated continuation for equilibria of PDEs, SIAM Journal on Numerical Analysis, 45 (2007), 1398-1424.
doi: 10.1137/050645968. |
[11] |
J. P. Desi, H. Edrees, J. Price, E. Sander and T. Wanner,
The dynamics of nucleation in stochastic Cahn-Morral systems, SIAM Journal on Applied Dynamical Systems, 10 (2011), 707-743.
doi: 10.1137/100801378. |
[12] |
A. Dhooge, W. Govaerts and Y. A. Kuznetsov,
MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs, Association for Computing Machinery. Transactions on Mathematical Software, 29 (2003), 141-164.
doi: 10.1145/779359.779362. |
[13] |
E. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, in Proceedings of the Tenth Manitoba Conference on Numerical Mathematics and Computing, Vol. I (Winnipeg, Man., 1980), 30 (1981), 265–284. |
[14] |
M. Gameiro, J.-P. Lessard and K. Mischaikow,
Validated continuation over large parameter ranges for equilibria of PDEs, Mathematics and Computers in Simulation, 79 (2008), 1368-1382.
doi: 10.1016/j.matcom.2008.03.014. |
[15] |
Z. G. Huseynov and A. M. Shykhammedov,
On bases of sines and cosines in Sobolev spaces, Applied Mathematics Letters, 25 (2012), 275-278.
doi: 10.1016/j.aml.2011.08.026. |
[16] |
I. Johnson, E. Sander and T. Wanner,
Branch interactions and long-term dynamics for the diblock copolymer model in one dimension, Discrete and Continuous Dynamical Systems. Series A, 33 (2013), 3671-3705.
doi: 10.3934/dcds.2013.33.3671. |
[17] |
T. Kinoshita, Y. Watanabe and M. T. Nakao,
An alternative approach to norm bound computation for inverses of linear operators in Hilbert spaces, Journal of Differential Equations, 266 (2019), 5431-5447.
doi: 10.1016/j.jde.2018.10.027. |
[18] |
J.-P. Lessard, E. Sander and T. Wanner,
Rigorous continuation of bifurcation points in the diblock copolymer equation, Journal of Computational Dynamics, 4 (2017), 71-118.
doi: 10.3934/jcd.2017003. |
[19] |
S. Maier-Paape, U. Miller, K. Mischaikow and T. Wanner,
Rigorous numerics for the Cahn-Hilliard equation on the unit square, Revista Matematica Complutense, 21 (2008), 351-426.
doi: 10.5209/rev_REMA.2008.v21.n2.16380. |
[20] |
S. Maier-Paape, K. Mischaikow and T. Wanner,
Structure of the attractor of the Cahn-Hilliard equation on a square, International Journal of Bifurcation and Chaos, 17 (2007), 1221-1263.
doi: 10.1142/S0218127407017781. |
[21] |
S. Maier-Paape and T. Wanner,
Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: Nonlinear dynamics, Archive for Rational Mechanics and Analysis, 151 (2000), 187-219.
doi: 10.1007/s002050050196. |
[22] |
T. R. Muradov and V. F. Salmanov,
On the basis property of trigonometric systems with linear phase in a weighted Sobolev space, Dokl. Math., 90 (2014), 611-612.
doi: 10.1134/s1064562414060301. |
[23] |
M. T. Nakao, M. Plum and Y. Watanabe, Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations, Springer-Verlag, Berlin, 2019.
doi: 10.1007/978-981-13-7669-6. |
[24] |
T. Ohta and K. Kawasaki,
Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632.
doi: 10.1021/ma00164a028. |
[25] |
M. Plum,
Existence and enclosure results for continua of solutions of parameter-dependent nonlinear boundary value problems, Journal of Computational and Applied Mathematics, 60 (1995), 187-200.
doi: 10.1016/0377-0427(94)00091-E. |
[26] |
M. Plum, Enclosures for two-point boundary value problems near bifurcation points, in Scientific Computing and Validated Numerics (Wuppertal, 1995), vol. 90 of Mathematical Research, Akademie Verlag, Berlin, 1996,265–279. |
[27] |
M. Plum,
Computer-assisted proofs for semilinear elliptic boundary value problems, Japan Journal of Industrial and Applied Mathematics, 26 (2009), 419-442.
doi: 10.1007/BF03186542. |
[28] |
S. M. Rump, INTLAB - INTerval LABoratory, in Developments in Reliable Computing (ed. T. Csendes), Kluwer Academic Publishers, Dordrecht, 1999, 77–104, http://www.ti3.tuhh.de/rump/. Google Scholar |
[29] |
S. M. Rump,
Verification methods: Rigorous results using floating-point arithmetic, Acta Numerica, 19 (2010), 287-449.
doi: 10.1017/S096249291000005X. |
[30] |
E. Sander and T. Wanner,
Validated saddle-node bifurcations and applications to lattice dynamical systems, SIAM Journal on Applied Dynamical Systems, 15 (2016), 1690-1733.
doi: 10.1137/16M1061011. |
[31] |
L. N. Trefethen and M. Embree, Spectra and Pseudospectra, Princeton University Press, Princeton, NJ, 2005.
![]() |
[32] |
J. B. van den Berg and J. F. Williams,
Validation of the bifurcation diagram in the 2D Ohta-Kawasaki problem, Nonlinearity, 30 (2017), 1584-1638.
doi: 10.1088/1361-6544/aa60e8. |
[33] |
J. B. van den Berg and J. F. Williams, Optimal periodic structures with general space group symmetries in the Ohta-Kawasaki problem, arXiv: 1912.00059. Google Scholar |
[34] |
J. B. van den Berg and J. F. Williams,
Rigorously computing symmetric stationary states of the Ohta-Kawasaki problem in three dimensions, SIAM Journal on Mathematical Analysis, 51 (2019), 131-158.
doi: 10.1137/17M1155624. |
[35] |
T. Wanner, Topological analysis of the diblock copolymer equation, in Mathematical Challenges in a New Phase of Materials Science (eds. Y. Nishiura and M. Kotani), vol. 166 of Springer Proceedings in Mathematics & Statistics, Springer-Verlag, 2016, 27–51.
doi: 10.1007/978-4-431-56104-0_2. |
[36] |
T. Wanner,
Computer-assisted equilibrium validation for the diblock copolymer model, Discrete and Continuous Dynamical Systems, Series A, 37 (2017), 1075-1107.
doi: 10.3934/dcds.2017045. |
[37] |
T. Wanner,
Computer-assisted bifurcation diagram validation and applications in materials science, Proceedings of Symposia in Applied Mathematics, 74 (2018), 123-174.
|
[38] |
T. Wanner,
Validated bounds on embedding constants for Sobolev space Banach algebras, Mathematical Methods in the Applied Sciences, 41 (2018), 9361-9376.
doi: 10.1002/mma.5294. |
[39] |
Y. Watanabe, T. Kinoshita and M. T. Nakao,
An improved method for verifying the existence and bounds of the inverse of second-order linear elliptic operators mapping to dual space, Japan Journal of Industrial and Applied Mathematics, 36 (2019), 407-420.
doi: 10.1007/s13160-019-00344-8. |
[40] |
Y. Watanabe, K. Nagatou, M. Plum and M. T. Nakao,
Norm bound computation for inverses of linear operators in Hilbert spaces, Journal of Differential Equations, 260 (2016), 6363-6374.
doi: 10.1016/j.jde.2015.12.041. |
[41] |
N. Yamamoto,
A numerical verification method for solutions of boundary value problems with local uniqueness by Banach's fixed-point theorem, SIAM Journal on Numerical Analysis, 35 (1998), 2004-2013.
doi: 10.1137/S0036142996304498. |
[42] |
N. Yamamoto, M. T. Nakao and Y. Watanabe,
A theorem for numerical verification on local uniqueness of solutions to fixed-point equations, Numerical Functional Analysis and Optimization, 32 (2011), 1190-1204.
doi: 10.1080/01630563.2011.594348. |




Dimension |
|||
Sobolev Embedding Constant |
|||
Sobolev Embedding Constant |
|||
Banach Algebra Constant |
Dimension |
|||
Sobolev Embedding Constant |
|||
Sobolev Embedding Constant |
|||
Banach Algebra Constant |
6.2575 | 89 | 0.0016 | 0.0056 | ||
2.9259e-04 | 0.0056 | ||||
2.8705e-06 | 0.0044 | ||||
6.4590 | 104 | 0.0011 | 0.0050 | ||
2.5369e-04 | 0.0050 | ||||
2.5579e-06 | 0.0041 | ||||
3.1030 | 74 | 0.0052 | 0.0107 | ||
0.0011 | 0.0106 | ||||
1.2871e-05 | 0.0092 |
6.2575 | 89 | 0.0016 | 0.0056 | ||
2.9259e-04 | 0.0056 | ||||
2.8705e-06 | 0.0044 | ||||
6.4590 | 104 | 0.0011 | 0.0050 | ||
2.5369e-04 | 0.0050 | ||||
2.5579e-06 | 0.0041 | ||||
3.1030 | 74 | 0.0052 | 0.0107 | ||
0.0011 | 0.0106 | ||||
1.2871e-05 | 0.0092 |
21.1303 | 28 | 1.6124e-04 | 0.0020 | ||
6.1338e-05 | 0.0020 | ||||
5.9914e-07 | 0.0016 | ||||
30.1656 | 72 | 1.1833e-05 | 4.7710e-04 | ||
5.1514e-06 | 4.7858e-04 | ||||
4.4558e-08 | 4.2316e-04 |
21.1303 | 28 | 1.6124e-04 | 0.0020 | ||
6.1338e-05 | 0.0020 | ||||
5.9914e-07 | 0.0016 | ||||
30.1656 | 72 | 1.1833e-05 | 4.7710e-04 | ||
5.1514e-06 | 4.7858e-04 | ||||
4.4558e-08 | 4.2316e-04 |
22.6527 | 22 | 0.1143e-04 | 0.5917e-03 | ||
0.1707e-04 | 0.5955e-03 | ||||
0.0010e-04 | 0.4901e-03 |
22.6527 | 22 | 0.1143e-04 | 0.5917e-03 | ||
0.1707e-04 | 0.5955e-03 | ||||
0.0010e-04 | 0.4901e-03 |
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