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Equilibrium validation in models for pattern formation based on Sobolev embeddings. (English) Zbl 1465.35050
Summary: 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.
35B36 Pattern formations in context of PDEs
35B32 Bifurcations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35B41 Attractors
35K35 Initial-boundary value problems for higher-order parabolic equations
35K58 Semilinear parabolic equations
37M20 Computational methods for bifurcation problems in dynamical systems
65G20 Algorithms with automatic result verification
65G30 Interval and finite arithmetic
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
Full Text: DOI
[1] G. Arioli; 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, 1033-1051 (2010) · Zbl 1231.35016
[2] S. Cai and Y. Watanabe, A computer-assisted method for the diblock copolymer model, Zeitschrift für Angewandte Mathematik und Mechanik, 99 (2019), e201800125, 14pp.
[3] L. Chierchia, KAM lectures, in Dynamical Systems. Part I, Scuola Normale Superiore, Pisa, Italy, 2003, 1-55.
[4] R. Choksi; M. Maras; J. F. Williams, 2D phase diagram for minimizers of a Cahn-Hilliard functional with long-range interactions, SIAM Journal on Applied Dynamical Systems, 10, 1344-1362 (2011) · Zbl 1236.49074
[5] R. Choksi; M. A. Peletier; 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, 1712-1738 (2009) · Zbl 1400.74089
[6] R. Choksi; X. Ren, On the derivation of a density functional theory for microphase separation of diblock copolymers, Journal of Statistical Physics, 113, 151-176 (2003) · Zbl 1034.82037
[7] R. Choksi; X. Ren, Diblock copolymer/homopolymer blends: Derivation of a density functional theory, Physica D, 203, 100-119 (2005) · Zbl 1120.82307
[8] J. Cyranka; T. Wanner, Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model, SIAM Journal on Applied Dynamical Systems, 17, 694-731 (2018) · Zbl 1415.35057
[9] S. Day; J.-P. Lessard; K. Mischaikow, Validated continuation for equilibria of PDEs, SIAM Journal on Numerical Analysis, 45, 1398-1424 (2007) · Zbl 1151.65074
[10] J. P. Desi; H. Edrees; J. Price; E. Sander; T. Wanner, The dynamics of nucleation in stochastic Cahn-Morral systems, SIAM Journal on Applied Dynamical Systems, 10, 707-743 (2011) · Zbl 1367.74041
[11] A. Dhooge; W. Govaerts; Y. A. Kuznetsov, MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs, Association for Computing Machinery. Transactions on Mathematical Software, 29, 141-164 (2003) · Zbl 1070.65574
[12] 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.
[13] M. Gameiro; J.-P. Lessard; K. Mischaikow, Validated continuation over large parameter ranges for equilibria of PDEs, Mathematics and Computers in Simulation, 79, 1368-1382 (2008) · Zbl 1166.65379
[14] Z. G. Huseynov; A. M. Shykhammedov, On bases of sines and cosines in Sobolev spaces, Applied Mathematics Letters, 25, 275-278 (2012) · Zbl 1248.46027
[15] I. Johnson; E. Sander; T. Wanner, Branch interactions and long-term dynamics for the diblock copolymer model in one dimension, Discrete and Continuous Dynamical Systems. Series A, 33, 3671-3705 (2013) · Zbl 1280.35012
[16] T. Kinoshita; Y. Watanabe; M. T. Nakao, An alternative approach to norm bound computation for inverses of linear operators in Hilbert spaces, Journal of Differential Equations, 266, 5431-5447 (2019) · Zbl 07041840
[17] J.-P. Lessard; E. Sander; T. Wanner, Rigorous continuation of bifurcation points in the diblock copolymer equation, Journal of Computational Dynamics, 4, 71-118 (2017) · Zbl 1404.35030
[18] S. Maier-Paape; U. Miller; K. Mischaikow; T. Wanner, Rigorous numerics for the Cahn-Hilliard equation on the unit square, Revista Matematica Complutense, 21, 351-426 (2008) · Zbl 1160.37418
[19] S. Maier-Paape; K. Mischaikow; T. Wanner, Structure of the attractor of the Cahn-Hilliard equation on a square, International Journal of Bifurcation and Chaos, 17, 1221-1263 (2007) · Zbl 1148.35008
[20] S. Maier-Paape; T. Wanner, Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: Nonlinear dynamics, Archive for Rational Mechanics and Analysis, 151, 187-219 (2000) · Zbl 0954.35089
[21] T. R. Muradov; V. F. Salmanov, On the basis property of trigonometric systems with linear phase in a weighted Sobolev space, Dokl. Math., 90, 611-612 (2014) · Zbl 1307.42001
[22] M. T. Nakao, M. Plum and Y. Watanabe, Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations, Springer-Verlag, Berlin, 2019. · Zbl 1462.65004
[23] T. Ohta; K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19, 2621-2632 (1986)
[24] M. Plum, Existence and enclosure results for continua of solutions of parameter-dependent nonlinear boundary value problems, Journal of Computational and Applied Mathematics, 60, 187-200 (1995) · Zbl 0834.65119
[25] 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. · Zbl 0849.65060
[26] M. Plum, Computer-assisted proofs for semilinear elliptic boundary value problems, Japan Journal of Industrial and Applied Mathematics, 26, 419-442 (2009) · Zbl 1186.35073
[27] 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/. · Zbl 0949.65046
[28] S. M. Rump, Verification methods: Rigorous results using floating-point arithmetic, Acta Numerica, 19, 287-449 (2010) · Zbl 1323.65046
[29] E. Sander; T. Wanner, Validated saddle-node bifurcations and applications to lattice dynamical systems, SIAM Journal on Applied Dynamical Systems, 15, 1690-1733 (2016) · Zbl 1414.35115
[30] J. B. van den Berg; J. F. Williams, Validation of the bifurcation diagram in the 2D Ohta-Kawasaki problem, Nonlinearity, 30, 1584-1638 (2017) · Zbl 1366.65068
[31] 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.
[32] J. B. van den Berg; J. F. Williams, Rigorously computing symmetric stationary states of the Ohta-Kawasaki problem in three dimensions, SIAM Journal on Mathematical Analysis, 51, 131-158 (2019) · Zbl 1409.35080
[33] 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.
[34] T. Wanner, Computer-assisted equilibrium validation for the diblock copolymer model, Discrete and Continuous Dynamical Systems, Series A, 37, 1075-1107 (2017) · Zbl 1364.35116
[35] T. Wanner, Computer-assisted bifurcation diagram validation and applications in materials science, Proceedings of Symposia in Applied Mathematics, 74, 123-174 (2018) · Zbl 1406.35040
[36] T. Wanner, Validated bounds on embedding constants for Sobolev space Banach algebras, Mathematical Methods in the Applied Sciences, 41, 9361-9376 (2018) · Zbl 1414.46027
[37] Y. Watanabe; T. Kinoshita; 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, 407-420 (2019) · Zbl 07094577
[38] Y. Watanabe; K. Nagatou; M. Plum; M. T. Nakao, Norm bound computation for inverses of linear operators in Hilbert spaces, Journal of Differential Equations, 260, 6363-6374 (2016) · Zbl 1398.65094
[39] 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, 2004-2013 (1998) · Zbl 0972.65084
[40] N. Yamamoto; M. T. Nakao; Y. Watanabe, A theorem for numerical verification on local uniqueness of solutions to fixed-point equations, Numerical Functional Analysis and Optimization, 32, 1190-1204 (2011) · Zbl 1232.47059
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