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February  2017, 37(2): 1075-1107. doi: 10.3934/dcds.2017045

Computer-assisted equilibrium validation for the diblock copolymer model

Thomas Wanner

Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA

Received  March 2015 Revised  August 2015 Published  November 2016

The diblock copolymer model is a fourth-order parabolic partial differential equation which models phase separation with fine structure. The equation is a gradient flow with respect to an extension of the standard van der Waals free energy functional which involves nonlocal interactions. Thus, the long-term dynamical behavior of the diblock copolymer model is described by its finite-dimensional attractor. However, even on one-dimensional domains, the full structure of the underlying equilibrium set is not fully understood. In this paper, we develop a rigorous computational approach for establishing the existence of equilibrium solutions of the diblock copolymer model. We consider both individual solutions, as well as pieces of solution branches in a parameter-dependent situation. The results are presented for the case of one-dimensional domains, and can easily be implemented using standard interval arithmetic packages.

Keywords: Diblock copolymers, phase field model, modified Cahn-Hilliard equation, numerical continuation, bifurcation, computer-assisted proof.
Mathematics Subject Classification: Primary:35B40, 35B41;Secondary:35K55, 60F10, 60H15, 74N9.
Citation: Thomas Wanner. Computer-assisted equilibrium validation for the diblock copolymer model. Discrete & Continuous Dynamical Systems, 2017, 37 (2) : 1075-1107. doi: 10.3934/dcds.2017045
References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, San Diego -London, 1975.   Google Scholar
[2]

M. Bahiana and Y. Oono, Cell dynamical system approach to block copolymers, Physical Review A, 41 (1990), 6763-6771.  doi: 10.1103/PhysRevA.41.6763.  Google Scholar

[3]

P. W. Bates and P. C. Fife, Spectral comparison principles for the Cahn-Hilliard and phase-field equations, and time scales for coarsening, Physica D, 43 (1990), 335-348.  doi: 10.1016/0167-2789(90)90141-B.  Google Scholar

[4]

P. W. Bates and P. C. Fife, The dynamics of nucleation for the Cahn-Hilliard equation, SIAM Journal on Applied Mathematics, 53 (1993), 990-1008.  doi: 10.1137/0153049.  Google Scholar

[5]

D. Blömker, B. Gawron and T. Wanner, Nucleation in the one-dimensional stochastic Cahn-Hilliard model, Discrete and Continuous Dynamical Systems, Series A, 27 (2010), 25-52.  doi: 10.3934/dcds.2010.27.25.  Google Scholar

[6]

C.-K. Chen and P. C. Fife, Nonlocal models of phase transitions in solids, Advances in Mathematical Sciences and Applications, 10 (2000), 821-849.   Google Scholar

[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

E. B. Davies, Linear Operators and Their Spectra, vol. 106 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2007. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

E. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Proceedings of the Tenth Manitoba Conference on Numerical Mathematics and Computing, Vol. Ⅰ (Winnipeg, Man., 1980), 30 (1981), 265-284.   Google Scholar

[13]

P. C. Fife and M. Kowalczyk, A class of pattern-forming models, Journal of Nonlinear Science, 9 (1999), 641-669.  doi: 10.1007/s003329900081.  Google Scholar

[14]

P. C. Fife, Models for phase separation and their mathematics, Electronic Journal of Differential Equations, 2000 (2000), 1-26.   Google Scholar

[15]

P. C. Fife, Pattern formation in gradient systems, in Handbook of dynamical systems, North-Holland, Amsterdam, 2 (2002), 677-722. Google Scholar

[16]

P. C. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in Trends in Nonlinear Analysis, Springer, Berlin, 2003,153-191. Google Scholar

[17]

P. C. Fife and D. Hilhorst, The Nishiura-Ohnishi free boundary problem in the 1D case, SIAM Journal on Mathematical Analysis, 33 (2001), 589-606.  doi: 10.1137/S0036141000372507.  Google Scholar

[18]

P. C. Fife, H. Kielhöfer, S. Maier-Paape and T. Wanner, Perturbation of doubly periodic solution branches with applications to the Cahn-Hilliard equation, Physica D, 100 (1997), 257-278.  doi: 10.1016/S0167-2789(96)00190-X.  Google Scholar

[19]

M. Gameiro and J.-P. Lessard, Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs, Journal of Differential Equations, 249 (2010), 2237-2268.  doi: 10.1016/j.jde.2010.07.002.  Google Scholar

[20]

M. Gameiro and J.-P. Lessard, Efficient rigorous numerics for higher-dimensional PDEs via one-dimensional estimates, SIAM Journal on Numerical Analysis, 51 (2013), 2063-2087.  doi: 10.1137/110836651.  Google Scholar

[21]

M. Grinfeld and A. Novick-Cohen, Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments, Proceedings of the Royal Society of Edinburgh, 125 (1995), 351-370.  doi: 10.1017/S0308210500028079.  Google Scholar

[22]

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.  Google Scholar

[23]

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.  Google Scholar

[24]

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.  Google Scholar

[25]

S. Maier-Paape and T. Wanner, Solutions of nonlinear planar elliptic problems with triangle symmetry, Journal of Differential Equations, 136 (1997), 1-34.  doi: 10.1006/jdeq.1996.3240.  Google Scholar

[26]

S. Maier-Paape and T. Wanner, Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions. Part Ⅰ: Probability and wavelength estimate, Communications in Mathematical Physics, 195 (1998), 435-464.  doi: 10.1007/s002200050397.  Google Scholar

[27]

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.  Google Scholar

[28]

J. T. Marti, Evaluation of the least constant in Sobolev's inequality for $H^{1}(0,\,s)$, SIAM Journal on Numerical Analysis, 20 (1983), 1239-1242.  doi: 10.1137/0720094.  Google Scholar

[29]

R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, SIAM, Philadelphia, 2009. Google Scholar

[30]

K. Nagatou, Validated computation for infinite dimensional eigenvalue problems, in 12th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics, IEEE Computer Society, 2007, 3-13. Google Scholar

[31]

K. Nagatou, N. Yamamoto and M. T. Nakao, An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness, Numerical Functional Analysis and Optimization, 20 (1999), 543-565.  doi: 10.1080/01630569908816910.  Google Scholar

[32]

Y. Nishiura, Far-from-Equilibrium Dynamics, vol. 209 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 2002. Google Scholar

[33]

Y. Nishiura and I. Ohnishi, Some mathematical aspects of the micro-phase separation in diblock copolymers, Physica D, 84 (1995), 31-39.  doi: 10.1016/0167-2789(95)00005-O.  Google Scholar

[34]

T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632.  doi: 10.1021/ma00164a028.  Google Scholar

[35]

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.  Google Scholar

[36]

M. Plum, Existence and multiplicity proofs for semilinear elliptic boundary value problems by computer assistance, Jahresbericht der Deutschen Mathematiker-Vereinigung, 110 (2008), 19-54.   Google Scholar

[37]

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.  Google Scholar

[38]

W. Richardson, Steepest descent and the least $C$ for Sobolev's inequality, Bulletin of the London Mathematical Society, 18 (1986), 478-484.  doi: 10.1112/blms/18.5.478.  Google Scholar

[39]

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

[40]

S. M. Rump, Verification methods: Rigorous results using floating-point arithmetic, Acta Numerica, 19 (2010), 287-449.  doi: 10.1017/S096249291000005X.  Google Scholar

[41]

S. M. Rump, Verified bounds for singular values, in particular for the spectral norm of a matrix and its inverse, BIT. Numerical Mathematics, 51 (2011), 367-384.  doi: 10.1007/s10543-010-0294-0.  Google Scholar

[42]

E. Sander and T. Wanner, Monte Carlo simulations for spinodal decomposition, Journal of Statistical Physics, 95 (1999), 925-948.  doi: 10.1023/A:1004550416829.  Google Scholar

[43]

E. Sander and T. Wanner, Unexpectedly linear behavior for the Cahn-Hilliard equation, SIAM Journal on Applied Mathematics, 60 (2000), 2182-2202.  doi: 10.1137/S0036139999352225.  Google Scholar

[44]

T. Stephens and T. Wanner, Rigorous validation of isolating blocks for flows and their Conley indices, SIAM Journal on Applied Dynamical Systems, 13 (2014), 1847-1878.  doi: 10.1137/140971075.  Google Scholar

[45]

T. Wanner, Maximum norms of random sums and transient pattern formation, Transactions of the American Mathematical Society, 356 (2004), 2251-2279.  doi: 10.1090/S0002-9947-03-03480-9.  Google Scholar

[46]

K. Watanabe, Y. Kametaka, A. Nagai, K. Takemura and H. Yamagishi, The best constant of Sobolev inequality on a bounded interval, Journal of Mathematical Analysis and Applications, 340 (2008), 699-706.  doi: 10.1016/j.jmaa.2007.08.054.  Google Scholar

[47]

Y. Watanabe, K. Nagatou, M. Plum and M. T. Nakao, Verified computations of eigenvalue exclosures for eigenvalue problems in Hilbert spaces, SIAM Journal on Numerical Analysis, 52 (2014), 975-992.  doi: 10.1137/120894683.  Google Scholar

[48] E. Zeidler, Nonlinear Functional Analysis and its Applications. Ⅰ: Fixed-Point Theorems, Springer-Verlag, New York -Berlin -Heidelberg, 1986.  doi: 10.1007/978-1-4612-4838-5.  Google Scholar
[49]

P. Zgliczyński and K. Mischaikow, Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation, Foundations of Computational Mathematics, 1 (2001), 255-288.  doi: 10.1007/s002080010010.  Google Scholar

show all references

References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, San Diego -London, 1975.   Google Scholar
[2]

M. Bahiana and Y. Oono, Cell dynamical system approach to block copolymers, Physical Review A, 41 (1990), 6763-6771.  doi: 10.1103/PhysRevA.41.6763.  Google Scholar

[3]

P. W. Bates and P. C. Fife, Spectral comparison principles for the Cahn-Hilliard and phase-field equations, and time scales for coarsening, Physica D, 43 (1990), 335-348.  doi: 10.1016/0167-2789(90)90141-B.  Google Scholar

[4]

P. W. Bates and P. C. Fife, The dynamics of nucleation for the Cahn-Hilliard equation, SIAM Journal on Applied Mathematics, 53 (1993), 990-1008.  doi: 10.1137/0153049.  Google Scholar

[5]

D. Blömker, B. Gawron and T. Wanner, Nucleation in the one-dimensional stochastic Cahn-Hilliard model, Discrete and Continuous Dynamical Systems, Series A, 27 (2010), 25-52.  doi: 10.3934/dcds.2010.27.25.  Google Scholar

[6]

C.-K. Chen and P. C. Fife, Nonlocal models of phase transitions in solids, Advances in Mathematical Sciences and Applications, 10 (2000), 821-849.   Google Scholar

[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

E. B. Davies, Linear Operators and Their Spectra, vol. 106 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2007. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

E. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Proceedings of the Tenth Manitoba Conference on Numerical Mathematics and Computing, Vol. Ⅰ (Winnipeg, Man., 1980), 30 (1981), 265-284.   Google Scholar

[13]

P. C. Fife and M. Kowalczyk, A class of pattern-forming models, Journal of Nonlinear Science, 9 (1999), 641-669.  doi: 10.1007/s003329900081.  Google Scholar

[14]

P. C. Fife, Models for phase separation and their mathematics, Electronic Journal of Differential Equations, 2000 (2000), 1-26.   Google Scholar

[15]

P. C. Fife, Pattern formation in gradient systems, in Handbook of dynamical systems, North-Holland, Amsterdam, 2 (2002), 677-722. Google Scholar

[16]

P. C. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in Trends in Nonlinear Analysis, Springer, Berlin, 2003,153-191. Google Scholar

[17]

P. C. Fife and D. Hilhorst, The Nishiura-Ohnishi free boundary problem in the 1D case, SIAM Journal on Mathematical Analysis, 33 (2001), 589-606.  doi: 10.1137/S0036141000372507.  Google Scholar

[18]

P. C. Fife, H. Kielhöfer, S. Maier-Paape and T. Wanner, Perturbation of doubly periodic solution branches with applications to the Cahn-Hilliard equation, Physica D, 100 (1997), 257-278.  doi: 10.1016/S0167-2789(96)00190-X.  Google Scholar

[19]

M. Gameiro and J.-P. Lessard, Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs, Journal of Differential Equations, 249 (2010), 2237-2268.  doi: 10.1016/j.jde.2010.07.002.  Google Scholar

[20]

M. Gameiro and J.-P. Lessard, Efficient rigorous numerics for higher-dimensional PDEs via one-dimensional estimates, SIAM Journal on Numerical Analysis, 51 (2013), 2063-2087.  doi: 10.1137/110836651.  Google Scholar

[21]

M. Grinfeld and A. Novick-Cohen, Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments, Proceedings of the Royal Society of Edinburgh, 125 (1995), 351-370.  doi: 10.1017/S0308210500028079.  Google Scholar

[22]

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.  Google Scholar

[23]

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.  Google Scholar

[24]

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.  Google Scholar

[25]

S. Maier-Paape and T. Wanner, Solutions of nonlinear planar elliptic problems with triangle symmetry, Journal of Differential Equations, 136 (1997), 1-34.  doi: 10.1006/jdeq.1996.3240.  Google Scholar

[26]

S. Maier-Paape and T. Wanner, Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions. Part Ⅰ: Probability and wavelength estimate, Communications in Mathematical Physics, 195 (1998), 435-464.  doi: 10.1007/s002200050397.  Google Scholar

[27]

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.  Google Scholar

[28]

J. T. Marti, Evaluation of the least constant in Sobolev's inequality for $H^{1}(0,\,s)$, SIAM Journal on Numerical Analysis, 20 (1983), 1239-1242.  doi: 10.1137/0720094.  Google Scholar

[29]

R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, SIAM, Philadelphia, 2009. Google Scholar

[30]

K. Nagatou, Validated computation for infinite dimensional eigenvalue problems, in 12th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics, IEEE Computer Society, 2007, 3-13. Google Scholar

[31]

K. Nagatou, N. Yamamoto and M. T. Nakao, An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness, Numerical Functional Analysis and Optimization, 20 (1999), 543-565.  doi: 10.1080/01630569908816910.  Google Scholar

[32]

Y. Nishiura, Far-from-Equilibrium Dynamics, vol. 209 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 2002. Google Scholar

[33]

Y. Nishiura and I. Ohnishi, Some mathematical aspects of the micro-phase separation in diblock copolymers, Physica D, 84 (1995), 31-39.  doi: 10.1016/0167-2789(95)00005-O.  Google Scholar

[34]

T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632.  doi: 10.1021/ma00164a028.  Google Scholar

[35]

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.  Google Scholar

[36]

M. Plum, Existence and multiplicity proofs for semilinear elliptic boundary value problems by computer assistance, Jahresbericht der Deutschen Mathematiker-Vereinigung, 110 (2008), 19-54.   Google Scholar

[37]

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.  Google Scholar

[38]

W. Richardson, Steepest descent and the least $C$ for Sobolev's inequality, Bulletin of the London Mathematical Society, 18 (1986), 478-484.  doi: 10.1112/blms/18.5.478.  Google Scholar

[39]

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

[40]

S. M. Rump, Verification methods: Rigorous results using floating-point arithmetic, Acta Numerica, 19 (2010), 287-449.  doi: 10.1017/S096249291000005X.  Google Scholar

[41]

S. M. Rump, Verified bounds for singular values, in particular for the spectral norm of a matrix and its inverse, BIT. Numerical Mathematics, 51 (2011), 367-384.  doi: 10.1007/s10543-010-0294-0.  Google Scholar

[42]

E. Sander and T. Wanner, Monte Carlo simulations for spinodal decomposition, Journal of Statistical Physics, 95 (1999), 925-948.  doi: 10.1023/A:1004550416829.  Google Scholar

[43]

E. Sander and T. Wanner, Unexpectedly linear behavior for the Cahn-Hilliard equation, SIAM Journal on Applied Mathematics, 60 (2000), 2182-2202.  doi: 10.1137/S0036139999352225.  Google Scholar

[44]

T. Stephens and T. Wanner, Rigorous validation of isolating blocks for flows and their Conley indices, SIAM Journal on Applied Dynamical Systems, 13 (2014), 1847-1878.  doi: 10.1137/140971075.  Google Scholar

[45]

T. Wanner, Maximum norms of random sums and transient pattern formation, Transactions of the American Mathematical Society, 356 (2004), 2251-2279.  doi: 10.1090/S0002-9947-03-03480-9.  Google Scholar

[46]

K. Watanabe, Y. Kametaka, A. Nagai, K. Takemura and H. Yamagishi, The best constant of Sobolev inequality on a bounded interval, Journal of Mathematical Analysis and Applications, 340 (2008), 699-706.  doi: 10.1016/j.jmaa.2007.08.054.  Google Scholar

[47]

Y. Watanabe, K. Nagatou, M. Plum and M. T. Nakao, Verified computations of eigenvalue exclosures for eigenvalue problems in Hilbert spaces, SIAM Journal on Numerical Analysis, 52 (2014), 975-992.  doi: 10.1137/120894683.  Google Scholar

[48] E. Zeidler, Nonlinear Functional Analysis and its Applications. Ⅰ: Fixed-Point Theorems, Springer-Verlag, New York -Berlin -Heidelberg, 1986.  doi: 10.1007/978-1-4612-4838-5.  Google Scholar
[49]

P. Zgliczyński and K. Mischaikow, Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation, Foundations of Computational Mathematics, 1 (2001), 255-288.  doi: 10.1007/s002080010010.  Google Scholar

Figure 1.  Numerically computed bifurcation diagrams for the diblock copolymer model (2) on the domain $\Omega = (0,1)$ and with $\sigma = 6$. The left diagram is for total mass $\mu = 0$, while the right diagram is for $\mu = 0.3$. The solution measure for both diagrams is the $L^2(0,1)$-norm of the solutions. The solution branches are color-coded by the numerically determined Morse index of the solutions, and the colors black, red, blue, green, magenta, and cyan correspond to indices $0$, $1$, $2$, $3$, $4$, and $5$, respectively. The bifurcation parameter is $\lambda$.
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Figure 2.  Numerically computed bifurcation diagrams for the diblock copolymer model (2) on the domain $\Omega = (0,1)$ and with $\sigma = 6$. The left diagram is for total mass $\mu = 0$, while the right diagram is for $\mu = 0.3$. The solution measure for both diagrams is the energy of the solutions as defined in (1. The solution branches are color-coded by the numerically determined Morse index of the solutions, and the colors black, red, blue, green, magenta, and cyan correspond to indices $0$, $1$, $2$, $3$, $4$, and $5$, respectively. The bifurcation parameter is $\lambda$.
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Figure 1, i.e., black curves are stable solutions, red curves have index $1$, and blue functions have index $2$.">Figure 3.  Equilibrium solutions for the diblock copolymer model on the domain $\Omega = (0,1)$ for mass $\mu = 0$ and $\sigma = 6$. The left diagram shows solutions for $\lambda = 100$, while the right diagram is for $\lambda = 150$. In each case, the color of the solution curve indicates its stability as in Figure 1, i.e., black curves are stable solutions, red curves have index $1$, and blue functions have index $2$.
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Table 1 below, while the two solutions on the right are validated in Table 2. In each case, the color of the solution curve indicates its stability as in Figure 1, i.e., black curves are stable solutions, red curves have index $1$, blue functions have index $2$, and the green solution has index $3$.">Figure 4.  Equilibrium solutions for the diblock copolymer model on the domain $\Omega = (0,1)$ for mass $\mu = 0$ and $\sigma = 6$. Both diagrams shows solutions for $\lambda = 200$. The ones on the left correspond to the solutions which are validated in Table 1 below, while the two solutions on the right are validated in Table 2. In each case, the color of the solution curve indicates its stability as in Figure 1, i.e., black curves are stable solutions, red curves have index $1$, blue functions have index $2$, and the green solution has index $3$.
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Figure 1, i.e., black curves are stable solutions, red curves have index $1$, and blue functions have index $2$.">Figure 5.  Equilibrium solutions for the diblock copolymer model on the domain $\Omega = (0,1)$ for mass $\mu = 0.3$ and $\sigma = 6$. The left diagram shows solutions for $\lambda = 100$, while the right diagram is for $\lambda = 200$. In each case, the color of the solution curve indicates its stability as in Figure 1, i.e., black curves are stable solutions, red curves have index $1$, and blue functions have index $2$.
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Figure 6.  Choice of the constant $\delta_\lambda$ in Theorem 2.2. In both diagrams, the blue lines indicate the identities $\delta_\lambda = (\delta_u - \delta_1) / (2 M_3 K)$ and $\delta_\lambda = M_1 (\delta_3 - \delta_u) / M_2$, which form the upper boundary of the region of admissible pairs $(\delta_u, \delta_\lambda)$. The left diagram depicts the normal situation, in which $\delta_3<d_1$. In the right image we illustrate the case $\delta_1<d_1<\delta_3$, in which the admissible region is reduced due to hypothesis (H3). For both diagrams, we assume that $\delta_{\lambda,opt}<d_2 \le d_3$, which is usually satisfied.
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Table 1.  Rigorous numerical results for the diblock copolymer model with $\mu = 0$. For the $\lambda$-values $100$, $150$, and $200$ we consider representative solutions from most of the branches shown in the left diagram of Figure 1, numbered from top to bottom, i.e., according to decreasing $L^2$-norm. The table lists the index of each solution, the value $\varrho $ of the residual in (H1), as well as the values $\delta_1$ and $\delta_2$ from Theorem 2.1. The last column indicates which value of $N$ is chosen in Theorem 2.3. All computations use Proposition 3.5 with $K = 100$.
Equilibrium Validation for $\mu = 0$ with $K = 100$
$\lambda$ Number Index $\varrho $ $\delta_1$ $\delta_2$ $N$
100 #1 0 2.020897e-14 2.020897e-12 7.512488e-06 190
#2 0 2.222804e-14 2.222804e-12 1.547854e-05 106
#3 1 4.444820e-14 4.444820e-12 1.394691e-05 112
#4 2 3.276268e-15 3.276268e-13 1.625587e-05 59
150 #1 0 3.632609e-14 3.632611e-12 4.268987e-06 332
#2 1 4.705001e-14 4.705003e-12 8.092680e-06 584
#3 1 1.814983e-14 1.814984e-12 3.929213e-06 508
#4 2 2.672918e-14 2.672918e-12 5.189605e-06 232
#5 2 2.310413e-14 2.310413e-12 7.824295e-06 140
200 #1 0 4.799846e-14 4.799850e-12 2.932214e-06 479
#4 1 6.153889e-14 6.153893e-12 5.080967e-06 314
#5 2 6.674118e-14 6.674124e-12 3.937197e-06 301
#6 2 3.750452e-14 3.750453e-12 6.304030e-06 142
#7 3 7.652149e-15 7.652150e-13 3.130001e-06 676
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Equilibrium Validation for $\mu = 0$ with $K = 100$
$\lambda$ Number Index $\varrho $ $\delta_1$ $\delta_2$ $N$
100 #1 0 2.020897e-14 2.020897e-12 7.512488e-06 190
#2 0 2.222804e-14 2.222804e-12 1.547854e-05 106
#3 1 4.444820e-14 4.444820e-12 1.394691e-05 112
#4 2 3.276268e-15 3.276268e-13 1.625587e-05 59
150 #1 0 3.632609e-14 3.632611e-12 4.268987e-06 332
#2 1 4.705001e-14 4.705003e-12 8.092680e-06 584
#3 1 1.814983e-14 1.814984e-12 3.929213e-06 508
#4 2 2.672918e-14 2.672918e-12 5.189605e-06 232
#5 2 2.310413e-14 2.310413e-12 7.824295e-06 140
200 #1 0 4.799846e-14 4.799850e-12 2.932214e-06 479
#4 1 6.153889e-14 6.153893e-12 5.080967e-06 314
#5 2 6.674118e-14 6.674124e-12 3.937197e-06 301
#6 2 3.750452e-14 3.750453e-12 6.304030e-06 142
#7 3 7.652149e-15 7.652150e-13 3.130001e-06 676
Table 2.  Rigorous numerical results for the diblock copolymer model with $\mu = 0$. For $\lambda = 200$ we consider representative solutions from the second and third branches shown in the left diagram of Figure 1, numbered from top to bottom. The table lists the index of each solution, the value $\varrho $ in (H1), as well as $\delta_1$ and $\delta_2$ from Theorem 2.1. The last column indicates which value of $N$ is chosen in Theorem 2.3. The computations use $K = 500$.
Equilibrium Validation for $\mu = 0$ with $K = 500$
$\lambda$ Number Index $\varrho $ $\delta_1$ $\delta_2$ $N$
200 #2 0 2.642564e-14 1.321300e-11 4.858832e-07 1828
#3 1 6.084200e-14 3.042193e-11 4.959956e-07 1108
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Equilibrium Validation for $\mu = 0$ with $K = 500$
$\lambda$ Number Index $\varrho $ $\delta_1$ $\delta_2$ $N$
200 #2 0 2.642564e-14 1.321300e-11 4.858832e-07 1828
#3 1 6.084200e-14 3.042193e-11 4.959956e-07 1108
Table 3.  Rigorous numerical results for the diblock copolymer model with $\mu = 0.3$. For the $\lambda$-values $100$ and $200$ we consider representative solutions from each of the branches shown in the right diagram of Figure 1, numbered from top to bottom, i.e., according to decreasing $L^2$-norm. The table lists the index of each solution, the value $\varrho $ of the residual in (H1), as well as the values $\delta_1$ and $\delta_2$ from Theorem 2.1. The last column indicates which value of $N$ is chosen in Theorem 2.3. All computations use Proposition 3.5 with $K = 100$.
Equilibrium Validation for $\mu = 0.3$ with $K = 100$
$\lambda$ Number Index $\varrho $ $\delta_1$ $\delta_2$ $N$
100 #1 0 5.728511e-14 5.728512e-12 1.047050e-05 106
#2 0 2.129870e-14 2.129870e-12 8.569644e-06 175
#3 1 2.129870e-14 2.129870e-12 8.569644e-06 811
#4 1 6.466218e-15 6.466218e-13 4.747448e-05 64
200 #1 0 8.153471e-14 8.153483e-12 2.813889e-06 644
#2 1 7.326258e-14 7.326268e-12 2.834757e-06 552
#3 1 1.075251e-13 1.075252e-11 4.071324e-06 180
#4 2 8.642886e-14 8.642896e-12 3.858723e-06 271
#5 2 2.696057e-14 2.696057e-12 1.384417e-05 113
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Equilibrium Validation for $\mu = 0.3$ with $K = 100$
$\lambda$ Number Index $\varrho $ $\delta_1$ $\delta_2$ $N$
100 #1 0 5.728511e-14 5.728512e-12 1.047050e-05 106
#2 0 2.129870e-14 2.129870e-12 8.569644e-06 175
#3 1 2.129870e-14 2.129870e-12 8.569644e-06 811
#4 1 6.466218e-15 6.466218e-13 4.747448e-05 64
200 #1 0 8.153471e-14 8.153483e-12 2.813889e-06 644
#2 1 7.326258e-14 7.326268e-12 2.834757e-06 552
#3 1 1.075251e-13 1.075252e-11 4.071324e-06 180
#4 2 8.642886e-14 8.642896e-12 3.858723e-06 271
#5 2 2.696057e-14 2.696057e-12 1.384417e-05 113
Table 4.  Rigorous numerical results for the diblock copolymer model with $\mu = 0$. For the $\lambda$-values $100$ and $150$ we consider representative solutions from each of the branches shown in the left diagram of Figure 1, numbered from top to bottom, i.e., according to decreasing $L^2$-norm. The table lists the values of $\delta_1$, $\delta_2$, $\delta_3$, and $\delta_{\lambda,opt}$ from Theorem 2.2. The last two columns indicate a numerical estimate $K_{est}$ for the optimal constant $K$ in (H2), as well as the value of $K$ used for the validation computation.
Branch Validation for $\mu = 0.0$ with estimated $K$
$\lambda$ No. $\delta_1$ $\delta_2$ $\delta_3$ $\delta_{\lambda,opt}$ $K_{est}$ $K$
100 #1 1.0446e-13 1.1065e-04 1.1079e-04 2.6068e-04 2.2 3.3
#2 5.1001e-13 8.9555e-05 8.9572e-05 2.6200e-05 5.7 8.6
#3 3.0910e-13 1.7520e-04 1.7527e-04 1.2525e-04 2.6 3.9
#4 8.1585e-14 5.3658e-05 5.3726e-05 9.5155e-05 10.0 15.1
150 #1 2.1895e-13 5.7574e-05 5.7619e-05 1.3199e-04 2.4 3.7
#2 5.7821e-12 8.6076e-06 8.6078e-06 4.4425e-07 31.3 46.9
#3 3.3317e-13 1.7562e-05 1.7569e-05 2.1107e-05 7.4 11.1
#4 1.5386e-13 9.3999e-05 9.4056e-05 1.5404e-04 1.8 2.7
#5 1.3800e-13 1.3328e-04 1.3336e-04 1.8173e-04 1.9 2.9
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Branch Validation for $\mu = 0.0$ with estimated $K$
$\lambda$ No. $\delta_1$ $\delta_2$ $\delta_3$ $\delta_{\lambda,opt}$ $K_{est}$ $K$
100 #1 1.0446e-13 1.1065e-04 1.1079e-04 2.6068e-04 2.2 3.3
#2 5.1001e-13 8.9555e-05 8.9572e-05 2.6200e-05 5.7 8.6
#3 3.0910e-13 1.7520e-04 1.7527e-04 1.2525e-04 2.6 3.9
#4 8.1585e-14 5.3658e-05 5.3726e-05 9.5155e-05 10.0 15.1
150 #1 2.1895e-13 5.7574e-05 5.7619e-05 1.3199e-04 2.4 3.7
#2 5.7821e-12 8.6076e-06 8.6078e-06 4.4425e-07 31.3 46.9
#3 3.3317e-13 1.7562e-05 1.7569e-05 2.1107e-05 7.4 11.1
#4 1.5386e-13 9.3999e-05 9.4056e-05 1.5404e-04 1.8 2.7
#5 1.3800e-13 1.3328e-04 1.3336e-04 1.8173e-04 1.9 2.9
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