# American Institute of Mathematical Sciences

## Equilibrium validation in models for pattern formation based on Sobolev embeddings

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

* Corresponding author: Thomas Wanner

Received  March 2020 Revised  August 2020 Published  August 2020

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.

Citation: Evelyn Sander, Thomas Wanner. Equilibrium validation in models for pattern formation based on Sobolev embeddings. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020260
##### References:

show all references

##### References:
Ten sample validated one-dimensional equilibrium solutions. For all solutions we choose $\lambda = 150$ and $\sigma = 6$. Three of the solutions have total mass $\mu = 0$, three are for mass $\mu = 0.1$, three for $\mu = 0.3$, and finally one for $\mu = 0.5$
There is a tradeoff between high-dimensional calculations and optimal results. The top left figure shows how the bound of $K$ varies with the dimension of the truncated approximation matrix used to calculate $K_N$. These calculations are for dimension one, but a similar effect occurs in higher dimensions as well. The top right figure shows the corresponding estimate for $\delta_x$, and the bottom panel shows the estimate for $\delta_\alpha$, where $\alpha$ is each of the three parameters. The size of the validated interval grows larger as the truncation dimension grows, but with diminishing returns on the computational investment
Six of the seventeen validated two-dimensional equilibrium solutions. For all seventeen solutions we use $\sigma = 6$. Five of these solutions are for $\lambda = 75$ and $\mu = 0$ (top left). The rest of them use $\lambda = 150$ and $\mu = 0$ (top middle and top right), $\mu = 0.1$ (bottom left), $\mu = 0.3$ (bottom middle), and $\mu = 0.5$ (bottom right)
A three-dimensional validated solution for the parameter values $\lambda = 75$, $\sigma = 6$, and $\mu = 0$
These values are rigorous upper bounds for the embedding constants in (11)
 Dimension $d$ $1$ $2$ $3$ Sobolev Embedding Constant $C_m$ $1.010947$ $1.030255$ $1.081202$ Sobolev Embedding Constant $\overline{C}_m$ $0.149072$ $0.248740$ $0.411972$ Banach Algebra Constant $C_b$ $1.471443$ $1.488231$ $1.554916$
 Dimension $d$ $1$ $2$ $3$ Sobolev Embedding Constant $C_m$ $1.010947$ $1.030255$ $1.081202$ Sobolev Embedding Constant $\overline{C}_m$ $0.149072$ $0.248740$ $0.411972$ Banach Algebra Constant $C_b$ $1.471443$ $1.488231$ $1.554916$
A sample of the one-dimensional solution validation parameters for three typical solutions. In each case, we use $\sigma = 6$ and $\lambda = 150$. If we had chosen a larger value of $N$, we could significantly improve the results
 $\mu$ $K$ $N$ $P$ $\delta_\alpha$ $\delta_x$ $0$ 6.2575 89 $\lambda$ 0.0016 0.0056 $\sigma$ 2.9259e-04 0.0056 $\mu$ 2.8705e-06 0.0044 $0.1$ 6.4590 104 $\lambda$ 0.0011 0.0050 $\sigma$ 2.5369e-04 0.0050 $\mu$ 2.5579e-06 0.0041 $0.5$ 3.1030 74 $\lambda$ 0.0052 0.0107 $\sigma$ 0.0011 0.0106 $\mu$ 1.2871e-05 0.0092
 $\mu$ $K$ $N$ $P$ $\delta_\alpha$ $\delta_x$ $0$ 6.2575 89 $\lambda$ 0.0016 0.0056 $\sigma$ 2.9259e-04 0.0056 $\mu$ 2.8705e-06 0.0044 $0.1$ 6.4590 104 $\lambda$ 0.0011 0.0050 $\sigma$ 2.5369e-04 0.0050 $\mu$ 2.5579e-06 0.0041 $0.5$ 3.1030 74 $\lambda$ 0.0052 0.0107 $\sigma$ 0.0011 0.0106 $\mu$ 1.2871e-05 0.0092
A sample of the two-dimensional validation parameters for a couple of typical solutions. In all cases, we use $\sigma = 6$. Again as in the previous table, we could improve results by choosing a larger value of $N$, but in this case since $N$ is only the linear dimension, the dimension of the calculation varies with $N^2$
 $(\lambda,\mu)$ $K$ $N$ $P$ $\delta_\alpha$ $\delta_x$ $(75,0)$ 21.1303 28 $\lambda$ 1.6124e-04 0.0020 $\sigma$ 6.1338e-05 0.0020 $\mu$ 5.9914e-07 0.0016 $(150,0.1)$ 30.1656 72 $\lambda$ 1.1833e-05 4.7710e-04 $\sigma$ 5.1514e-06 4.7858e-04 $\mu$ 4.4558e-08 4.2316e-04
 $(\lambda,\mu)$ $K$ $N$ $P$ $\delta_\alpha$ $\delta_x$ $(75,0)$ 21.1303 28 $\lambda$ 1.6124e-04 0.0020 $\sigma$ 6.1338e-05 0.0020 $\mu$ 5.9914e-07 0.0016 $(150,0.1)$ 30.1656 72 $\lambda$ 1.1833e-05 4.7710e-04 $\sigma$ 5.1514e-06 4.7858e-04 $\mu$ 4.4558e-08 4.2316e-04
Validation parameters for a three-dimensional sample solution
 $(\lambda,\sigma,\mu)$ $K$ $N$ $P$ $\delta_\alpha$ $\delta_x$ $(75,6,0)$ 22.6527 22 $\lambda$ 0.1143e-04 0.5917e-03 $\sigma$ 0.1707e-04 0.5955e-03 $\mu$ 0.0010e-04 0.4901e-03
 $(\lambda,\sigma,\mu)$ $K$ $N$ $P$ $\delta_\alpha$ $\delta_x$ $(75,6,0)$ 22.6527 22 $\lambda$ 0.1143e-04 0.5917e-03 $\sigma$ 0.1707e-04 0.5955e-03 $\mu$ 0.0010e-04 0.4901e-03
 [1] Ping Liu, Junping Shi, Yuwen Wang. A double saddle-node bifurcation theorem. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2923-2933. doi: 10.3934/cpaa.2013.12.2923 [2] Rui Dilão, András Volford. Excitability in a model with a saddle-node homoclinic bifurcation. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 419-434. doi: 10.3934/dcdsb.2004.4.419 [3] Ale Jan Homburg, Todd Young. Intermittency and Jakobson's theorem near saddle-node bifurcations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 21-58. doi: 10.3934/dcds.2007.17.21 [4] Piotr Zgliczyński. Steady state bifurcations for the Kuramoto-Sivashinsky equation: A computer assisted proof. Journal of Computational Dynamics, 2015, 2 (1) : 95-142. doi: 10.3934/jcd.2015.2.95 [5] Thomas Wanner. Computer-assisted equilibrium validation for the diblock copolymer model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 1075-1107. doi: 10.3934/dcds.2017045 [6] Xiao-Biao Lin, Changrong Zhu. Saddle-node bifurcations of multiple homoclinic solutions in ODES. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1435-1460. doi: 10.3934/dcdsb.2017069 [7] Flaviano Battelli. Saddle-node bifurcation of homoclinic orbits in singular systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 203-218. doi: 10.3934/dcds.2001.7.203 [8] A. Aschwanden, A. Schulze-Halberg, D. Stoffer. Stable periodic solutions for delay equations with positive feedback - a computer-assisted proof. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 721-736. doi: 10.3934/dcds.2006.14.721 [9] Chiara Caracciolo, Ugo Locatelli. Computer-assisted estimates for Birkhoff normal forms. Journal of Computational Dynamics, 2020, 7 (2) : 425-460. doi: 10.3934/jcd.2020017 [10] Maxime Breden, Jean-Philippe Lessard. Polynomial interpolation and a priori bootstrap for computer-assisted proofs in nonlinear ODEs. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2825-2858. doi: 10.3934/dcdsb.2018164 [11] István Balázs, Jan Bouwe van den Berg, Julien Courtois, János Dudás, Jean-Philippe Lessard, Anett Vörös-Kiss, JF Williams, Xi Yuan Yin. Computer-assisted proofs for radially symmetric solutions of PDEs. Journal of Computational Dynamics, 2018, 5 (1&2) : 61-80. doi: 10.3934/jcd.2018003 [12] Kie Van Ivanky Saputra, Lennaert van Veen, Gilles Reinout Willem Quispel. The saddle-node-transcritical bifurcation in a population model with constant rate harvesting. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 233-250. doi: 10.3934/dcdsb.2010.14.233 [13] Fatiha Alabau-Boussouira, Piermarco Cannarsa. A constructive proof of Gibson's stability theorem. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 611-617. doi: 10.3934/dcdss.2013.6.611 [14] Victoriano Carmona, Soledad Fernández-García, Antonio E. Teruel. Saddle-node of limit cycles in planar piecewise linear systems and applications. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5275-5299. doi: 10.3934/dcds.2019215 [15] W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351 [16] Qingyan Shi, Junping Shi, Yongli Song. Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 467-486. doi: 10.3934/dcdsb.2018182 [17] Majid Gazor, Mojtaba Moazeni. Parametric normal forms for Bogdanov--Takens singularity; the generalized saddle-node case. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 205-224. doi: 10.3934/dcds.2015.35.205 [18] Julien Barré, Pierre Degond, Diane Peurichard, Ewelina Zatorska. Modelling pattern formation through differential repulsion. Networks & Heterogeneous Media, 2020, 15 (3) : 307-352. doi: 10.3934/nhm.2020021 [19] Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions. Communications on Pure & Applied Analysis, 2005, 4 (3) : 665-682. doi: 10.3934/cpaa.2005.4.665 [20] Peter Rashkov. Remarks on pattern formation in a model for hair follicle spacing. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1555-1572. doi: 10.3934/dcdsb.2015.20.1555

2019 Impact Factor: 1.27

## Tools

Article outline

Figures and Tables