Equilibrium validation in models for pattern formation based on Sobolev embeddings

Evelyn Sander; Thomas Wanner

doi:10.3934/dcdsb.2020260

Article Contents

2021, Volume 26, Issue 1: 603-632. Doi: 10.3934/dcdsb.2020260

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$ P^* $

rule in the stochastic Holt-Lawton model of apparent competition

Equilibrium validation in models for pattern formation based on Sobolev embeddings

Evelyn Sander and
Thomas Wanner^,

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

^* Corresponding author: Thomas Wanner
^* Corresponding author: Thomas Wanner

Received: February 29, 2020

Revised: July 31, 2020

Early access: August 2020

Published: January 2021

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Abstract

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.

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Mathematics Subject Classification: Primary: 35B40, 35B41, 35K55, 37M20, 65G20; Secondary: 65G30, 65N35, 74N99.

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Figure 1. 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 $

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Figure 2. 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

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Figure 3. 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)

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Figure 4. A three-dimensional validated solution for the parameter values $ \lambda = 75 $, $ \sigma = 6 $, and $ \mu = 0 $

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Table 1. 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 $

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Table 2. 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

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Table 3. 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

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Table 4. 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

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