# American Institute of Mathematical Sciences

January  2021, 26(1): 603-632. doi: 10.3934/dcdsb.2020260

## 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, 2021, 26 (1) : 603-632. 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] Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020465 [2] Alexandra Köthe, Anna Marciniak-Czochra, Izumi Takagi. Hysteresis-driven pattern formation in reaction-diffusion-ODE systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3595-3627. doi: 10.3934/dcds.2020170 [3] Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317 [4] Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264 [5] Jinfeng Wang, Sainan Wu, Junping Shi. Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1273-1289. doi: 10.3934/dcdsb.2020162 [6] Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047 [7] Zhouchao Wei, Wei Zhang, Irene Moroz, Nikolay V. Kuznetsov. Codimension one and two bifurcations in Cattaneo-Christov heat flux model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020344 [8] Yukio Kan-On. On the limiting system in the Shigesada, Kawasaki and Teramoto model with large cross-diffusion rates. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3561-3570. doi: 10.3934/dcds.2020161 [9] Guojie Zheng, Dihong Xu, Taige Wang. A unique continuation property for a class of parabolic differential inequalities in a bounded domain. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020280 [10] Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032 [11] Raimund Bürger, Christophe Chalons, Rafael Ordoñez, Luis Miguel Villada. A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function. Networks & Heterogeneous Media, 2021  doi: 10.3934/nhm.2021004 [12] Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084 [13] Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3117-3142. doi: 10.3934/dcds.2019226 [14] Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation. Electronic Research Archive, 2021, 29 (1) : 1709-1734. doi: 10.3934/era.2020088 [15] Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021005 [16] Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392 [17] H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433 [18] Musen Xue, Guowei Zhu. Partial myopia vs. forward-looking behaviors in a dynamic pricing and replenishment model for perishable items. Journal of Industrial & Management Optimization, 2021, 17 (2) : 633-648. doi: 10.3934/jimo.2019126 [19] Ryuji Kajikiya. Existence of nodal solutions for the sublinear Moore-Nehari differential equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1483-1506. doi: 10.3934/dcds.2020326 [20] Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 909-918. doi: 10.3934/dcdss.2020391

2019 Impact Factor: 1.27

## Tools

Article outline

Figures and Tables