Rigorous Numerics for the Cahn-Hilliard Equation on the Unit Square

Abstract

While the structure of the set of stationary solutions of the Cahn-Hilliard equation on one-dimensional domains is completely understood, only partial results are available for two-dimensional base domains. In this paper, we demonstrate how rigorous computational techniques can be employed to establish computerassisted existence proofs for equilibria of the Cahn-Hilliard equation on the unit square. Our method is based on results by Mischaikow and Zgliczyński [22], and combines rigorous computations with Conley index techniques. We are able to establish branches of equilibria and, under more restrictive conditions, even the local uniqueness of specific equilibrium solutions. Sample computations for several branches are presented, which illustrate the resulting patterns.

Computation of Smooth Manifolds Via Rigorous Multi-parameter Continuation in Infinite Dimensions

“…The smooth singular value decomposition would be a useful tool for this project, see [22]. It would also be valuable to try to prove existence of cusp bifurcations in the two dimensional manifold of equilibria of the 2D Cahn-Hilliard model as numerically suggested in [4]. Another interesting problem would be to extend the work of [23] and compute two-dimensional manifolds of connecting include all formulas and proofs so that the paper is self-contained.…”

“…The analytic estimates are presented in Appendix A. 4. We demonstrate that for each σ ∈ S, the chart U σ is smooth.…”

“…In the context of infinite dimensional problems like PDEs and DDEs, the continuation algorithms have to be performed on a finite dimensional projection, which raises the natural question of the validity of the outputs. In order to address this fundamental question, rigorous one-parameter continuation methods have been proposed to compute global branches of solutions of PDEs and DDEs [1,2,3,4,5]. While these methods have been applied to compute one-dimensional manifolds, we are not aware of any rigorous method aiming at computing solution manifolds of dimension greater than one.…”

Steady state bifurcations for the Kuramoto-Sivashinsky equation: A computer assisted proof

“…Hence linking our method with a continuation approach should be enough for the task. This is the approach taken by Maier-Pappe and coauthors in [17], where using the method of self-consistent a priori bounds the authors were able to continue the branches of steady states for the Cahn-Hillard equation on the square (they did not treat bifurcations).…”

Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains

“…In the stochastic case the polynomial nonlinearity has been analyzed in [10,11,15,16,19,25], while in [22,21,31] a stochastic Cahn-Hilliard with reflection is considered. Numerical results for the Cahn-Hilliard equation on the unit square has been presented in [41].…”