Validation of the bifurcation diagram in the 2D Ohta–Kawasaki problem

Abstract

We develop a rigorous numerical method to compare local minimizers of the Ohta-Kawasaki functional in two dimensions. In particular, we validate the phase diagram identifying regions of parameter space where rolls are favorable, where hexagonally packed spots have lowest energy and finally where the constant mixed state does. More generally, we present a method to rigorously determine such features in problems where optimal domain sizes are not known a priori.

A general framework for validated continuation of periodic orbits in systems of polynomial ODEs

“…s and G @ s make it easy to "glue" subsequent steps of the continuation together to form a continuous (even smooth) curve. Indeed, following the arguments used in [11,9], we conclude that if we choose the datax s , 9…”

“…The functional analytic approach is particularly convenient for continuation problems. In the radii polynomial setting this was first introduced in [5,11], and further explored in [22,9,26], see also [32] for a closely related perspective on computerassisted continuation and bifurcation proofs. In the current paper we adopt the radii polynomial methodology within the functional analytic framework and develop it into a universal mathematically rigorous continuation tool for periodic orbits.…”

“…In some instances bifurcation problems can in fact be attacked in a similar framework, see e.g. [1,33,26,9], but many cases remain unexplored. In a forthcoming paper [6] we build on the current general continuation framework to analyze saddle-node bifurcations of branches of periodic orbits, as well as the branches emanating from equilibria at Hopf bifurcations.…”

“…In order not to overburden the notation, we did not pursue such weighted norms in the current paper. We refer to [9] for an extensive discussion of an application where weighted norms are essential.…”

Dimension Reduction and Optimality of the Uniform State in a Phase-Field-Crystal Model Involving a Higher-Order Functional

“…Finding analytically the exact curves separating such parameter regions is a big challenge and remains still open, despite some rigorous attempts such as the paper [24] where some bounds on the order-disorder phase transition were obtained by a numerical algorithm. An attempt was also conducted in this direction for a similar type energy, that is the Ohta-Kawasaki problem [26] using numerics that take into account the impact of domain size optimization. In [5], the authors studied the existence of bifurcation branches from the trivial solution with a constraint on the Hamiltonian in the one dimensional case.…”