On Validated Equilibria and Bifurcations in Materials Science and Stochastic Dynamics

Abstract

Partial Differential Equations (PDEs) and their solutions provide a theoretically rich, widely applicable, and analytically challenging subfield of mathematics and physics. In the applied setting, analytical solutions are rarely known, and one must resort to numerical methods. These methods are, at their best, approximations to the exact solution of the PDE and quickly exceed the computational ability of humans, making a computer necessary to execute them efficiently. While many of these methods, such as the Finite Element Method (FEM) and the Spectral Method, produce good quality approximations, problems exist where the solution to the numerical formulation is in fact not close to a solution of the PDE. The proliferation of computing in scientific applications demands exploration of verification methods for numerical solutions. We explore two special cases: the Fokker-Planck-Kolmogorov Equation and a variant of the Ohta-Kawasaki model of block copolymers. The Fokker-Planck-Kolmogorov Equation stems from the study of Stochastic Differential Equations (SDEs) and, as a Partial Differential Equation (PDE), provides a fundamental link between SDEs and PDEs. The principal object of study in this case is the transition probability density associated with the stochastic system. Under suitable conditions, there exists a distribution that is invariant through time even though the paths of the SDEmove. We provide theoretical estimates for the application of a constructive version of the Implicit Function Theorem to verify numerical solutions to the eigenvalue form of the Fokker-Planck-Kolmogorov Equation. While our numerical examples focus on the invariant distribution (i.e., zero eigenvalue), the theory extends without modification to the nonzero eigenvalue case. Block copolymers play an important role in materials sciences and have found widespread use in many applications. From a mathematical perspective, they are governed by a nonlinear fourth-order partial differential equation which is a suitable gradient of the Ohta-Kawasaki energy. While the equilibrium states associated with this equation are of central importance for the description of the dynamics of block copolymers, their mathematical study remains challenging. We develop computer-assisted proof methods which can be used to study equilibrium solutions in block copolymers consisting of more than two monomer chains, with a focus on triblock copolymers. This is achieved by establishing a computer-assisted proof technique for bounding the norm of the inverses of certain fourth-order elliptic operators, in combination with an application of a constructive version of the implicit function theorem. While these results are only applied to the triblock copolymer case, we demonstrate that the obtained norm estimates can also be directly used in other contexts such as the rigorous verification of bifurcation points, or pseudo-arclength continuation in fourth-order parabolic problems.