Continuation and Bifurcation in Nonlinear PDEs – Algorithms, Applications, and Experiments

Numerical continuation and bifurcation methods can be used to explore the set of steady and time–periodic solutions of parameter dependent nonlinear ODEs or PDEs. For PDEs, a basic idea is to first convert the PDE into a system of algebraic equations or ODEs via a spatial discretization. However, the large class of possible PDE bifurcation problems makes developing a general and user–friendly software a challenge, and the often needed large number of degrees of freedom, and the typically large set of solutions, often require adapted methods. Here we review some of these methods, and illustrate the approach by application of the package to some advanced pattern formation problems, including the interaction of Hopf and Turing modes, patterns on disks, and an experimental setting of dead core pattern formation.

The Navier–Stokes equations on the plane with time-dependent external forces

We are concerned with the non-stationary Navier–Stokes equations on the whole plane with external forces which are non-decaying in time, and give a sufficient condition on the external forces for the existence of a solution which exists for whole time. Typical examples are time-periodic solutions and solutions almost periodic in time. The stability under perturbation is also verified.

Weak Solutions and Their Kinetic Energy Regarding Time-Periodic Navier–Stokes Equations in Three Dimensional Whole-Space

The existence of weak time-periodic solutions to Navier–Stokes equations in three dimensional whole-space with time-periodic forcing terms are established. The solutions are constructed in such a way that the structural properties of their kinetic energy are obtained. No restrictions on either the size or structure of the external force are required.

Time-periodic solutions of symmetric hyperbolic systems

We prove the unique existence of time-periodic solutions to general hyperbolic equations with periodic external forces autonomous or nonautonomous over a domain bounded by two parallel planes, provided that all the characteristics with respect to the direction normal to the planes have the same sign. It is also shown that global-in-time solutions to initial-boundary value problems coincide with the solutions to corresponding time-periodic problems after a finite time. We devote one section to the reformulation of several realistic problems and see our results have wide applicability.