Scientific report: Computational Proofs for Dynamics in PDEs

Jan Bouwe van den Berg, Jean-Philippe Lessard

Description and aims

The past decade has seen enormous advances in the development of rigorously verified computing. For questions related to nonlinear dynamics the most significant results are associated with finite dimensional systems. In this workshop we explored the challenges that lie ahead in applying these techniques to fully fledged problems in the theory of infinite dimensional nonlinear dynamical systems, with a particular emphasis on nonlinear partial differential equations.

Beforehand, we declared that we would consider the workshop successful if we compiled a list of challenging problems, developed initial ideas for solving these, and started new collaborations to examine them further.

Format

The workshop had a very limited number of lectures: four on the first day to set the stage, but only one plenary lecture on each of the following days. On the first day we had two lectures on the state-of-the-art of computer-assisted proofs in dynamical systems, as well as two lectures dedicated to formulating challenging PDE problems. Afterwards we had a plenary discussion about the problems we wanted to work on during the week. This list was revisited every subsequent day of the workshop. The rest of the week we broke up into groups to collaborate on these problems and then reported back to all participants about progress. Each day we also had a plenary lecture where another type of open problem was discussed, which led to amendments of open-problems list. Throughout the week we had several smaller time slots for spontaneous lectures, and indeed one or two of these were used every day. Additionally, on Tuesday, all PhD students gave short presentations to introduce themselves and their research.

Scientific developments and Aha-insights

The format of the workshop was very beneficial for achieving these aims. In particular:

• We generalised the existing methods from polynomial vector fields to general vector field.

• We identified a startup problem in the field of stochastic differential equations, with a link to finite element methods.

• We discussed how to prove existence and smoothness of invariant tori.

• We identified fully nonlinear reaction-diffusion problems (with cross-diffusion) that seem challenging but achievable.

• We discussed several concrete radially symmetric PDE problems which are currently just beyond the scope of existing methods, and we started working on them.

• We pinpointed Kolmogorov flow as a natural starting point for investigating the applicability of the computer-assisted proof methods to the Navier-Stokes equations.

• We made significant progress on applications to delay equations.

After the workshop we received a lot of positive feedback, especially from the participants who were relative outsider: they remarked that this type of hands-on workshop introduced them much better to the field than a regular “talks-based” workshop would have done.