**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.