Singularities and patterns in evolution equations

4–8 September 2023, University of Bath

This event is part of the activities of the EPSRC network in Generalised and low-regularity solutions for nonlinear PDEs.

Location

Wolfson Lecture Theatre 4W 1.7
Department of Mathematical Sciences
University of Bath
Claverton Down
Bath
BA2 7AY

University travel advice

Mini courses taught by

• Thierry Gallay (Université Grenoble Alpes)

  Stability of Vortex Rings at High Reynolds Number
  The goal of these lectures is to present in a recent result in collaboration with V. Sverak, which is devoted to the vanishing viscosity limit for vortex rings originating from circular filaments. The whole analysis is carried out in the framework of axisymmetric flows without swirl, which is both mathematically convenient and physically relevant for the phenomena we want to study. We first review a few standard properties of the axisymmetric solutions without swirl of the incompressible Navier-Stokes equations. We then explain how to construct an approximate solution of our problem, which is accurate enough in the high Reynolds number regime. Finally, we establish the stability of our approximation using carefully designed energy estimates, which partially rely on Arnold's geometric approach to the stability of stationary flows for the two-dimensional Euler equations.

• Gustav Holzegel (WWU Münster)

  Non-linear wave equations on black hole spacetimes
  This mini course will be an introduction to the dynamics of linear and non-linear waves in various geometric settings, focussing on (but not restricted to) black hole spacetimes. Starting with linear problems, we will outline the proofs of the relevant decay estimates emphasising the role played by various geometric phenomena associated with black holes. We finally present a novel and unifying framework to prove small data global existence results for quasi-linear wave equations in these geometric settings, which will include the case of the (near) Kerr black hole geometry. The last part will be based on recent joint work with Dafermos, Rodnianski and Taylor, arXiv:2212.14093.

• Jens Rademacher (Universität Hamburg)

  Geometric singular perturbation theory in pattern formation
  Spatial patterns far from homogeneous steady states can sometimes be studied with the help of singular perturbation theory. A singular limit in parameters is typically an infinite temporal or spatial scale separation, such as an infinite diffusion ratio in reaction diffusion systems. The lectures will discuss elements of geometric singular perturbation theory and its application to existence, stability and reduced dynamics of resulting patterns. This in particular entails the role of the singular perturbation in the associated eigenvalue problem. The FitzHugh-Nagumo model will serve as a basic example, and further cases will be outlined with focus on reaction-diffusion systems. Some time will be used for exercises.

Invited speakers

• Tobias Barker(University of Bath)
• Juan Davila Bonczos (University of Bath)
• Arjen Doelman (Leiden University)
• Jean Dolbeault (Université Paris Dauphine)
• Claudia Garcia (Universidad de Granada)
• Michael Herrmann (TU Braunschweig)
• John King (University of Nottingham)
• Camilla Nobili (University of Surrey)
• Wolfgang Reichel (Karlsruher Institut für Technologie)
• Guido Schneider (Universität Stuttgart)
• Philippe Souplet (Université Sorbonne Paris Nord)
• Martin Taylor (Imperial College London)
• Peter Topping (University of Warwick)
• Miles Wheeler (University of Bath)
• Michael Winkler (Universität Paderborn)
• Ewelina Zatorska (Imperial College London)

Posters

• There will be a poster session.

Organisers

Manuel del Pino | Karsten Matthies | Monica Musso

Registration is free. Please email mm2683@bath.ac.uk with name, affiliation, food allergies/dietary requirements, and title and abstract if you are interested to present a poster. If you are a PhD student, please specify that in your registration email.