Seminar on Analysis, Geometry and PDEs - Dan Hill (Saarland University)
Going Around in Circles: Developing a Radial Spatial Dynamic Theory for Fully Localised 2D Patterns
Speaker: Dan Hill (Saarland University)
The emergence of small-amplitude localised patterns is rigorously well understood in one spatial dimension, thanks in part to the theory of spatial dynamics developed by Kirchgässner in the 1980s. A key concept in this approach is treating an unbounded spatial direction as a `time-like' variable and using tools from dynamical systems to prove the existence of localised (in `time') solutions. However, a key question emerges when considering higher spatial dimensions; how can one use such an approach to study patterns that are localised in multiple directions? Such patterns are often found in experiments and yet there is very little analytic theory regarding them, beyond a handful of examples.
In this talk, we will focus on 2D patterns that are localised in the radial direction and explore the theory of radial spatial dynamics, introduced by Scheel in 2003. We will review prior studies of axisymmetric patterns, before presenting recent results for approximating fully localised patterns with dihedral symmetries. Throughout we will highlight connections to the 1D problem, how new classes of solutions can emerge in this framework, as well as some of the difficulties encountered when considering nonlinear PDEs in polar coordinates.
Tid: 2022-12-05 10:30 till 11:30
erik [dot] wahlen [at] math [dot] lu [dot] se