KTH/SU Mathematics Colloquium
November 5, 2008
Jens Eggers, University of Bristol
The role of self-similarity in singularities of PDE's
Singularities lie at the heart of many physical phenomena like
shock formation, drop formation, air entrainment, or flow separation.
Here we survey results on the formation of point-like singularities
(or blow-up) in evolution equations. We use a similarity
transformation of the original equation with
respect to the blow-up point, such that self-similar behaviour is mapped to
the fixed point of a dynamical system. We point out that analysing
the dynamics close to the fixed point is a useful way of characterising
the singularity, in that the dynamics frequently reduces to very few
dimensions. As far as we are aware,
examples from the literature either correspond to stable fixed points,
low-dimensional centre-manifold dynamics, limit cycles, or travelling waves.
For each ``class'' of singularity, we give detailed examples, and
we emphasise the physical meaning of singularities.