In this talk I will establish the well-posedness of the transport equation in
the case when the coecients are irregular (e.g. have only Sobolev regularity)
and additionally have 'geometrically complicated' sets of singularities. We
will see that the well-posedness results hold when the set of singularities has
a sucient small anisotropic fractal dimension, which is encoded in a 'codi-
mension print'. I will relate this esoteric notion of dimension to the more
familiar box-counting dimensions to provide straightforward criteria for the
well-posedness of the transport equation.
Further, I will consider some qualitative properties of the 'generalised' ow
solution of the corresponding ODE, which denes a dynamical system in this
irregular setting. This work extends the renormalization theory of DiPerna &
Lions and Ambrosio to account for geometrically complicated singularities.