  We consider the regularity of the extremal solution of the nonlinear
eigenvalue problem (S)_\lambda \qquad {rcr} -\Delta u + c(x) \cdot \nabla u &=&
\frac{\lambda}{(1-u)^2} \qquad {in $ \Omega$}, u &=& 0 \qquad {on $ \pOm$},
where $ \Omega $ is a smooth bounded domain in $ \IR^N$ and $ c(x)$ is a smooth
bounded vector field on $\bar \Omega$. We show that, just like in the
advection-free model ($c\equiv 0$), all semi-stable solutions are smooth if
(and only if) the dimension $N\leq 7$. The novelty here comes from the lack of
a suitable variational characterization for the semi-stability assumption. We
overcome this difficulty by using a general version of Hardy's inequality. In a
forthcoming paper \cite{CG2}, we indicate how this method applies to many other
nonlinear eigenvalue problems involving advection (including the Gelfand
problem), showing that they all essentially have the same critical dimension as
their advection-free counterparts.
