  We prove strong ill-posedness in $L^{\infty}$ for linear perturbations of the
2d Euler equations of the form: \[\partial_t \omega + u\cdot\nabla\omega =
R(\omega),\] where $R$ is any non-trivial second order Riesz transform. Namely,
we prove that there exist smooth solutions that are initially small in
$L^{\infty}$ but become arbitrarily large in short time. Previous works in this
direction relied on the strong ill-posedness of the linear problem, viewing the
transport term perturbatively, which only led to mild growth. In this work we
derive a nonlinear model taking all of the leading order effects into account
to determine the precise pointwise growth of solutions for short time.
Interestingly, the Euler transport term does counteract the linear growth so
that the full nonlinear equation grows an order of magnitude less than the
linear one. In particular, the (sharp) growth rate we establish is consistent
with the global regularity of smooth solutions.
