  We investigate Kato's method for parabolic equations with a quadratic
non-linearity in an abstract form. We extract several properties known from
linear systems theory which turn out to be the essential ingredients for the
method. We give necessary and sufficient conditions for these conditions and
provide new and more general proofs, based on real interpolation. In
application to the Navier-Stokes equations, our approach unifies several
results known in the literature, partly with different proofs. Moreover, we
establish new existence and uniqueness results for rough initial data on
arbitrary domains in ${\mathbb R}^3$ and irregular domains in ${\mathbb R}^n$.
