  The flow equations of the renormalisation group permit to analyse the
perturbative $n$-point functions of renormalisable quantum field theories.
Rigorous bounds implying renormalisablility allow to control large momentum
behaviour, infrared singularities and large order behaviour in the number of
loops and the number of arguments $n\,$. Gauge symmetry which is broken by the
flow in momentum or position space, can be shown to be restored in the
renormalised theory.
  In this paper we want to do a first but important step towards a rigorous
nonperturbative analysis of the flow equations (FEs). We restrict to massive
scalar fields and analyse the {\it mean field limit} where the Schwinger or 1PI
functions are considered to be momentum independent or, otherwise stated, are
replaced by their zero momentum values.
  We regard smooth solutions of the system of FEs for the $n$-point functions
for different sets of boundary conditions. We will realise that allowing for
non\-vanishing irrelevant terms permits to construct {\it asymptotically free}
and thus nontrivial {\it scalar field theories} in the mean field
approximation. We will also analyse the so-called trivial solution so far
generally believed to exhaust four-dimensional scalar field theory. The method
paves the way to a study of the system of FEs beyond the mean field limit.
