Mean field flow equations and asymptotically free scalar fields

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.