  Nonlinear $\sigma$-model is an ubiquitous model. In this paper, the $O(N)$
model where the $N$-component spin is a unit vector, ${\bf S}^2=1$,is
considered. The stability of this model with respect to gradient operators
$(\partial_{\mu}{\bf S}\cdot \partial_{\nu}{\bf S})^s$, where the degree $s$ is
arbitrary, is discussed. Explicit two-loop calculations within the scheme of
$\epsilon$-expansion, where $\epsilon=(d-2)$, leads to the surprising result
that these operators are relevant. In fact, the relevancy increases with the
degree $s$. We argue that this phenomenon in the $O(N)$-model actually reflects
the failure of the perturbative analysis, that is, the
$(2+\epsilon)$-expansion. It is likely that it is necessary to take into
account non-perturbative effects if one wants to describe the phase transition
of the Heisenberg model within the context of the non-linear $\sigma$-model.
Thus, uncritical use of the $(2+\epsilon)$-expansion may be misleading,
especially for those cases for which there are not many independent checks.
