  We consider the scalar Zakharov system in $\R^3$ for initial conditions
$(\psi(0), n(0), n_t(0)) \in H^{\ell+1/2} \times H^\ell \times H^{\ell-1} $,
  $0\leq\ell \leq 1$. Assuming that the solution blows up in a finite time $t^*
< \infty$, we establish a lower bound for the rate of blow-up of the
corresponding Sobolev norms in the form $$ \|\psi(t)\|_{H^{\ell+1/2}}
+\|n(t)\|_{H^{\ell}} + \|n_t(t)\|_{H^{\ell-1}} > C(t^*-t)^{-\theta_\ell} $$
with $\theta_\ell = \frac{1}{4}(1+ 2 \ell)^-$. The analysis is a reappraisal of
the local wellposedness theory of Ginibre, Tsutsumi and Velo (1997) combined
with an argument developed by Cazenave and Weissler (1990) in the context of
nonlinear Schr\"odinger equations.
