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.
Categories: math.AP