Lower bound for the rate of blow-up of singular solutions of the Zakharov system in $\R^3$

TL;DR

This paper establishes a lower bound on the rate at which solutions to the 3D Zakharov system blow up in Sobolev norms, refining understanding of singularity formation in this nonlinear PDE system.

Contribution

It provides the first explicit lower bound for blow-up rates of Sobolev norms in the Zakharov system in three dimensions, combining well-posedness theory with blow-up analysis techniques.

Findings

Lower bound for blow-up rate of Sobolev norms established

Quantitative relation between blow-up time and Sobolev norm divergence

Extension of techniques from nonlinear Schrödinger equations to Zakharov system

Abstract

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.