Lower bounds for the blowup rate of solutions of the Zakharov equation in dimension two

We consider the blowup solution ( u,n,v )( t ) of the Zakharov equations $$\cases{iu_t = - \Delta u + n u\cr n_t = - \nabla \cdot \nu \cr {1\over{c_{0}^{2}}} \nu_t = - \nabla n - \nabla \mid u \mid^2\cr (u(0),n(0),\nu(0))=(u_0,n_0,\nu_0) }\leqno(1)$$ where u : R 2 → C, n : R 2 → R, v : R 2 → R 2 in the energy space H 1 = {( u,n,v ) η H 1 × L 2 × L 2 }. We show that there is a constant c depending on the L 2 ‐norm of u 0 such that $$\mid(u,n,\nu)(t)\mid_{H_1} \geq \mid \nabla u(t)\mid_{L^2} \geq {c \over {(T-t)}},$$ where T is the blowup time. We check that this estimate is optimal and give further applications. © 1996 John Wiley & Sons, Inc.