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

Abstract

We consider the blowup solution (u, n, v)(t) of the Zakharov equations { iut = -Δu + nu nt = - ∇ · v (I) 1/c20vt = -∇n - ∇|u|2 (u(0), n(0), v(0)) = (u0, n0, v0) where u : ℝ2 → ℂ, n : ℝ2 → ℝ, v : ℝ2 → ℝ2 in the energy space H1 = {(u, n, v) ∈ H1 × L2 × L2}. We show that there is a constant c depending on the L2-norm of u0 such that |(u, n, v)(t)|H1 ≧ |∇ u(t)|L2 ≧ c/(T - t), where T is the blowup time. We check that this estimate is optimal and give further applications. © 1996 John Wiley & Sons, Inc.