Local ill-posedness of the 1D Zakharov system
Texas State University-San Marcos, Department of Mathematics
Ginibre-Tsutsumi-Velo (1997) proved local well-posedness for the Zakharov system i∂tu + Δu = nu ∂2tn - Δn = Δ|u|2 u(x, 0) = u0(x), n(x, 0) = n0(x), ∂tn(x, 0) = n1(x) where u = u(x, t) ∈ ℂ, n = n(x, t) ∈ ℝ, x ∈ ℝ, and t ∈ ℝ. The proof was made for any dimension d, in the inhomogeneous Sobolev spaces (u, n) ∈ Hk (ℝd) x Hs (ℝd) for a range of exponents k, s depending on d. Here we restrict to dimension d = 1 and present a few results establishing local ill-posedness for exponent pairs (k, s) outside of the well-posedness regime. The techniques employed are rooted in the work of Bourgain (1993), Birnir-Kenig-Ponce-Svanstedt-Vega (1996), and Christ-Colliander-Tao (2003) applied to the nonlinear Schrödinger equation.
Zakharov system, Cauchy problem, Local well-posedness, Local ill-posedness
Holmer, J. (2007). Local ill-posedness of the 1D Zakharov system. <i>Electronic Journal of Differential Equations, 2007</i>(24), pp. 1-22.