Ginibre-Tsutsumi-Velo (1997) proved local well-posedness for the Zakharov system
where , , , and . The proof was made for any dimension , in the inhomogeneous Sobolev spaces for a range of exponents , depending on . Here we restrict to dimension and present a few results establishing local ill-posedness for exponent pairs 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 Schrodinger equation.
Submitted March 24, 2006. Published February 12, 2007.
Math Subject Classifications: 35Q55, 35Q51, 35R25.
Key Words: Zakharov system; Cauchy problem; local well-posedness; local ill-posedness.
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| Justin Holmer |
Department of Mathematics
University of California
Berkeley, CA 94720-3840, USA
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