local ill-posedness of the 1d zakharov system

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ID: 257459
2007
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Ranked #214 of 219 articles by views in icsoft 2006 - 1st international conference on software and data technologies, proceedings

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Abstract
Ginibre-Tsutsumi-Velo (1997) proved local well-posedness for the Zakharov system$$displaylines{ ipartial_tu + Delta u = nu cr partial_t^2 n - Delta n = Delta |u|^2 cr u(x,0)=u_0(x), cr n(x,0)=n_0(x), quad partial_tn(x,0)=n_1(x)}$$ where $u=u(x,t)in mathbb{C}$, $n=n(x,t)in mathbb{R}$, $xin mathbb{R}$, and $tin mathbb{R}$. The proof was made for any dimension $d$, in the inhomogeneous Sobolev spaces $(u,n)in H^k(mathbb{R}^d)imes H^s(mathbb{R}^d)$ 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 Schrodinger equation.
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Authors Justin Holmer
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Journal icsoft 2006 - 1st international conference on software and data technologies, proceedings
Year 2007
DOI
DOI not found
URL
http://ejde.math.txstate.edu/Volumes/2007/24/abstr.html
Keywords
mathematics

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