An improved local well-posedness result for the one-dimensional Zakharov system

Details An improved local well-posedness result for the one-dimensional Zakharov system An improved local well-posedness result for the one-dimensional Zakharov system

2006-11-27

arxiv.org/abs/math/0611811v2

Mathematics

Analysis of PDEs

17 pages. Final version to appear in Journal of Mathematical Analysis and Applications

Scientific paper

The 1D Cauchy problem for the Zakharov system is shown to be locally well-posed for low regularity Schr\"odinger data u_0 \in \hat{H^{k,p}} and wave data (n_0,n_1) \in \hat{H^{l,p}} \times \hat{H^{l-1,p}} under certain assumptions on the parameters k,l and 1^k \hat{u_0}\|_{L^{p'}}, generalizing the results for p=2 by Ginibre, Tsutsumi, and Velo. Especially we are able to improve the results from the scaling point of view, and also allow suitable k<0, l<-1/2, i.e. data u_0 \not\in L^2 and (n_0,n_1)\not\in H^{-1/2}\times H^{-3/2}, which was excluded in the case p=2.

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