Mathematics – Analysis of PDEs
Scientific paper
2005-11-19
Mathematics
Analysis of PDEs
Scientific paper
We prove two new mixed sharp bilinear estimates of Schr\"odinger-Airy type. In particular, we obtain the local well-posedness of the Cauchy problem of the Schr\"odinger - Kortweg-deVries (NLS-KdV) system in the \emph{periodic setting}. Our lowest regularity is $H^{1/4}\times L^2$, which is somewhat far from the naturally expected endpoint $L^2\times H^{-1/2}$. This is a novel phenomena related to the periodicity condition. Indeed, in the continuous case, Corcho and Linares proved local well-posedness for the natural endpoint $L^2\times H^{-{3/4}+}$. Nevertheless, we conclude the global well-posedness of the NLS-KdV system in the energy space $H^1\times H^1$ using our local well-posedness result and three conservation laws discovered by M. Tsutsumi.
Arbieto Alexander
Corcho Adan
Matheus Carlos
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