Rough solutions for the periodic Schrödinger - Korteweg-deVries system

Carlos Matheus | Arbieto, Alexander | Corcho, Adán

Keywords: Local and global well-posedness | Schrodinger - Korteweg-deVries system

We prove two new mixed sharp bilinear estimates of Schródinger-Airy type. In particular, we obtain the local well-posedness of the Cauchy problem of the Schrodinger - 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 Lineares proved local well-posedness for the natural endpoint $L^2\times H^{-\frac{3}{4}+}$. 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.

Anexos:

• NLS-KdV.pdf