Global well-posedness for KdV in Sobolev Spaces of negative index

The initial value problem for the Korteweg-deVries equation on the line is shown to be globally well-posed for rough data. In particular, we show global well-posedness for initial data in H^s({\mathbb{R}), -3/10<s.Comment: 5 pages. Electronic Journal of Differential equations (submitted

VLT/VIMOS Observations of an Occulting Galaxy Pair: Redshifts and Effective Extinction Curve

We present VLT/VIMOS IFU observations of an occulting galaxy pair previously discovered in HST observations. The foreground galaxy is a low-inclination spiral disk, which causes clear attenuation features seen against the bright bulge and disk of the background galaxy. We find redshifts of $z=0.064 \pm0.003$ and z=0.065 for the foreground and background galaxy respectively. This relatively small difference does not rule out gravitational interaction between the two galaxies. Emission line ratios point to a star-forming, not AGN-dominated foreground galaxy. We fit the Cardelli, Clayton & Mathis (CCM) extinction law to the spectra of individual fibres to derive slope ($R_V$) and normalization ($A_V$). The normalization agrees with the HST attenuation map and the slope is lower than the Milky Way relation ($R_V<3.1$), which is likely linked to the spatial sampling of the disk. We speculate that the values of $R_V$ point to either coherent ISM structures in the disk larger than usual ($\sim9$ kpc) or higher starting values of $R_V$, indicative of recent processing of the dust. The foreground galaxy is a low stellar mass spiral ($M_* \sim 3 \times 10^9 M_\odot$) with a high dust content ($M_{\rm dust} \sim 0.5 \times 10^6 M_\odot$). The dust disk geometry visible in the HST image would explain the observed SED properties of smaller galaxies: a lower mean dust temperature, a high dust-to-stellar mass ratio but relatively little optical attenuation. Ongoing efforts to find occulting pairs with a small foreground galaxies will show how common this geometry is.Comment: 16 pages, 3 tables, 13 figures, accepted for publication in MNRA