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

Characteristics of liquid cluster ion beam for surface treatment

A liquid cluster ion source, which is an ion source for the cluster beams produced with liquid materials, has been developed for the surface treatment of solid materials. The electrodes were designed for increasing the cluster beam intensity by a computer simulation of beam trajectories. The peaks of the cluster size distributions of the water and ethanol cluster ion beams of 3 atm vapor pressure were approximately at 2.4×103 and 1.6×103 molecules, respectively. The cluster size distributions of ethanol clusters were not sensitive to the variations of the acceleration voltages (Ve) and currents (Ie) of the electrons for ionization when the Ve and Ie were larger than approximately 200 V and 200 mA, respectively