> TITLE: Kato's smoothing effect for solutions to the  capillary water-wave problem.
>
>
> ABSTRACT:  In
> collaboration with Hans Cristianson and Vera Hur we proved that the solutions to the
> Cauchy problem for exact free-surface water waves in
> presence of surface tension, as t>0, gain 1/4 derivative smoothness compared to the initial profile, this is what we call the 1/4 Kato's smoothing effect.
> The major
> difficulty in proving this result  is severe nonlinearity on free surface. To deal with
> nonlinearity, first, we reformulate the problem as a nonlinear dispersive
> equation for a modified velocity on the free surface, whose linear part may
> be recognized as a hybrid of the wave equation and the Schroedinger or the
> Korteweg-de Vries equation. Our novel formulation exhibits strong
> dispersive property due to surface tension, and indeed, smoothing effects.
> Dispersion allows us to treat nonlinear terms with first or second
> spatial derivatives by means of techniques of oscillatory integrals.  But this would not be enough.
> Secondly, we view the most severe nonlinear term as a "linear component" of the
> equation, but with a variable coefficient which happens to depend on the
> solution itself. That is, we reduce the size of the nonlinear terms at the
> cost of making the linear part more complicated. A sophisticated microlocal
> analysis approach is to establish smoothing effects for this 'water-wave
> operator' with variable coefficient. We provide more refined analysis than
> classical energy estimates, which is the only estimate known so far in the
> analysis of the Cauchy problem for water waves. Our result requires less
> number of derivatives in the choice of Sobolev spaces, which is a major
> improvement of this project and proves some new  Kato type smoothing effect for the solutions.