Structural resolvent estimates and derivative nonlinear Schrodinger equations

A refinement of uniform resolvent estimate is given and several smoothing estimates for Schrodinger equations in the critical case are induced from it. The relation between this resolvent estimate and radiation condition is discussed. As an application of critical smoothing estimates, we show a global existence results for derivative nonlinear Schrodinger equations.Comment: 21 page

Recent progress in smoothing estimates for evolution equations

This paper is a survey article of results and arguments from several of authors' papers, and it describes a new approach to global smoothing problems for dispersive and non-dispersive evolution equations based on ideas of comparison principle and canonical transforms. For operators $a(D_x)$ of order $m$ satisfying the dispersiveness condition $\nabla a(\xi)\neq0$, a range of smoothing estimates is established. Especially, time-global smoothing estimates for the operator $a(D_x)$ with lower order terms are the benefit of our new method. These estimates are known to fail for general non-dispersive operators. For the case when the dispersiveness breaks, we suggest a modification of the smoothing estimate. It is equivalent to the usual estimate in the dispersive case and is also invariant under canonical transformations for the operator $a(D_x)$. Moreover, it does continue to hold for a variety of non-dispersive operators $a(D_x)$, where $\nabla a(\xi)$ may become zero on some set. It is interesting that this method allows us to carry out a global microlocal reduction of equations to the translation invariance property of the Lebesgue measure.Comment: 13 page

Sharp Morawetz estimates

We prove sharp Morawetz estimates - global in time with a singular weight in the spatial variables - for the linear waves. Klein Gordon and Schrodinger equations, for which we can characterise the maximisers. We also prove refined inequalities with respect to the angular integrability.Partially supported by the ERC grant 277778.Peer reviewe