We prove, for the energy critical, focusing NLS, that for data whose energy is smaller than that of the standing wave, and whose homogeneous Sobolev norm H^1 is smaller than that of the standing wave… Expand

u(x, 0) = u0(x), where u0 ∈ H(R). Our principal aim here is to lower the best index s for which one has local well posedness in H(R), i.e. existence, uniqueness, persistence and continuous dependence… Expand

(1.1) &ItU + axu + U1xU = O, x, t E R { u(x, 0) = uo(x). The KdV equation, which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves [21], has been… Expand

This paper is concerned with oscillatory integrals and their relationship with the local and global smoothing properties of dispersive equations. Also we shall study some applications of these… Expand

and the pi are additionally restricted by the requirement r > 23 (instead of r > 12 ). The condition r > 2 3 is a consequence of the methods used in their proof, and may not be necessary; these… Expand

Introduction Divergence form elliptic equations Some classes of examples and their perturbation theory Epilogue: Some further results and open problems References.