Some topics in hyperbolic and dispersive PDE

This thesis is devoted to the study of some problems in the theory of nonlinear partial differential equations of hyperbolic and dispersive type.;The first part is dedicated to free boundary Euler equations. We study a two-phase problem modeling the motion of two fluids with different constant densities, separated by a surface of discontinuity. We establish convergence to solutions of the one phase-problem for the motion of a single fluid in vacuum. This convergence is obtained by letting the density of one of the fluids and the surface tension on the interface tend to zero, under a suitable rate constraint.;In the rest of this thesis we investigate global existence and asymptotic behavior of solutions of some weakly dispersive equations. Our framework is the method of space-time resonances, introduced by Germain, Masmoudi, and Shatah. We apply and further develop this method, studying, in particular, the connection between resonances and null structures.;We treat two different problems. Firstly, we study some classes of first order systems of wave equations in three space dimensions. The problem we analyze originates from a classical result of Klainerman about global existence of solutions to quadratic wave equations in three space dimensions, under the assumption of the null condition. Here, we provide a new interpretation, and an extension to first order systems, of the null condition, that we call "null condition in the sense of space-time resonances". Under this condition we show global existence and scattering of small solutions. Since the method of vector fields does not seem to be applicable to our problem, we rely instead on a delicate analysis performed in frequency space.;Finally, we give a new simple proof of global existence and long range scattering for the gauge-invariant NLS equation on the real line, and for Hartree equations in dimension larger than one. The interesting feature of our approach is the identification of the phase correction term---which is necessary to prove sharp pointwise bounds and scattering---through a very natural stationary phase argument.