Motivated by applications to stochastic differential equations, an extension of H{o}rmanders hypoellipticity theorem is proved for second-order degenerate elliptic operators with non-smooth coefficients. The main results are established using point-w
ise Bessel kernel estimates and a weighted Sobolev inequality of Stein and Weiss. Of particular interest is that our results apply to operators with quite general first-order terms.
We consider small amplitude wave packet-like solutions to the 3D inviscid incompressible irrotational infinite depth water wave problem neglecting surface tension. Formal multiscale calculations suggest that the modulation of such a solution is descr
ibed by a profile traveling at group velocity and governed by a hyperbolic cubic nonlinear Schrodinger equation. In this paper we show that, given wave packet initial data, the corresponding solution exists and retains the form of a wave packet on natural NLS time scales. Moreover, we give rigorous error estimates between the true and formal solutions on the appropriate time scale in Sobolev spaces using the energy method. The proof proceeds by directly applying modulational analysis to the formulation of the 3D water wave problem developed by Sijue Wu.
We consider the 2D inviscid incompressible irrotational infinite depth water wave problem neglecting surface tension. Given wave packet initial data, we show that the modulation of the solution is a profile traveling at group velocity and governed by
a focusing cubic nonlinear Schrodinger equation, with rigorous error estimates in Sobolev spaces. As a consequence, we establish existence of solutions of the water wave problem in Sobolev spaces for times in the NLS regime provided the initial data is suitably close to a wave packet of sufficiently small amplitude in Sobolev spaces.
