Global well-posedness of the Euler-Korteweg system for small irrotational data

The Euler-Korteweg equations are a modification of the Euler equations that takes into account capillary effects. In the general case they form a quasi-linear system that can be recast as a degenerate Schr\"odinger type equation. Local well-posedness (in subcritical Sobolev spaces) was obtained by Benzoni-Danchin-Descombes in any space dimension, however, except in some special case (semi-linear with particular pressure) no global well-posedness is known. We prove here that under a natural stability condition on the pressure, global well-posedness holds in dimension $d\geq 3$ for small irrotational initial data. The proof is based on a modified energy estimate, standard dispersive properties if $d\geq 5$, and a careful study of the nonlinear structure of the quadratic terms in dimension $3$ and $4$ involving the theory of space time resonance.