GlobalWell-Posedness of the Euler-Korteweg System for Small Irrotational Data

The Euler-Korteweg equations are a modification of the Euler equations that take into account capillary effects. In the general case they form a quasi-linear system that can be recast as a degenerate Schrodinger 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 >= 3 for small irrotational initial data. The proof is based on a modified energy estimate, standard dispersive properties if d >= 5, and a careful study of the structure of quadratic nonlinearities in dimension 3 and 4, involving the method of space time resonances.