  Under a natural stability condition on the pressure, it is known that for
small irrotational initial data, the solutions of the Euler-Korteweg system are
global in time. When the initial velocity has a small rotational part, we
obtain a lower bound on the time of existence that depends only on the
rotational part. In the zero vorticity limit we recover the previous global
well-posedness result. Independently of this analysis, we also provide (in a
special case) a simple example of solution that blows up in finite time.
