  We study the inhomogeneous incompressible Navier-Stokes system endowed with a
general capillary term. Thanks to recent methods based on Lagrangian change of
variables, we obtain local well-posedness in critical Besov spaces (even if the
integration index p is different from 2) and for variable viscosity and
capillary terms. In the case of constant coefficients and for initial data that
are perturbations of a constant state, we are able to prove that the lifespan
goes to infinity as the capillary coefficient goes to zero, connecting our
result to the global existence result obtained by Danchin and Mucha for the
incompressible Navier-Stokes system with constant coefficients.
