Lagrangian Methods for a General Inhomogeneous Incompressible Navier--Stokes--Korteweg System with Variable Capillarity and Viscosity Coefficients

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 $p\neq 2$) 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.