The purpose of this work is to investigate the initial value problem for a general isothermal model of capillary fluids derived by Dunn and Serrin [12],which can be used as a phase transition model.Motivated by [9],we aim at extending the work by Danchin-Desjardins [11] to a critical framework which is not related to the energy space.For small perturbations of a stable equilibrium state in the sense of suitable Lp-type Besov norms,we establish the global existence.As a consequence,like for incompressible flows,one may exhibit a class of large highly oscillating initial velocity fields for which global existence and uniqueness holds true.