This work is devoted to prove the existence of global strong solution in dimension N ≥ 2 for a general isothermal model of capillary fluids derived by J.E Dunn and J.Serrin (1985) (see [16]), which can be used as a phase transition model. In a first part we prove the existence of strong solution in finite time for large initial data with a precise bound by below on the life span T * of existence of the solution. This one depends on the norm of the initial data (ρ 0 , v 0). The second part consists in proving the existence of global strong solution with particular choice on the capillary coefficient κ(ρ) = µ 2 ρ and on the viscosity tensor which corresponds to the shallow water case −2µdiv(ρDu). To do this we derivate different energy estimate on the density and the effective velocity v which ensures that the strong solution can be extended beyond T * . The main difficulty consists in controlling the vacuum or in other words to estimate the L ∞ norm of 1 ρ . The proof relies mostly on a method introduced by De Giorgi [14] (see also Ladyzhenskaya et al in [30] for the parabolic case) to obtain regularity results for elliptic equations with discontinuous diffusion coefficients and a suitable bootstrap argument.