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This work is devoted to prove the existence of global strong solution indimension $N\geq 2$ for a isothermal model of capillary fluids derived by J.EDunn and J.Serrin (1985) (see \cite{fDS}), which can be used as a phasetransition model. We will restrict us to the case of the so called compressibleNavier-Stokes system with quantum pressure which corresponds to consider thecapillary coefficient $\kappa(\rho)=\frac{\kappa_1}{\rho}$ with $\kappa_1>0$.In a first part we prove the existence of strong solution in finite time forlarge initial data with a precise bound by below on the life span $T^*$. Thisone depends on the norm of the initial data $(\rho_0,v_0)$. The second partconsists in proving the existence of global strong solution with particularchoice on the capillary coefficient ( where $\kappa_1=\mu^2$) and on theviscosity tensor which corresponds to the viscous shallow water case $-2\mu{\rmdiv}(\rho Du)$. To do this we derivate different energy estimate on the densityand the effective velocity $v$ which ensures that the strong solution can beextended beyond $T^*$. The main difficulty consists in controlling the vacuumor in other words to estimate the $L^\infty$ norm of $\frac{1}{\rho}$. Theproof relies mostly on a method introduced by De Giorgi \cite{DG} (see alsoLadyzhenskaya et al in \cite{La} for the parabolic case) to obtain regularityresults for elliptic equations with discontinuous diffusion coefficients and asuitable bootstrap argument.

Global strong solution for the Korteweg system with quantum pressure in dimension $N\geq 2$