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

This work is devoted to prove the existence of global strong solution in dimension $N\geq 2$ for a isothermal model of capillary fluids derived by J.E Dunn and J.Serrin (1985) (see \cite{fDS}), which can be used as a phase transition model. We will restrict us to the case of the so called compressible Navier-Stokes system with quantum pressure which corresponds to consider the capillary 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 for large initial data with a precise bound by below on the life span $T^*$. This one depends on the norm of the initial data $(\rho_0,v_0)$. The second part consists in proving the existence of global strong solution with particular choice on the capillary coefficient ( where $\kappa_1=\mu^2$) and on the viscosity tensor which corresponds to the viscous shallow water case $-2\mu{\rm div}(\rho Du)$. 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 $L^\infty$ norm of $\frac{1}{\rho}$. The proof relies mostly on a method introduced by De Giorgi \cite{DG} (see also Ladyzhenskaya et al in \cite{La} for the parabolic case) to obtain regularity results for elliptic equations with discontinuous diffusion coefficients and a suitable bootstrap argument.