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This work is devoted to the study of the initial boundary value problem for ageneral isothermal model of capillary fluids derived by J.E Dunn and J.Serrin(1985), which can be used as a phase transition model. We will prove theexistence of local and global (under a condition of smallness on the initialdata) strong solutions with discontinuous initial density when $\ln\rho_{0}$belongs in the Besov space $B^{\N}_{2,\infty}(\R^{N})$.The main difficultyconcerns the proof of new estimate of maximum principle type for the linearsystem associated to the Korteweg system, the proof is based on acharacterization of the Besov space in terms of the semi group associated tothis linear system. Let also point out that we prove the existence of globalstrong solution with a smallness hypothesis which is subcritical in terms ofthe scaling of the equations, it allows us to exhibit a family of large energyinitial data for the scaling of the equations providing global strong solution.In particular for the first time up our knowledge we show the existence ofglobal strong solution for some large energy initial data when N=2. We finishthis paper by introducing the notion of quasi-solutions for the Korteweg'ssystem (a tool which has been developed in the framework of the compressibleNavier-Stokes equations \cite{arxiv,arxiv1,hal,cras1,cras2}) which enables usto improve the previous result and to obtain the existence of global strongsolution with large initial velocity in $B^{\N-1}_{2,\infty}$. As a corollary,we get global existence (and uniqueness) for highly compressible Kortewegsystem when $N\geq2$.

Existence of global strong solution for Korteweg system with large infinite energy initial data