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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 strong solutions in finite time with discontinuous initialdensity, more precisely $\ln\rho_{0}$ is in $B^{\N}_{2,\infty}(\R^{N})$. Ouranalysis improves the results of \cite{fDD} and \cite{fH1}, \cite{fH2} byworking in space of infinite energy. In passing our result allow to considerinitial data with discontinuous interfaces, whereas in all the literature theresults of existence of strong solutions consider always initial density thatare continuous. More precisely we investigate the existence of strong solutionfor Korteweg's system when we authorize jump in the pressure across somehypersurface. We obtain also a result of ill-posedness for Korteweg system andwe derive a new blow-up criterion which is the main result of this paper. Moreprecisely we show that if we control the vacuum (i.e $\frac{1}{\rho}\inL^{\infty}_{T}(\dot{B}^{0}_{N+\e,1}(\R^{N}))$ with $\e>0$) then we can extendthe strong solutions in finite time. It extends substantially previous resultsobtained for compressible equations.

Blow-up criterion, ill-posedness and existence of strong solution for Korteweg system with infinite energy