Résumé en anglais

This work is devoted to the study of the initial boundary value problem for a general isothermal model of capillary fluids derived by J.E. Dunn and J. Serrin (1985) (see [18]), which can be used as a phase transition model. We aim at proving the existence of local and global (under a condition of smallness on the initial data) strong solutions with initial density ln⁡ρ0ln⁡ρ0 belonging to the Besov space View the MathML sourceB2,∞N2. It implies in particular that some classes of discontinuous initial density generate strong solutions. The proof relies on the fact that the density can be written as the sum of the solution ρLρL of the associated linear system and a remainder term View the MathML sourceρ¯; this last term is more regular than ρLρL provided that we have regularizing effects induced on the bilinear convection term. The main difficulty consists in obtaining new estimates of maximum principle type for the associated linear system; this is based on a characterization of the Besov space in terms of the semi-group associated with this linear system. We show in particular the existence of global strong solution for small initial data in View the MathML source(B˜2,∞N2−1,N2∩L∞)×B2,∞N2−1; it allows us to exhibit a family of large energy initial data when N=2N=2 providing global strong solution. In conclusion we introduce the notion of quasi-solutions for the Korteweg's system (a tool which has been developed in the framework of the compressible Navier–Stokes equations [31], [30], [32], [26] and [27]) which enables to obtain the existence of global strong solution with a smallness condition which is subcritical. Indeed we can deal with large initial velocity in View the MathML sourceB2,1N2−1. As a corollary, we get global strong solution for highly compressible Korteweg system when N≥2N≥2. It means that for any large initial data (under an irrotational condition on the initial velocity) we have the existence of global strong solution provided that the Mach number is sufficiently large.