Existence of solutions for compressible fluid models of Korteweg type

The purpose of this work is to prove existence and uniqueness results of suitably smooth solutions for an isothermal model of capillary compressible fluids derived by J.E. Dunn and J. Serrin (1985), which can be used as a phase transition model. We first study the well-posedness of the model in spaces with critical regularity indices with respect to the scaling of the associated equations. In a functional setting as close as possible to the physical energy spaces, we prove global existence of solutions close to a stable equilibrium, and local in time existence for solutions when the pressure law may present spinodal regions. Uniqueness is also obtained. Assuming a lower and upper control of the density, we also show the existence of weak solutions in dimension 2 near equilibrium. Finally, referring to the work of Z. Xin (1998) in the non-capillary case, we describe some blow-up properties of smooth solutions with finite total mass, (C) 2001 Editions scientifiques et medicales Elsevier SAS.