Propagating phase boundaries and capillary fluids

The aim is to give an overview of recent advancements in the theory of Euler–Korteweg model for liquid-vapour mixtures. This model takes into account the surface tension of interfaces by means of a capillarity coefficient. The interfaces are not sharp fronts. Their width, even though extremely small for values of the capillarity compatible with the measured, physical surface tension, is nonzero. We are especially interested in nondissipative isothermal models, in which the viscosity of the fluid is neglected and therefore the (extended) free energy, depending on the density and its gradient, is a conserved quantity. From the mathematical point of view, the resulting conservation law for the momentum of the fluid involves a third order, dispersive term but no parabolic smoothing effect. We present recent results about well-posedness and propagation of solitary waves. Acknowledgements These notes have been prepared for the International Summer School on “Mathematical Fluid Dynamics”, held at Levico Terme (Trento), June 27th-July 2nd, 2010. They are based for a large part on a joint work with R. Danchin (Paris 12), S. Descombes (Nice), and D. Jamet (physicist at CEA Grenoble), on the doctoral thesis of C. Audiard (Lyon 1), and on discussions with J.-F. Coulombel (Lille 1), F. Rousset (Rennes), and N. Tzvetkov (Cergy-Pontoise).