RESUMEN

We show for the p-system with viscosity-capillarity that all heteroclinic phase transitions lying sufficiently close to the Maxwell line are (linearly) dynamically stable under localized perturbations, independent of the form of the flux. On the other hand, we show that nearby homoclinic phase transitions (involving two or more changes in phase) are exponentially unstable. These results together imply that stationary and slowly moving phase transitions are dynamically stable if and only if they are monotone, in analogy with results obtained by Carr, Gurtin, and Slemrod in the variational setting [Arch. Rational Mech. Anal., 86 (1984), pp. 317--351]. This research is significant as the first proof of stability of a strong viscous shock wave for systems. Our analysis uses the technical framework developed in [R. Gardner and K. Zumbrun, Comm. Pure Appl. Math., 51 (1998), pp. 797--855; K. Zumbrun and P. Howard, Indiana Univ. Math. J., 47 (1998), pp. 741--871] together with a nonstandard energy estimate.