**Abstract:** In this lecture, we discuss stability and asymptotic behavior of solutions of phase-transitional models arising in van der Waals gas dynamics and elasticity and in multiphase flow in porous media. Such models present a rich solution structure resembling that found in their cousin Cahn--Hilliard or Allen--Cahn models (reaction--diffusion models, vs. the convection--diffusion models we study), including pulse-type and spatially periodic traveling waves along with the more usual front-type (shock) solutions expected in compressible flow. Likewise, they exhibit behaviors of pattern formation, nucleation, etc. similar to that found in the reaction-diffusion setting. On the other hand, the equations retain the technical difficulty of compressible gas dynamics; in particular, we cannot determine stability by variational principles alone, as for the gradient flow models mentioned above. Instead, we make use of a periodic Evans function introduced recently by R.A. Gardner to study directly the stability of spatially periodic traveling waves. However, we use it in a different way than it was used by Gardner, exploring stability by geometric considerations rather than an asymptotic limit, in the spirit of the original work of J. Evans in the context of traveling front- or pulse-type solutions.