Linear stability of propagating phase boundaries in capillary fluids

Dynamical phase changes occurring along planar traveling waves in capillary fluids are considered. Their multi-dimensional stability, which is encoded in the spectrum of a third order variational differential operator, is addressed by two approaches. On one hand, an original energy method is used to show that these diffuse interfaces are weakly stable. More precisely, the spectrum of the involved operator is found to coincide with the imaginary axis. On the other hand, an Evans function technique reveals a bifurcation phenomenon about zero, the interpretation of which still needs some clarification. It is the counterpart of the surface waves exhibited in an earlier work for the corresponding sharp interfaces.