This study considers the spatially periodic initial value problem of 2 × 2 quasilinear
parabolic systems in one space dimension having quadratic polynomial
flux functions. These systems arise physically in the interfacial dynamics
of viscous immiscible multilayer channel flows. The equations describe
the spatiotemporal evolution of phase-separating interfaces with dissipation
arising from surface tension (fourth-order) and/or stable stratification effects
(second-order). A crucial mathematical aspect of these systems is the presence
of mixed hyperbolic–elliptic flux functions that provide the only source of
instability. The study concentrates on scaled spatially 2π-periodic solutions
as the dissipation vanishes, and in particular the behaviour of such limits
when generalized dissipation operators (spanning second to fourth-order) are
considered. Extensive numerical computations and asymptotic analysis suggest
that the existence (or not) of bounded vanishing viscosity solutions depends
crucially on the structure of the flux function. In the absence of linear
terms (i.e. homogeneous flux functions) the vanishing viscosity limit does
not exist in the L∞-norm. On the other hand, if linear terms in the flux
function are present the computations strongly suggest that the solutions exist
and are bounded in the L∞-norm as the dissipation vanishes. It is found
that the key mechanism that provides such boundedness centres on persistent
spatiotemporal hyperbolic–elliptic transitions. Strikingly, as the dissipation
decreases, the flux function becomes almost everywhere hyperbolic except on
a fractal set of elliptic regions, whose dimension depends on the order of the
regularized operator. Furthermore, the spatial structures of the emerging weak
solutions are found to support an increasing number of discontinuities (measurevalued
solutions) located in the vicinity of the fractally distributed elliptic
regions. For the unscaled problem, such spatially oscillatory solutions can be realized as extensive dynamics analogous to those found in the Kuramoto–
Sivashinsky equation.