Vanishing viscosity limits of mixed hyperbolic-elliptic systems arising in multilayer channel flows

摘要

This study considers the spatially periodic initial value problem of 2x2 quasi-linear 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 2p-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-infinity-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-infinity-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 (measure-valued 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.