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Undercompressive shocks in thin film flows. (English) Zbl 1076.76509
Summary: Equations of the type ${h}_{t}+{\left({h}^{2}-{h}^{3}\right)}_{x}=-{\epsilon }^{3}{\left({h}^{3}{h}_{xxx}\right)}_{x}$ arise in the context of thin liquid films driven by the competing effects of a thermally induced surface tension gradient and gravity. In this paper, we focus on the interaction between the fourth-order regularization and the nonconvex flux. Jump initial data, from a moderately thick film to a thin precurser layer, is shown to give rise to a double wave structure that includes an undercompressive wave. This wave, which approaches an undercompressive shock as $\epsilon \to 0$, is an accumulation point for a countable family of compressive waves having the same speed. The family appears through a series of bifurcations parameterized by the shock speed. At each bifurcation, a pair of traveling waves is produced, one being stable for the PDE, the other unstable. The conclusions are based primarily on numerical results for the PDE, and on numerical investigations of the ODE describing traveling waves. Fourth-order linear regularization is observed to produce a similar bifurcation structure of traveling waves.
##### MSC:
 76A20 Thin fluid films (fluid mechanics) 76L05 Shock waves; blast waves (fluid mechanics) 76D45 Capillarity (surface tension)