Non-Newtonian two-phase thin-film problem: local existence, uniqueness, and stability

We study the flow of two immiscible fluids located on a solid bottom, where the lower fluid is Newtonian and the upper fluid is a non-Newtonian Ellis fluid. Neglecting gravitational effects, we consider the formal asymptotic limit of small film heights in the two-phase Navier–Stokes system. This leads to a strongly coupled system of two parabolic equations of fourth order with merely Hölder-continuous dependence on the coefficients. For the case of strictly positive initial film heights we prove local existence of strong solutions by abstract semi-group theory. Uniqueness is proved by energy methods. Under additional regularity assumptions, we investigate asymptotic stability of the unique equilibrium solution, which is given by constant film heights.