We study the dynamic behaviour of solutions to a fourth-order quasilinear degenerate
parabolic equation for large times arising in fluid dynamical applications. The
degeneracy occurs both with respect to the unknown and with respect to the sum of
its first and third spatial derivative. The modelling equation appears as a
thin-film limit for the interface separating two immiscible viscous fluid films
confined between two cylinders rotating at small relative angular velocity.
More precisely, the fluid occupying the layer next to the outer cylinder is
considered to be Newtonian, i.e., it has constant viscosity, while we assume that
the layer next to the inner cylinder is filled by a shear-thinning power-law
fluid.
Using energy methods, Fourier analysis and suitable regularity estimates for
higher-order parabolic equations, we prove global existence of positive weak solutions
in the case of low initial energy. Moreover, these global solutions are polynomially stable,
in the sense that interfaces which are initially close to a circle, converge at rate
for
some
to a circle, as time tends to infinity.
In addition, we provide regularity estimates for general nonlinear degenerate
parabolic equations of fourth order.
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