Long-time asymptotics and regularity estimates for weak solutions to a doubly degenerate thin-film equation in the Taylor-Couette setting

Abstract

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 1∕ 1∕ for some > 0 to a circle, as time tends to infinity.In addition, we provide regularity estimates for general nonlinear degenerate parabolic equations of fourth order.

Long-time behaviour and stability for quasilinear doubly degenerate parabolic equations of higher order

“…In the cylindrical Taylor-Couette setting, statement (iii) has first been shown in [PV20] in the framework of stable center manifolds. Similarly, the results in (i) and (ii) have been obtained in [LPV22] and [LV22], also in the cylindrical Taylor-Couette geometry.…”

“…Then, we derive 1 -in-time regularity estimates for the weak solution to (3.1). The results are the same as in the cylindrical Taylor-Couette setting in [LPV22;LV22]. However, since the present paper deals with the flat case, the proofs cannot rely on Fourier analysis.…”

“…Note that convergence to equilibrium has already been proved for the global non-negative weak solutions constructed in [AG04], but with no rate of convergence. The result on the rate of convergence is the same as in [LV22] for the cylindrical Taylor-Couette setting. The proof relies again on the differential inequality for the energy, derived in Proposition 4.1.…”

“…Next, we turn to 1 -in-time bounds for the dissipation functional in terms of the energy. The proof is a simplified version of the one in [LV22] for general degenerate parabolic problems of fourth order. In our case, it relies on testing the partial differential equation with a time cut-off of the second spatial derivative.…”