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

Christina Lienstromberg and Juan J. L. Velázquez

Vol. 6 (2024), No. 1, 187–236
DOI: 10.2140/paa.2024.6.187

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 1t1β for some β > 0 to a circle, as time tends to infinity.

In addition, we provide regularity estimates for general nonlinear degenerate parabolic equations of fourth order.

Taylor–Couette flow, non-Newtonian fluid, power-law fluid, degenerate-parabolic equation, weak-solution, long-time asymptotics, polynomial stability, thin-film equation, regularity estimates
Mathematical Subject Classification
Primary: 35B40, 35K35, 35Q35, 76A05, 76A20
Secondary: 35K65
Received: 14 February 2023
Revised: 6 October 2023
Accepted: 18 November 2023
Published: 22 February 2024
Christina Lienstromberg
Institute for Analysis, Dynamics and Modeling
University of Stuttgart
Juan J. L. Velázquez
Institute for Applied Mathematics
University of Bonn