Long-Time Behavior and Stability for Quasilinear Doubly Degenerate Parabolic Equations of Higher Order

Jansen, Jonas; Lienstromberg, Christina; Nik, Katerina

doi:10.1137/22m1491137

SIAM J. Math. Anal.

2023

DOI: 10.1137/22m1491137

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Long-Time Behavior and Stability for Quasilinear Doubly Degenerate Parabolic Equations of Higher Order

Jonas Jansen

Christina Lienstromberg

Katerina Nik

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2024

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Cited by 1 publication

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A Thin Film Model for Meniscus Evolution

Ghosh,

Velázquez

2024

J. Math. Fluid Mech.

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In this paper, we discuss a particular model arising from sinking of a rigid solid into a thin film of liquid, i.e. a liquid contained between two solid surfaces and part of the liquid surface is in contact with the air. The liquid is governed by Navier–Stokes equation, while the contact point, i.e. where the gas, liquid and solid meet, is assumed to be given by a constant, non-zero contact angle. We consider a scaling limit of the liquid thickness (lubrication approximation) and the contact angle between the liquid–solid and the liquid–gas interfaces close to $$\pi $$ π . This resulting model is a free boundary problem for the equation $$h_t + (h^3h_{xxx})_x = 0$$ h t + ( h 3 h xxx ) x = 0 , for which we have $$h>0$$ h > 0 at the contact point (different from the usual thin film equation with $$h=0$$ h = 0 at the contact point). We show that this fourth order quasilinear (non-degenerate) parabolic equation, together with the so-called partial wetting condition at the contact point, is well-posed. Furthermore, the contact point in our thin film equation can actually move, contrary to the classical thin film equation for a droplet arising from the no-slip condition. Additionally, we show the global stability of steady state solutions in a periodic setting.

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A Thin Film Model for Meniscus Evolution

Ghosh,

Velázquez

2024

J. Math. Fluid Mech.

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Long-Time Behavior and Stability for Quasilinear Doubly Degenerate Parabolic Equations of Higher Order

Cited by 1 publication

References 22 publications

A Thin Film Model for Meniscus Evolution

A Thin Film Model for Meniscus Evolution

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