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Steady states and dynamics of a thin-film-type equation with non-conserved mass

Part of: General higher-order equations and systems Parabolic equations and systems Foundations, constitutive equations, rheology Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  22 November 2019

HANGJIE JI Open the ORCID record for HANGJIE JI [Opens in a new window]  and
THOMAS P. WITELSKI Open the ORCID record for THOMAS P. WITELSKI [Opens in a new window]
HANGJIE JI
Affiliation:
Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095, USA email: hangjie@math.ucla.edu
THOMAS P. WITELSKI
Affiliation:
Department of Mathematics, Duke University, Durham, NC 27708-0320, USA email: witelski@math.duke.edu
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Abstract

We study the steady states and dynamics of a thin-film-type equation with non-conserved mass in one dimension. The evolution equation is a non-linear fourth-order degenerate parabolic partial differential equation (PDE) motivated by a model of volatile viscous fluid films allowing for condensation or evaporation. We show that by changing the sign of the non-conserved flux and breaking from a gradient flow structure, the problem can exhibit novel behaviours including having two distinct classes of co-existing steady-state solutions. Detailed analysis of the bifurcation structure for these steady states and their stability reveals several possibilities for the dynamics. For some parameter regimes, solutions can lead to finite-time rupture singularities. Interestingly, we also show that a finite-amplitude limit cycle can occur as a singular perturbation in the nearly conserved limit.

Keywords

Thin-film equationmodified Allen–Cahn/Cahn–Hilliard equationnon-conserved modelfourth-order parabolic partial differential equations

MSC classification

Primary: 35K25: Higher-order parabolic equations
Secondary: 35K55: Nonlinear parabolic equations 76A20: Thin fluid films 35Q35: PDEs in connection with fluid mechanics 35G20: Nonlinear higher-order equations
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 6 , December 2020 , pp. 968 - 1001
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Copyright
© Cambridge University Press 2019

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