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Nonnegative weak solutions of thin-film equations related to viscous flows in cylindrical geometries

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Abstract

Motivated by models for thin films coating cylinders in two physical cases proposed in Kerchman (J Fluid Mech 290:131–166, 1994) and Kerchman and Frenkel (Theor Comput Fluid Dyn 6:235–254, 1994), we analyze the dynamics of corresponding thin-film models. The models are governed by nonlinear, fourth-order, degenerate, parabolic PDEs. We prove, given positive and suitably regular initial data, the existence of weak solutions in all length scales of the cylinder, where all solutions are only local in time. We also prove that given a length constraint on the cylinder, long time and global in time weak solutions exist. This analytical result is motivated by numerical work on related models of Reed Ogrosky (Modeling liquid film flow inside a vertical tube, Ph.D. thesis, The University of North Carolina at Chapel Hill, 2013) in conjunction with the works (Camassa et al. in Phys Rev E 86(6):066305, 2012; Physica D Nonlinear Phenom 333:254–265, 2016; J Fluid Mech 745:682–715, 2014; J Fluid Mech 825:1056–1090, 2017).

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Acknowledgements

The authors thank Roberto Camassa and Reed Ogrosky for many helpful conversations about this family of thin-film models. We also thank the reviewers for their interest, insight, and comments in critiquing the work during the review process. JLM and SRS were supported in part by the NSF through JLM’s NSF CAREER Grant DMS-1352353.

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Authors and Affiliations

  1. Department of Mathematics, University of North Carolina at Chapel Hill, CB#3250 Phillips Hall, Chapel Hill, NC,  27599, USA

    Jeremy L. Marzuola & Sterling R. Swygert

  2. Institute of Applied Mathematics and Mechanics of the NASU, 1, Dobrovolśkogo Str., Sloviansk, 84100, Ukraine

    Roman Taranets

Authors
  1. Jeremy L. Marzuola
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  2. Sterling R. Swygert
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  3. Roman Taranets
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Correspondence to Sterling R. Swygert.

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Marzuola, J.L., Swygert, S.R. & Taranets, R. Nonnegative weak solutions of thin-film equations related to viscous flows in cylindrical geometries. J. Evol. Equ. 20, 1227–1249 (2020). https://doi.org/10.1007/s00028-019-00553-1

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