Ukrainian Mathematical Journal

, Volume 71, Issue 6, pp 956–969 | Cite as

Finite Speed of Propagation for the Thin-Film Equation in Spherical Geometry

  • R. M. TaranetsEmail author

We show that a doubly degenerate thin-film equation obtained in modeling the flows of viscous coatings on spherical surfaces has a finite speed of propagation for nonnegative strong solutions and, hence, there exists an interface or a free boundary separating the regions, where the solution u > 0 and u = 0. By using local entropy estimates, we also establish the upper bound for the rate of propagation of the interface.


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  1. 1.
    E. Beretta, M. Bertsch, and R. Dal Passo, “Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation,” Arch. Ration. Mech. Anal., 129, No. 2, 175–200 (1995).MathSciNetCrossRefGoogle Scholar
  2. 2.
    F. Bernis and A. Friedman, “Higher order nonlinear degenerate parabolic equations,” J. Different. Equat., 83, No. 1, 179–206 (1990).MathSciNetCrossRefGoogle Scholar
  3. 3.
    F. Bernis, “Finite speed of propagation and continuity of the interface for thin viscous flows,” Adv. Different. Equat., 1, No. 3, 337–368 (1996).MathSciNetzbMATHGoogle Scholar
  4. 4.
    F. Bernis, “Finite speed of propagation for thin viscous flows when 2 ≤ n < 3,” Compt. Rendus Acad. Sci. Math., 322, No. 12, 1169–1174 (1996).MathSciNetzbMATHGoogle Scholar
  5. 5.
    F. Bernis, L. A. Peletier, and S. M. Williams, “Source type solutions of a fourth order nonlinear degenerate parabolic equation,” Nonlin. Anal., 18, 217–234 (1992).MathSciNetCrossRefGoogle Scholar
  6. 6.
    A. L. Bertozzi, et al., “Singularities and similarities in interface flows,” in: Trends and Perspectives in Applied Mathematics, Springer, New York (1994), pp. 155–208.CrossRefGoogle Scholar
  7. 7.
    M. Chugunova, M. C. Pugh, and R. M. Taranets, “Nonnegative solutions for a long-wave unstable thin film equation with convection,” SIAM J. Math. Anal., 42, No. 4, 1826–1853 (2010).MathSciNetCrossRefGoogle Scholar
  8. 8.
    M. Chugunova and R. M. Taranets, “Qualitative analysis of coating flows on a rotating horizontal cylinder,” Int. J. Different. Equat., Article ID 570283 (2012), 30 p.Google Scholar
  9. 9.
    A. Friedman, “Interior estimates for parabolic systems of partial differential equations,” J. Math. Mech., 7, No. 3, 393–417 (1958).MathSciNetzbMATHGoogle Scholar
  10. 10.
    J. Hulshof and A. E. Shishkov, “The thin film equation with 2 ≤ n < 3: finite speed of propagation in terms of the L 1-norm,” Adv. Different. Equat., 3, No. 5, 625–642 (1998).zbMATHGoogle Scholar
  11. 11.
    D. Kang, A. Nadim, and M. Chugunova, “Dynamics and equilibria of thin viscous coating films on a rotating sphere,” J. Fluid Mech., 791, 495–518 (2016).MathSciNetCrossRefGoogle Scholar
  12. 12.
    D. Kang, A. Nadim, and M. Chugunova, “Marangoni effects on a thin liquid film coating a sphere with axial or radial thermal gradients,” Phys. Fluids, 29, 072106-1–072106-15 (2017).Google Scholar
  13. 13.
    D. Kang, T. Sangsawang, and J. Zhang, “Weak solution of a doubly degenerate parabolic equation,” arXiv:1610.06303v2 (2017).Google Scholar
  14. 14.
    L. Nirenberg, “An extended interpolation inequality,” Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 20, No. 3, 733–737 (1966).Google Scholar
  15. 15.
    A. E. Shishkov, “Dynamics of the geometry of the support of the generalized solution of a higher-order quasilinear parabolic equation in the divergence form,” Differents. Uravn., 29, No. 3, 537–547 (1993).MathSciNetGoogle Scholar
  16. 16.
    R. M. Taranets, “Strong solutions of the thin-film equation in spherical geometry,” in: V. A. Sadovnichiy and M. Zgurovsky (eds.), Modern Mathematics and Mechanics, Understanding Complex Systems, Springer (2019), pp. 181–192.Google Scholar
  17. 17.
    D. Takagi and H. E. Huppert, “Flow and instability of thin films on a cylinder and sphere,” J. Fluid Mech., 647, 221–238 (2010).MathSciNetCrossRefGoogle Scholar
  18. 18.
    S. K. Wilson, “The onset of steady Marangoni convection in a spherical geometry,” J. Eng. Math., 28, 427–445 (1994).MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Institute of Applied Mathematics and MechanicsUkrainian National Academy of SciencesSlovianskUkraine

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