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Analytical and numerical results on the positivity of steady state solutions of a thin film equation
1.  Department of Mathematics, University of Toronto, Toronto, Canada 
2.  School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, United States 
References:
[1] 
R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," Academic Press, New YorkLondon, 2003 
[2] 
J. Ashmore, A. E. Hosoi and H. A. Stone, The effect of surface tension on rimming flows in a partially filled rotating cylinder, Journal of Fluid Mechanics, 479 (2003), 6598. 
[3] 
D. Badali, M. Chugunova, D. Pelinovsky and S. Pollack, Regularized shock solutions in coating flows with small surface tension, Physics of Fluids, 23 (2011), 093103. 
[4] 
B. Jürgen and G. Günther, The thinfilm equation: Recent advances and some new perspectives, Journal of Physics: Condensed Matter, 17 (2005), S291S306. 
[5] 
E. Benilov, M. Benilov and N. Kopteva, Steady rimming flows with surface tension, Journal of Fluid Mechanics, 597 (2008), 91118. doi: 10.1017/S0022112007009585. 
[6] 
E. Beretta, M. Bertsch and R. Dal Passo, Nonnegative solutions of a fourthorder nonlinear degenerate parabolic equation, Arch. Rational Mech. Anal., 129 (1995), 175200. doi: 10.1007/BF00379920. 
[7] 
F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, Journal of Diff. Equations, 83 (1990), 179206. doi: 10.1016/00220396(90)90074Y. 
[8] 
A. L. Bertozzi and M. C. Pugh, The lubrication approximation for thin viscous films: The moving contact line with a "porous media'' cutoff of van der Waals interactions, Nonlinearity, 7 (1994), 15351564. 
[9] 
A. Burchard, M. Chugunova and B. Stephens, Convergence to equilibrium for a thinfilm equation on a cylindrical surface, Comm. Partial Diff. Equations, 37 (2012), 585609. doi: 10.1080/03605302.2011.648704. 
[10] 
M. Chugunova, M. C. Pugh and R. M. Taranets, Nonnegative solutions for a longwave unstable thin film equation with convection, SIAM J. on Math. Anal., 42 (2010), 18261853. doi: 10.1137/090777062. 
[11] 
R. V. Craster and O. K. Matar, Dynamics and stability of thin liquid films, Rev. Modern Physics, 81 (2009), 11311198. 
[12] 
E. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Cong. Num., 30 (1981), 265284. 
[13] 
L. C. Evans, "Partial Differential Equations," 2nd edition, American Mathematical Society, Providence, 2010. 
[14] 
A. Oron and S. G. Bankoff, Longscale evolution of thin liquid films, Rev. Modern Physics, 69 (1997), 931980. 
[15] 
K. Pougatch and I. Frigaard, Thin film flow on the inside surface of a horizontally rotating cylinder: Steady state solutions and their stability, Physics of Fluids, 23 (2011), 022102. 
[16] 
V. V. Pukhnachev, Motion of a liquid film on the surface of a rotating cylinder in a gravitational field, J. of App. Mech. and Tech. Physics, 18 (1977), 244351. 
[17] 
V. V. Pukhnachev, Asymptotic solution of the rotating film problem, Izv. Vyssh. Uchebn. Zaved. SeveroKavkaz. Reg. Estestv. Nauk, "Mathematics and Continuum Mechanics", (2004), 191199. 
[18] 
V. V. Pukhnachev, On the equation of a rotating film, Siberian Math. J., 46 (2005), 913924. doi: 10.1007/s1120200500889. 
[19] 
A. E. Shishkov and R. M. Taranets, On the equation of the flow of thin films with nonlinear convection in multidimensional domains, Ukr. Math. Bull., 1 (2004), 402444. 
[20] 
R. M. Taranets and A. E. Shishkov, A singular Cauchy problem for the equation of the flow of thin viscous films with nonlinear convection, Ukr. Math. J., 58 (2006), 250271. doi: 10.1007/s1125300600669. 
[21] 
L. N. Trefethen, "Spectral Methods in MATLAB," SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719598. 
show all references
References:
[1] 
R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," Academic Press, New YorkLondon, 2003 
[2] 
J. Ashmore, A. E. Hosoi and H. A. Stone, The effect of surface tension on rimming flows in a partially filled rotating cylinder, Journal of Fluid Mechanics, 479 (2003), 6598. 
[3] 
D. Badali, M. Chugunova, D. Pelinovsky and S. Pollack, Regularized shock solutions in coating flows with small surface tension, Physics of Fluids, 23 (2011), 093103. 
[4] 
B. Jürgen and G. Günther, The thinfilm equation: Recent advances and some new perspectives, Journal of Physics: Condensed Matter, 17 (2005), S291S306. 
[5] 
E. Benilov, M. Benilov and N. Kopteva, Steady rimming flows with surface tension, Journal of Fluid Mechanics, 597 (2008), 91118. doi: 10.1017/S0022112007009585. 
[6] 
E. Beretta, M. Bertsch and R. Dal Passo, Nonnegative solutions of a fourthorder nonlinear degenerate parabolic equation, Arch. Rational Mech. Anal., 129 (1995), 175200. doi: 10.1007/BF00379920. 
[7] 
F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, Journal of Diff. Equations, 83 (1990), 179206. doi: 10.1016/00220396(90)90074Y. 
[8] 
A. L. Bertozzi and M. C. Pugh, The lubrication approximation for thin viscous films: The moving contact line with a "porous media'' cutoff of van der Waals interactions, Nonlinearity, 7 (1994), 15351564. 
[9] 
A. Burchard, M. Chugunova and B. Stephens, Convergence to equilibrium for a thinfilm equation on a cylindrical surface, Comm. Partial Diff. Equations, 37 (2012), 585609. doi: 10.1080/03605302.2011.648704. 
[10] 
M. Chugunova, M. C. Pugh and R. M. Taranets, Nonnegative solutions for a longwave unstable thin film equation with convection, SIAM J. on Math. Anal., 42 (2010), 18261853. doi: 10.1137/090777062. 
[11] 
R. V. Craster and O. K. Matar, Dynamics and stability of thin liquid films, Rev. Modern Physics, 81 (2009), 11311198. 
[12] 
E. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Cong. Num., 30 (1981), 265284. 
[13] 
L. C. Evans, "Partial Differential Equations," 2nd edition, American Mathematical Society, Providence, 2010. 
[14] 
A. Oron and S. G. Bankoff, Longscale evolution of thin liquid films, Rev. Modern Physics, 69 (1997), 931980. 
[15] 
K. Pougatch and I. Frigaard, Thin film flow on the inside surface of a horizontally rotating cylinder: Steady state solutions and their stability, Physics of Fluids, 23 (2011), 022102. 
[16] 
V. V. Pukhnachev, Motion of a liquid film on the surface of a rotating cylinder in a gravitational field, J. of App. Mech. and Tech. Physics, 18 (1977), 244351. 
[17] 
V. V. Pukhnachev, Asymptotic solution of the rotating film problem, Izv. Vyssh. Uchebn. Zaved. SeveroKavkaz. Reg. Estestv. Nauk, "Mathematics and Continuum Mechanics", (2004), 191199. 
[18] 
V. V. Pukhnachev, On the equation of a rotating film, Siberian Math. J., 46 (2005), 913924. doi: 10.1007/s1120200500889. 
[19] 
A. E. Shishkov and R. M. Taranets, On the equation of the flow of thin films with nonlinear convection in multidimensional domains, Ukr. Math. Bull., 1 (2004), 402444. 
[20] 
R. M. Taranets and A. E. Shishkov, A singular Cauchy problem for the equation of the flow of thin viscous films with nonlinear convection, Ukr. Math. J., 58 (2006), 250271. doi: 10.1007/s1125300600669. 
[21] 
L. N. Trefethen, "Spectral Methods in MATLAB," SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719598. 
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