-
Previous Article
The evolution of the orbit distance in the double averaged restricted 3-body problem with crossing singularities
- DCDS-B Home
- This Issue
-
Next Article
Traveling wave solutions for a diffusive sis epidemic model
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 York-London, 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), 65-98. |
[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 thin-film equation: Recent advances and some new perspectives, Journal of Physics: Condensed Matter, 17 (2005), S291-S306. |
[5] |
E. Benilov, M. Benilov and N. Kopteva, Steady rimming flows with surface tension, Journal of Fluid Mechanics, 597 (2008), 91-118.
doi: 10.1017/S0022112007009585. |
[6] |
E. Beretta, M. Bertsch and R. Dal Passo, Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation, Arch. Rational Mech. Anal., 129 (1995), 175-200.
doi: 10.1007/BF00379920. |
[7] |
F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, Journal of Diff. Equations, 83 (1990), 179-206.
doi: 10.1016/0022-0396(90)90074-Y. |
[8] |
A. L. Bertozzi and M. C. Pugh, The lubrication approximation for thin viscous films: The moving contact line with a "porous media'' cut-off of van der Waals interactions, Nonlinearity, 7 (1994), 1535-1564. |
[9] |
A. Burchard, M. Chugunova and B. Stephens, Convergence to equilibrium for a thin-film equation on a cylindrical surface, Comm. Partial Diff. Equations, 37 (2012), 585-609.
doi: 10.1080/03605302.2011.648704. |
[10] |
M. Chugunova, M. C. Pugh and R. M. Taranets, Nonnegative solutions for a long-wave unstable thin film equation with convection, SIAM J. on Math. Anal., 42 (2010), 1826-1853.
doi: 10.1137/090777062. |
[11] |
R. V. Craster and O. K. Matar, Dynamics and stability of thin liquid films, Rev. Modern Physics, 81 (2009), 1131-1198. |
[12] |
E. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Cong. Num., 30 (1981), 265-284. |
[13] |
L. C. Evans, "Partial Differential Equations," 2nd edition, American Mathematical Society, Providence, 2010. |
[14] |
A. Oron and S. G. Bankoff, Long-scale evolution of thin liquid films, Rev. Modern Physics, 69 (1997), 931-980. |
[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), 244-351. |
[17] |
V. V. Pukhnachev, Asymptotic solution of the rotating film problem, Izv. Vyssh. Uchebn. Zaved. Severo-Kavkaz. Reg. Estestv. Nauk, "Mathematics and Continuum Mechanics", (2004), 191-199. |
[18] |
V. V. Pukhnachev, On the equation of a rotating film, Siberian Math. J., 46 (2005), 913-924.
doi: 10.1007/s11202-005-0088-9. |
[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), 402-444. |
[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), 250-271.
doi: 10.1007/s11253-006-0066-9. |
[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 York-London, 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), 65-98. |
[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 thin-film equation: Recent advances and some new perspectives, Journal of Physics: Condensed Matter, 17 (2005), S291-S306. |
[5] |
E. Benilov, M. Benilov and N. Kopteva, Steady rimming flows with surface tension, Journal of Fluid Mechanics, 597 (2008), 91-118.
doi: 10.1017/S0022112007009585. |
[6] |
E. Beretta, M. Bertsch and R. Dal Passo, Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation, Arch. Rational Mech. Anal., 129 (1995), 175-200.
doi: 10.1007/BF00379920. |
[7] |
F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, Journal of Diff. Equations, 83 (1990), 179-206.
doi: 10.1016/0022-0396(90)90074-Y. |
[8] |
A. L. Bertozzi and M. C. Pugh, The lubrication approximation for thin viscous films: The moving contact line with a "porous media'' cut-off of van der Waals interactions, Nonlinearity, 7 (1994), 1535-1564. |
[9] |
A. Burchard, M. Chugunova and B. Stephens, Convergence to equilibrium for a thin-film equation on a cylindrical surface, Comm. Partial Diff. Equations, 37 (2012), 585-609.
doi: 10.1080/03605302.2011.648704. |
[10] |
M. Chugunova, M. C. Pugh and R. M. Taranets, Nonnegative solutions for a long-wave unstable thin film equation with convection, SIAM J. on Math. Anal., 42 (2010), 1826-1853.
doi: 10.1137/090777062. |
[11] |
R. V. Craster and O. K. Matar, Dynamics and stability of thin liquid films, Rev. Modern Physics, 81 (2009), 1131-1198. |
[12] |
E. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Cong. Num., 30 (1981), 265-284. |
[13] |
L. C. Evans, "Partial Differential Equations," 2nd edition, American Mathematical Society, Providence, 2010. |
[14] |
A. Oron and S. G. Bankoff, Long-scale evolution of thin liquid films, Rev. Modern Physics, 69 (1997), 931-980. |
[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), 244-351. |
[17] |
V. V. Pukhnachev, Asymptotic solution of the rotating film problem, Izv. Vyssh. Uchebn. Zaved. Severo-Kavkaz. Reg. Estestv. Nauk, "Mathematics and Continuum Mechanics", (2004), 191-199. |
[18] |
V. V. Pukhnachev, On the equation of a rotating film, Siberian Math. J., 46 (2005), 913-924.
doi: 10.1007/s11202-005-0088-9. |
[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), 402-444. |
[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), 250-271.
doi: 10.1007/s11253-006-0066-9. |
[21] |
L. N. Trefethen, "Spectral Methods in MATLAB," SIAM, Philadelphia, 2000.
doi: 10.1137/1.9780898719598. |
[1] |
Kersten Schmidt, Ralf Hiptmair. Asymptotic boundary element methods for thin conducting sheets. Discrete and Continuous Dynamical Systems - S, 2015, 8 (3) : 619-647. doi: 10.3934/dcdss.2015.8.619 |
[2] |
Jitraj Saha, Nilima Das, Jitendra Kumar, Andreas Bück. Numerical solutions for multidimensional fragmentation problems using finite volume methods. Kinetic and Related Models, 2019, 12 (1) : 79-103. doi: 10.3934/krm.2019004 |
[3] |
Z. Jackiewicz, B. Zubik-Kowal, B. Basse. Finite-difference and pseudo-spectral methods for the numerical simulations of in vitro human tumor cell population kinetics. Mathematical Biosciences & Engineering, 2009, 6 (3) : 561-572. doi: 10.3934/mbe.2009.6.561 |
[4] |
Emmanuel Frénod. Homogenization-based numerical methods. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : i-ix. doi: 10.3934/dcdss.201605i |
[5] |
Takeshi Saito, Kazuyuki Yagasaki. Chebyshev spectral methods for computing center manifolds. Journal of Computational Dynamics, 2021, 8 (2) : 165-181. doi: 10.3934/jcd.2021008 |
[6] |
Ching-Shan Chou, Yong-Tao Zhang, Rui Zhao, Qing Nie. Numerical methods for stiff reaction-diffusion systems. Discrete and Continuous Dynamical Systems - B, 2007, 7 (3) : 515-525. doi: 10.3934/dcdsb.2007.7.515 |
[7] |
Emmanuel Frénod. An attempt at classifying homogenization-based numerical methods. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : i-vi. doi: 10.3934/dcdss.2015.8.1i |
[8] |
Sebastián J. Ferraro, David Iglesias-Ponte, D. Martín de Diego. Numerical and geometric aspects of the nonholonomic SHAKE and RATTLE methods. Conference Publications, 2009, 2009 (Special) : 220-229. doi: 10.3934/proc.2009.2009.220 |
[9] |
Yue Qiu, Sara Grundel, Martin Stoll, Peter Benner. Efficient numerical methods for gas network modeling and simulation. Networks and Heterogeneous Media, 2020, 15 (4) : 653-679. doi: 10.3934/nhm.2020018 |
[10] |
Abdon Atangana, José Francisco Gómez-Aguilar, Jordan Y. Hristov, Kolade M. Owolabi. Preface on "New trends of numerical and analytical methods". Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : i-ii. doi: 10.3934/dcdss.20203i |
[11] |
Timothy Blass, Rafael de la Llave. Perturbation and numerical methods for computing the minimal average energy. Networks and Heterogeneous Media, 2011, 6 (2) : 241-255. doi: 10.3934/nhm.2011.6.241 |
[12] |
Joseph A. Connolly, Neville J. Ford. Comparison of numerical methods for fractional differential equations. Communications on Pure and Applied Analysis, 2006, 5 (2) : 289-307. doi: 10.3934/cpaa.2006.5.289 |
[13] |
Miguel Ángel Evangelista-Alvarado, José Crispín Ruíz-Pantaleón, Pablo Suárez-Serrato. On computational Poisson geometry II: Numerical methods. Journal of Computational Dynamics, 2021, 8 (3) : 273-307. doi: 10.3934/jcd.2021012 |
[14] |
Yin Yang, Yunqing Huang. Spectral Jacobi-Galerkin methods and iterated methods for Fredholm integral equations of the second kind with weakly singular kernel. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 685-702. doi: 10.3934/dcdss.2019043 |
[15] |
Chao Xing, Ping Zhou, Hong Luo. The steady state solutions to thermohaline circulation equations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3709-3722. doi: 10.3934/dcdsb.2016117 |
[16] |
Youcef Amirat, Kamel Hamdache. Steady state solutions of ferrofluid flow models. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2329-2355. doi: 10.3934/cpaa.2016039 |
[17] |
Torsten Trimborn, Stephan Gerster, Giuseppe Visconti. Spectral methods to study the robustness of residual neural networks with infinite layers. Foundations of Data Science, 2020, 2 (3) : 257-278. doi: 10.3934/fods.2020012 |
[18] |
Z. Foroozandeh, Maria do rosário de Pinho, M. Shamsi. On numerical methods for singular optimal control problems: An application to an AUV problem. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2219-2235. doi: 10.3934/dcdsb.2019092 |
[19] |
Giacomo Albi, Lorenzo Pareschi, Mattia Zanella. Opinion dynamics over complex networks: Kinetic modelling and numerical methods. Kinetic and Related Models, 2017, 10 (1) : 1-32. doi: 10.3934/krm.2017001 |
[20] |
Martin Benning, Elena Celledoni, Matthias J. Ehrhardt, Brynjulf Owren, Carola-Bibiane Schönlieb. Deep learning as optimal control problems: Models and numerical methods. Journal of Computational Dynamics, 2019, 6 (2) : 171-198. doi: 10.3934/jcd.2019009 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]