American Institute of Mathematical Sciences

January  2017, 37(1): 405-434. doi: 10.3934/dcds.2017017

Long-time behavior of a fully discrete Lagrangian scheme for a family of fourth order equations

 Zentrum Mathematik, TU München Boltzmannstr. 3, D-85748 Garching, Germany

Received  January 2015 Revised  August 2016 Published  November 2016

Fund Project: This research was supported by the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics”.

A fully discrete Lagrangian scheme for solving a family of fourth order equations numerically is presented. The discretization is based on the equations' underlying gradient flow structure with respect to the Wasserstein metric, and preserves numerous of their most important structural properties by construction, like conservation of mass and entropy-dissipation.

In this paper, the long-time behavior of our discretization is analysed: We show that discrete solutions decay exponentially to equilibrium at the same rate as smooth solutions of the original problem. Moreover, we give a proof of convergence of discrete entropy minimizers towards Barenblatt-profiles or Gaussians, respectively, using $Γ$-convergence.

Citation: Horst Osberger. Long-time behavior of a fully discrete Lagrangian scheme for a family of fourth order equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 405-434. doi: 10.3934/dcds.2017017
References:
 [1] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. doi: 978-3-7643-2428-5. Google Scholar [2] L. Ambrosio, S. Lisini and G. Savaré, Stability of flows associated to gradient vector fields and convergence of iterated transport maps, Manuscripta Mathematica, 121 (2006), 1-50. doi: 10.1007/s00229-006-0003-0. Google Scholar [3] J. Becker and G. Grün, The thin-film equation: Recent advances and some new perspectives, Journal of Physics: Condensed Matter, 17 (2015), 291-307. doi: 10.1088/0953-8984/17/9/002. Google Scholar [4] F. Bernis and F. Avner, Higher order nonlinear degenerate parabolic equations, Journal of Differential Equations, 83 (1990), 179-206. doi: 10.1016/0022-0396(90)90074-Y. Google Scholar [5] M. Bertsch, R. Dal Passo, H. Garcke and G. Grün, The thin viscous flow equation in higher space dimensions, Journal of Differential Equations, 3 (1998), 417-440. Google Scholar [6] A. Blanchet, V. Calvez and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel mode, SIAM Journal on Numerical Analysis, 46 (2008), 691-721. doi: 10.1137/070683337. Google Scholar [7] P. M. Bleher, J. L. Lebowitz and E. R. Speer, Existence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations, Communications on Pure and Applied Mathematics, 47 (1994), 923-942. doi: 10.1002/cpa.3160470702. Google Scholar [8] A. Braides, $Γ$ -convergence for Beginners Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001. Google Scholar [9] C. J. Budd, G. J. Collins, W. Z. Huang and R. D. Russell, Self-similar numerical solutions of the porous-medium equation using moving mesh methods, The Royal Society of London. Philosophical Transactions. Series A. Mathematical, Physical and Engineering Sciences, 357 (1999), 1047-1077. doi: 10.1098/rsta.1999.0364. Google Scholar [10] M. Bukal, E. Emmrich and A. Jüngel, Entropy-stable and entropy-dissipative approximations of a fourth-order quantum diffusion equation, Numerische Mathematik, 127 (2014), 365-396. doi: 10.1007/s00211-013-0588-7. Google Scholar [11] M. Burger, J. A. Carrillo and M.-T. Wolfram, A mixed finite element method for nonlinear diffusion equations, Kinetic and Related Models, 3 (2010), 59-83. doi: 10.3934/krm.2010.3.59. Google Scholar [12] M. J. Cáceres, J. A. Carrillo and G. Toscani, Long-time behavior for a nonlinear fourth-order parabolic equation, Transactions of the American Mathematical Society, 357 (2005), 1161-1175. doi: 10.1090/S0002-9947-04-03528-7. Google Scholar [13] E. A. Carlen and S. Ulusoy, Asymptotic equipartition and long time behavior of solutions of a thin-film equation, Journal of Differential Equations, 241 (2007), 279-292. doi: 10.1016/j.jde.2007.07.005. Google Scholar [14] J. A. Carrillo, J. Dolbeault, I. Gentil and A. Jüngel, Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 6 (2006), 1027-1050. doi: 10.3934/dcdsb.2006.6.1027. Google Scholar [15] J. A. Carrillo, A. Jüngel, P. A. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized {S}obolev inequalities, Monatshefte für Mathematik, 133 (2001), 1-82. doi: 10.1007/s006050170032. Google Scholar [16] J. A. Carrillo, A. Jüngel and S. Tang, Positive entropic schemes for a nonlinear fourth-order parabolic equation, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 3 (2003), 1-20. Google Scholar [17] J. A. Carrillo and J. S. Moll, Numerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms, SIAM Journal on Scientific Computing, 31 (2009/10), 4305-4329. doi: 10.1137/080739574. Google Scholar [18] J. A. Carrillo and G. Toscani, Long-time asymptotics for strong solutions of the thin film equation, Communications in Mathematical Physics, 225 (2002), 551-571. doi: 10.1007/s002200100591. Google Scholar [19] J. A. Carrillo and M. -T. Wolfram, A finite element method for nonlinear continuity equations in Lagrangian coordinates, work in progress.Google Scholar [20] F. Cavalli and G. Naldi, A Wasserstein approach to the numerical solution of the one-dimensional Cahn-Hilliard equation, Kinetic and Related Models, 3 (2010), 123-142. doi: 10.3934/krm.2010.3.123. Google Scholar [21] R. Dal Passo, H. Garcke and G. Grün, On a fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions, SIAM Journal on Mathematical Analysis, 29 (1998), 321-342. doi: 10.1137/S0036141096306170. Google Scholar [22] J. Denzler and R. J. McCann, Nonlinear diffusion from a delocalized source: affine self-similarity, time reversal, & nonradial focusing geometries, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, 25 (2008), 865-888. doi: 10.1016/j.anihpc.2007.05.002. Google Scholar [23] B. Derrida, J. L. Lebowitz, E. R. Speer and H. Spohn, Dynamics of an anchored Toom interface, Journal of Physics. A. Mathematical and General, 24 (1991), 4805-4834. doi: 10.1088/0305-4470/24/20/015. Google Scholar [24] B. Derrida, J. L. Lebowitz, E. R. Speer and H. Spohn, Fluctuations of a stationary nonequilibrium interface, Physical Review Letters, 67 (1991), 165-168. doi: 10.1103/PhysRevLett.67.165. Google Scholar [25] B. Düring, D. Matthes and J. P. Milišić, A gradient flow scheme for nonlinear fourth order equations, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 14 (2010), 935-959. doi: 10.3934/dcdsb.2010.14.935. Google Scholar [26] L. C. Evans, O. Savin and W. Gangbo, Diffeomorphisms and nonlinear heat flows, SIAM Journal on Mathematical Analysis, 37 (2005), 737-751. doi: 10.1137/04061386X. Google Scholar [27] J. Fischer, Uniqueness of solutions of the Derrida-Lebowitz-Speer-Spohn equation and quantum drift-diffusion models, SIAM Journal on Mathematical Analysis, 38 (2013), 2004-2047. doi: 10.1080/03605302.2013.823548. Google Scholar [28] L. Giacomelli and F. Otto, Variational formulation for the lubrication approximation of the Hele-Shaw flow, Calculus of Variations and Partial Differential Equations, 13 (2001), 377-403. doi: 10.1007/s005260000077. Google Scholar [29] U. Gianazza, G. Savaré and G. Toscani, The {W}asserstein gradient flow of the Fisher information and the quantum drift-diffusion equation, Archive for Rational Mechanics and Analysis, 194 (2009), 133-220. doi: 10.1007/s00205-008-0186-5. Google Scholar [30] E. Giusti, Minimal Surfaces and Functions of Bounded Variation Monographs in Mathematics, Birkhäuser Verlag, Basel, 1984. doi: 10.1007/978-1-4684-9486-0. Google Scholar [31] L. Gosse and G. Toscani, Identification of asymptotic decay to self-similarity for one-dimensional filtration equations, SIAM Journal on Numerical Analysis, 43 (2006), 2590-2606. doi: 10.1137/040608672. Google Scholar [32] L. Gosse and G. Toscani, Lagrangian numerical approximations to one-dimensional convolution-diffusion equations, SIAM Journal on Scientific Computing, 28 (2006), 1203-1227. doi: 10.1137/050628015. Google Scholar [33] G. Grün, Droplet spreading under weak slippage-existence for the Cauchy problem, Communications in Partial Differential Equations, 29 (2004), 1697-1744. doi: 10.1081/PDE-200040193. Google Scholar [34] M. P. Gualdani, A. Jüngel and G. Toscani, A nonlinear fourth-order parabolic equation with nonhomogeneous boundary conditions, SIAM Journal on Mathematical Analysis, 37 (2006), 1761-1779. doi: 10.1137/S0036141004444615. Google Scholar [35] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM Journal on Mathematical Analysis, 29 (1998), 1-17. doi: 10.1137/S0036141096303359. Google Scholar [36] A. Jüngel and D. Matthes, An algorithmic construction of entropies in higher-order nonlinear PDEs, Nonlinearity, 19 (2006), 633-659. doi: 10.1088/0951-7715/19/3/006. Google Scholar [37] A. Jüngel and D. Matthes, The Derrida-Lebowitz-Speer-Spohn equation: Existence, nonuniqueness, and decay rates of the solutions, SIAM Journal on Mathematical Analysis, 39 (2008), 1996-2015. doi: 10.1137/060676878. Google Scholar [38] A. Jüngel and R. Pinnau, Global nonnegative solutions of a nonlinear fourth-order parabolic equation for quantum systems, SIAM Journal on Mathematical Analysis, 32 (2000), 760-777. doi: 10.1137/S0036141099360269. Google Scholar [39] A. Jüngel and R. Pinnau, A positivity-preserving numerical scheme for a nonlinear fourth order parabolic system, SIAM Journal on Numerical Analysis, 39 (2001), 385-406. doi: 10.1137/S0036142900369362. Google Scholar [40] A. Jüngel and G. Toscani, Exponential time decay of solutions to a nonlinear fourth-order parabolic equation, Journal of Applied Mathematics and Physics, 54 (2003), 377-386. doi: 10.1007/s00033-003-1026-y. Google Scholar [41] A. Jüngel and I. Violet, First-order entropies for the Derrida-Lebowitz-Speer-Spohn equation, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 8 (2007), 861-877. doi: 10.3934/dcdsb.2007.8.861. Google Scholar [42] D. Kinderlehrer and N. J. Walkington, Approximation of parabolic equations using the Wasserstein metric, M2AN. Mathematical Modelling and Numerical Analysis, 33 (1999), 837-852. doi: 10.1051/m2an:1999166. Google Scholar [43] R. C. MacCamy and E. Socolovsky, A numerical procedure for the porous media equation, Computers & Mathematics with Applications. An International Journal, 11 (1985), 315-319. doi: 10.1016/0898-1221(85)90156-7. Google Scholar [44] D. Matthes, R. J. McCann and G. Savaré, A family of nonlinear fourth order equations of gradient flow type, Communications in Partial Differential Equations, 34 (2009), 1352-1397. doi: 10.1080/03605300903296256. Google Scholar [45] D. Matthes and H. Osberger, Convergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation, ESAIM. Mathematical Modelling and Numerical Analysis, 48 (2014), 697-726. doi: 10.1051/m2an/2013126. Google Scholar [46] D. Matthes and H. Osberger, A convergent Lagrangian discretization for a nonlinear fourth-order equation, Foundations of Computational Mathematics, 17 (2015), 1-54. doi: 10.1007/s10208-015-9284-6. Google Scholar [47] A. Oron, S. H. Davis and S. G. Bankoff, Long-scale evolution of thin liquid films, American Physical Society, 69 (1997), 931-980. doi: 10.1103/RevModPhys.69.931. Google Scholar [48] G. Russo, Deterministic diffusion of particles, Communications on Pure and Applied Mathematics, 43 (1990), 697-733. doi: 10.1002/cpa.3160430602. Google Scholar [49] C. Villani, Topics in Optimal Transportation American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/058. Google Scholar

show all references

References:
 [1] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. doi: 978-3-7643-2428-5. Google Scholar [2] L. Ambrosio, S. Lisini and G. Savaré, Stability of flows associated to gradient vector fields and convergence of iterated transport maps, Manuscripta Mathematica, 121 (2006), 1-50. doi: 10.1007/s00229-006-0003-0. Google Scholar [3] J. Becker and G. Grün, The thin-film equation: Recent advances and some new perspectives, Journal of Physics: Condensed Matter, 17 (2015), 291-307. doi: 10.1088/0953-8984/17/9/002. Google Scholar [4] F. Bernis and F. Avner, Higher order nonlinear degenerate parabolic equations, Journal of Differential Equations, 83 (1990), 179-206. doi: 10.1016/0022-0396(90)90074-Y. Google Scholar [5] M. Bertsch, R. Dal Passo, H. Garcke and G. Grün, The thin viscous flow equation in higher space dimensions, Journal of Differential Equations, 3 (1998), 417-440. Google Scholar [6] A. Blanchet, V. Calvez and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel mode, SIAM Journal on Numerical Analysis, 46 (2008), 691-721. doi: 10.1137/070683337. Google Scholar [7] P. M. Bleher, J. L. Lebowitz and E. R. Speer, Existence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations, Communications on Pure and Applied Mathematics, 47 (1994), 923-942. doi: 10.1002/cpa.3160470702. Google Scholar [8] A. Braides, $Γ$ -convergence for Beginners Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001. Google Scholar [9] C. J. Budd, G. J. Collins, W. Z. Huang and R. D. Russell, Self-similar numerical solutions of the porous-medium equation using moving mesh methods, The Royal Society of London. Philosophical Transactions. Series A. Mathematical, Physical and Engineering Sciences, 357 (1999), 1047-1077. doi: 10.1098/rsta.1999.0364. Google Scholar [10] M. Bukal, E. Emmrich and A. Jüngel, Entropy-stable and entropy-dissipative approximations of a fourth-order quantum diffusion equation, Numerische Mathematik, 127 (2014), 365-396. doi: 10.1007/s00211-013-0588-7. Google Scholar [11] M. Burger, J. A. Carrillo and M.-T. Wolfram, A mixed finite element method for nonlinear diffusion equations, Kinetic and Related Models, 3 (2010), 59-83. doi: 10.3934/krm.2010.3.59. Google Scholar [12] M. J. Cáceres, J. A. Carrillo and G. Toscani, Long-time behavior for a nonlinear fourth-order parabolic equation, Transactions of the American Mathematical Society, 357 (2005), 1161-1175. doi: 10.1090/S0002-9947-04-03528-7. Google Scholar [13] E. A. Carlen and S. Ulusoy, Asymptotic equipartition and long time behavior of solutions of a thin-film equation, Journal of Differential Equations, 241 (2007), 279-292. doi: 10.1016/j.jde.2007.07.005. Google Scholar [14] J. A. Carrillo, J. Dolbeault, I. Gentil and A. Jüngel, Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 6 (2006), 1027-1050. doi: 10.3934/dcdsb.2006.6.1027. Google Scholar [15] J. A. Carrillo, A. Jüngel, P. A. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized {S}obolev inequalities, Monatshefte für Mathematik, 133 (2001), 1-82. doi: 10.1007/s006050170032. Google Scholar [16] J. A. Carrillo, A. Jüngel and S. Tang, Positive entropic schemes for a nonlinear fourth-order parabolic equation, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 3 (2003), 1-20. Google Scholar [17] J. A. Carrillo and J. S. Moll, Numerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms, SIAM Journal on Scientific Computing, 31 (2009/10), 4305-4329. doi: 10.1137/080739574. Google Scholar [18] J. A. Carrillo and G. Toscani, Long-time asymptotics for strong solutions of the thin film equation, Communications in Mathematical Physics, 225 (2002), 551-571. doi: 10.1007/s002200100591. Google Scholar [19] J. A. Carrillo and M. -T. Wolfram, A finite element method for nonlinear continuity equations in Lagrangian coordinates, work in progress.Google Scholar [20] F. Cavalli and G. Naldi, A Wasserstein approach to the numerical solution of the one-dimensional Cahn-Hilliard equation, Kinetic and Related Models, 3 (2010), 123-142. doi: 10.3934/krm.2010.3.123. Google Scholar [21] R. Dal Passo, H. Garcke and G. Grün, On a fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions, SIAM Journal on Mathematical Analysis, 29 (1998), 321-342. doi: 10.1137/S0036141096306170. Google Scholar [22] J. Denzler and R. J. McCann, Nonlinear diffusion from a delocalized source: affine self-similarity, time reversal, & nonradial focusing geometries, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, 25 (2008), 865-888. doi: 10.1016/j.anihpc.2007.05.002. Google Scholar [23] B. Derrida, J. L. Lebowitz, E. R. Speer and H. Spohn, Dynamics of an anchored Toom interface, Journal of Physics. A. Mathematical and General, 24 (1991), 4805-4834. doi: 10.1088/0305-4470/24/20/015. Google Scholar [24] B. Derrida, J. L. Lebowitz, E. R. Speer and H. Spohn, Fluctuations of a stationary nonequilibrium interface, Physical Review Letters, 67 (1991), 165-168. doi: 10.1103/PhysRevLett.67.165. Google Scholar [25] B. Düring, D. Matthes and J. P. Milišić, A gradient flow scheme for nonlinear fourth order equations, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 14 (2010), 935-959. doi: 10.3934/dcdsb.2010.14.935. Google Scholar [26] L. C. Evans, O. Savin and W. Gangbo, Diffeomorphisms and nonlinear heat flows, SIAM Journal on Mathematical Analysis, 37 (2005), 737-751. doi: 10.1137/04061386X. Google Scholar [27] J. Fischer, Uniqueness of solutions of the Derrida-Lebowitz-Speer-Spohn equation and quantum drift-diffusion models, SIAM Journal on Mathematical Analysis, 38 (2013), 2004-2047. doi: 10.1080/03605302.2013.823548. Google Scholar [28] L. Giacomelli and F. Otto, Variational formulation for the lubrication approximation of the Hele-Shaw flow, Calculus of Variations and Partial Differential Equations, 13 (2001), 377-403. doi: 10.1007/s005260000077. Google Scholar [29] U. Gianazza, G. Savaré and G. Toscani, The {W}asserstein gradient flow of the Fisher information and the quantum drift-diffusion equation, Archive for Rational Mechanics and Analysis, 194 (2009), 133-220. doi: 10.1007/s00205-008-0186-5. Google Scholar [30] E. Giusti, Minimal Surfaces and Functions of Bounded Variation Monographs in Mathematics, Birkhäuser Verlag, Basel, 1984. doi: 10.1007/978-1-4684-9486-0. Google Scholar [31] L. Gosse and G. Toscani, Identification of asymptotic decay to self-similarity for one-dimensional filtration equations, SIAM Journal on Numerical Analysis, 43 (2006), 2590-2606. doi: 10.1137/040608672. Google Scholar [32] L. Gosse and G. Toscani, Lagrangian numerical approximations to one-dimensional convolution-diffusion equations, SIAM Journal on Scientific Computing, 28 (2006), 1203-1227. doi: 10.1137/050628015. Google Scholar [33] G. Grün, Droplet spreading under weak slippage-existence for the Cauchy problem, Communications in Partial Differential Equations, 29 (2004), 1697-1744. doi: 10.1081/PDE-200040193. Google Scholar [34] M. P. Gualdani, A. Jüngel and G. Toscani, A nonlinear fourth-order parabolic equation with nonhomogeneous boundary conditions, SIAM Journal on Mathematical Analysis, 37 (2006), 1761-1779. doi: 10.1137/S0036141004444615. Google Scholar [35] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM Journal on Mathematical Analysis, 29 (1998), 1-17. doi: 10.1137/S0036141096303359. Google Scholar [36] A. Jüngel and D. Matthes, An algorithmic construction of entropies in higher-order nonlinear PDEs, Nonlinearity, 19 (2006), 633-659. doi: 10.1088/0951-7715/19/3/006. Google Scholar [37] A. Jüngel and D. Matthes, The Derrida-Lebowitz-Speer-Spohn equation: Existence, nonuniqueness, and decay rates of the solutions, SIAM Journal on Mathematical Analysis, 39 (2008), 1996-2015. doi: 10.1137/060676878. Google Scholar [38] A. Jüngel and R. Pinnau, Global nonnegative solutions of a nonlinear fourth-order parabolic equation for quantum systems, SIAM Journal on Mathematical Analysis, 32 (2000), 760-777. doi: 10.1137/S0036141099360269. Google Scholar [39] A. Jüngel and R. Pinnau, A positivity-preserving numerical scheme for a nonlinear fourth order parabolic system, SIAM Journal on Numerical Analysis, 39 (2001), 385-406. doi: 10.1137/S0036142900369362. Google Scholar [40] A. Jüngel and G. Toscani, Exponential time decay of solutions to a nonlinear fourth-order parabolic equation, Journal of Applied Mathematics and Physics, 54 (2003), 377-386. doi: 10.1007/s00033-003-1026-y. Google Scholar [41] A. Jüngel and I. Violet, First-order entropies for the Derrida-Lebowitz-Speer-Spohn equation, Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, 8 (2007), 861-877. doi: 10.3934/dcdsb.2007.8.861. Google Scholar [42] D. Kinderlehrer and N. J. Walkington, Approximation of parabolic equations using the Wasserstein metric, M2AN. Mathematical Modelling and Numerical Analysis, 33 (1999), 837-852. doi: 10.1051/m2an:1999166. Google Scholar [43] R. C. MacCamy and E. Socolovsky, A numerical procedure for the porous media equation, Computers & Mathematics with Applications. An International Journal, 11 (1985), 315-319. doi: 10.1016/0898-1221(85)90156-7. Google Scholar [44] D. Matthes, R. J. McCann and G. Savaré, A family of nonlinear fourth order equations of gradient flow type, Communications in Partial Differential Equations, 34 (2009), 1352-1397. doi: 10.1080/03605300903296256. Google Scholar [45] D. Matthes and H. Osberger, Convergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation, ESAIM. Mathematical Modelling and Numerical Analysis, 48 (2014), 697-726. doi: 10.1051/m2an/2013126. Google Scholar [46] D. Matthes and H. Osberger, A convergent Lagrangian discretization for a nonlinear fourth-order equation, Foundations of Computational Mathematics, 17 (2015), 1-54. doi: 10.1007/s10208-015-9284-6. Google Scholar [47] A. Oron, S. H. Davis and S. G. Bankoff, Long-scale evolution of thin liquid films, American Physical Society, 69 (1997), 931-980. doi: 10.1103/RevModPhys.69.931. Google Scholar [48] G. Russo, Deterministic diffusion of particles, Communications on Pure and Applied Mathematics, 43 (1990), 697-733. doi: 10.1002/cpa.3160430602. Google Scholar [49] C. Villani, Topics in Optimal Transportation American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/058. Google Scholar
Evolution of a discrete solution $u_\Delta$, evaluated at different times $t = 0,0.05,0.1,0.15,0.175,0.25$ (from top left to bottom right). The red line is the corresponding Barenblatt-profile ${{\text{b}}_{\alpha ,\lambda }}$.
Left: Numerically observed decay of $H_{\alpha,\lambda}(t)-{\mathbf{H}_{\alpha ,\lambda }^{\min }}$ and $F_{\alpha,\lambda}(t)-{\mathbf{F}_{\alpha ,\lambda }^{\min }}$ along a time period of $t\in[0,0.8]$, using $K=25,50,100,200$, in comparison to the upper bounds $({\mathcal{H}_{\alpha ,\lambda }}(u^0)-{\mathcal{H}_{\alpha ,\lambda }}({{\text{b}}_{\alpha ,\lambda }}))\exp(-2\lambda t)$ and $({\mathcal{F}_{\alpha ,\lambda }}(u^0)-{\mathcal{F}_{\alpha ,\lambda }}({{\text{b}}_{\alpha ,\lambda }}))\exp(-2\lambda t)$, respectively. Right: Convergence of discrete minimizers $u_{\delta }^{\min }$ with a rate of $K^{-1.5}$.
Snapshots of the densities $\text{b}_{\alpha ,0}^{*}(t,\cdot)$ (red lines) and $u_\Delta$ (black lines) for the initial condition $\text{b}_{\alpha ,0}^{*}(0,\cdot)$ at times $t=0$ and $t= 0.1\cdot 10^{i}$, $i=0,\ldots,3$, using $K=50$ grid points and the time step size $\tau=10^{-3}$.
 [1] Eric A. Carlen, Süleyman Ulusoy. Localization, smoothness, and convergence to equilibrium for a thin film equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4537-4553. doi: 10.3934/dcds.2014.34.4537 [2] Daniel Ginsberg, Gideon Simpson. Analytical and numerical results on the positivity of steady state solutions of a thin film equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1305-1321. doi: 10.3934/dcdsb.2013.18.1305 [3] Marina Chugunova, Roman M. Taranets. New dissipated energy for the unstable thin film equation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 613-624. doi: 10.3934/cpaa.2011.10.613 [4] Richard S. Laugesen. New dissipated energies for the thin fluid film equation. Communications on Pure & Applied Analysis, 2005, 4 (3) : 613-634. doi: 10.3934/cpaa.2005.4.613 [5] Changchun Liu, Jingxue Yin, Juan Zhou. Existence of weak solutions for a generalized thin film equation. Communications on Pure & Applied Analysis, 2007, 6 (2) : 465-480. doi: 10.3934/cpaa.2007.6.465 [6] Jian-Guo Liu, Jinhuan Wang. Global existence for a thin film equation with subcritical mass. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1461-1492. doi: 10.3934/dcdsb.2017070 [7] Huiqiang Jiang. Energy minimizers of a thin film equation with born repulsion force. Communications on Pure & Applied Analysis, 2011, 10 (2) : 803-815. doi: 10.3934/cpaa.2011.10.803 [8] Andrey Shishkov. Waiting time of propagation and the backward motion of interfaces in thin-film flow theory. Conference Publications, 2007, 2007 (Special) : 938-945. doi: 10.3934/proc.2007.2007.938 [9] Karoline Disser, Matthias Liero. On gradient structures for Markov chains and the passage to Wasserstein gradient flows. Networks & Heterogeneous Media, 2015, 10 (2) : 233-253. doi: 10.3934/nhm.2015.10.233 [10] Lihua Min, Xiaoping Yang. Finite speed of propagation and algebraic time decay of solutions to a generalized thin film equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 543-566. doi: 10.3934/cpaa.2014.13.543 [11] Sergey Degtyarev. Classical solvability of the multidimensional free boundary problem for the thin film equation with quadratic mobility in the case of partial wetting. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3625-3699. doi: 10.3934/dcds.2017156 [12] Fausto Cavalli, Giovanni Naldi. A Wasserstein approach to the numerical solution of the one-dimensional Cahn-Hilliard equation. Kinetic & Related Models, 2010, 3 (1) : 123-142. doi: 10.3934/krm.2010.3.123 [13] Miguel Escobedo, Minh-Binh Tran. Convergence to equilibrium of a linearized quantum Boltzmann equation for bosons at very low temperature. Kinetic & Related Models, 2015, 8 (3) : 493-531. doi: 10.3934/krm.2015.8.493 [14] P. Álvarez-Caudevilla, J. D. Evans, V. A. Galaktionov. The Cauchy problem for a tenth-order thin film equation II. Oscillatory source-type and fundamental similarity solutions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 807-827. doi: 10.3934/dcds.2015.35.807 [15] Yohan Penel. An explicit stable numerical scheme for the $1D$ transport equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 641-656. doi: 10.3934/dcdss.2012.5.641 [16] Helge Holden, Xavier Raynaud. A convergent numerical scheme for the Camassa--Holm equation based on multipeakons. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 505-523. doi: 10.3934/dcds.2006.14.505 [17] Jian Zhai, Zhihui Cai. $\Gamma$-convergence with Dirichlet boundary condition and Landau-Lifshitz functional for thin film. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 1071-1085. doi: 10.3934/dcdsb.2009.11.1071 [18] Ronald E. Mickens. A nonstandard finite difference scheme for the drift-diffusion system. Conference Publications, 2009, 2009 (Special) : 558-563. doi: 10.3934/proc.2009.2009.558 [19] José Antonio Carrillo, Yanghong Huang, Francesco Saverio Patacchini, Gershon Wolansky. Numerical study of a particle method for gradient flows. Kinetic & Related Models, 2017, 10 (3) : 613-641. doi: 10.3934/krm.2017025 [20] Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic & Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056

2018 Impact Factor: 1.143

Metrics

• PDF downloads (15)
• HTML views (20)
• Cited by (0)

Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]