doi: 10.3934/dcds.2021001

Well-posedness and convergence of a numerical scheme for the corrected Derrida-Lebowitz-Speer-Spohn equation using the Hellinger distance

University of Zagreb, Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb, Croatia

Received  January 2020 Revised  October 2020 Published  January 2021

Fund Project: This work has been supported by the Croatian Science Foundation under Grant agreement No. UIP-05-2017-7249 (MANDphy) and in part by the bilaterial project No. HR 04/2018 between OeAD and MZO

In this paper we construct a unique global in time weak nonnegative solution to the corrected Derrida-Lebowitz-Speer-Spohn equation, which statistically describes the interface fluctuations between two phases in a certain spin system. The construction of the weak solution is based on the dissipation of a Lyapunov functional which equals to the square of the Hellinger distance between the solution and the constant steady state. Furthermore, it is shown that the weak solution converges at an exponential rate to the constant steady state in the Hellinger distance and thus also in the $ L^1 $-norm. Numerical scheme which preserves the variational structure of the equation is devised and its convergence in terms of a discrete Hellinger distance is demonstrated.

Citation: Mario Bukal. Well-posedness and convergence of a numerical scheme for the corrected Derrida-Lebowitz-Speer-Spohn equation using the Hellinger distance. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2021001
References:
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L. Ambrosio, N. Gigli and G. Savare, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhäuser Basel, 2008.  Google Scholar

[2]

H. Bae and R. Granero-Belinchón, Global existence and exponential decay to equilibrium for DLSS-Type equations, J. Dyn. Diff. Equat., (2020). doi: 10.1007/s10884-020-09852-5.  Google Scholar

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[4]

F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Diff. Eqs., 83 (1990), 179-206.  doi: 10.1016/0022-0396(90)90074-Y.  Google Scholar

[5]

A. Bertozzi, The mathematics of moving contact lines in thin liquid films, Notices Amer. Math. Soc., 45 (1998), 689-697.   Google Scholar

[6]

P. M. Bleher, J. L. Lebowitz and E. R. Speer, Existence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations, Commun. Pure Appl. Math., 47 (1994), 923–942. doi: 10.1002/cpa.3160470702.  Google Scholar

[7]

C. Bordenave, P. Germain and T. Trogdon, An extension of the Derrida–Lebowitz–Speer–Spohn equation, J. Phys. A: Math. Theor., 48 (2015), 485205. doi: 10.1088/1751-8113/48/48/485205.  Google Scholar

[8]

M. BukalE. 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

[9]

M. Bukal, A. Jüngel and D. Matthes, A multidimensional nonlinear sixth-order quantum diffusion equation, Annales de l'IHP Analyse Non Linéaire, 30 (2013), 337–365. doi: 10.1016/j.anihpc.2012.08.003.  Google Scholar

[10]

M. BurgerL. He and C.-B. Schönlieb, Cahn-Hilliard inpainting and a generalization for grayvalue images, SIAM Journal on Imaging Sciences, 2 (2009), 1129-1167.  doi: 10.1137/080728548.  Google Scholar

[11]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, The Journal of Chemical Physics, 28 (1958), 258-267.  doi: 10.1063/1.1744102.  Google Scholar

[12]

J. A. CarrilloJ. DolbeaultI. Gentil and A. Jüngel, Entropy-Energy inequalities and improved convergence rates for nonlinear parabolic equations, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1027-1050.  doi: 10.3934/dcdsb.2006.6.1027.  Google Scholar

[13]

J. A. CarrilloA. Jüngel and S. Tang, Positive entropic schemes for a nonlinear fourth-order equation, Discrete Contin. Dyn. Syst. B, 3 (2003), 1-20.  doi: 10.3934/dcdsb.2003.3.1.  Google Scholar

[14]

J. A. Carrillo and G. Toscani, Long-Time Asymptotics for Strong Solutions of the Thin Film Equation, Commun. Math. Phys., 225 (2002), 551-571.  doi: 10.1007/s002200100591.  Google Scholar

[15]

X. ChenA. Jüngel and J.-G. Liu, A note on Aubin-Lions-Dubinskii lemmas, Acta Appl. Math., 133 (2014), 33-43.  doi: 10.1007/s10440-013-9858-8.  Google Scholar

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[17]

R. Dal PassoH. Garcke and G. Grün, On a fourth order degenerate parabolic equation: Global entropy estimates and qualitative behaviour of solutions, SIAM J. Math. Anal., 29 (1998), 321-342.  doi: 10.1137/S0036141096306170.  Google Scholar

[18]

P. DegondF. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models, J. Stat. Phys., 118 (2005), 625-667.  doi: 10.1007/s10955-004-8823-3.  Google Scholar

[19]

B. DerridaJ. L. LebowitzE. R. Speer and H. Spohn, Dynamics of an anchored Toom interface, J. Phys. A: Math. Gen., 24 (1991), 4805-4834.  doi: 10.1088/0305-4470/24/20/015.  Google Scholar

[20]

B. DüringD. Matthes and J. P. Milišić, A gradient flow scheme for nonlinear fourth order equations, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 935-959.  doi: 10.3934/dcdsb.2010.14.935.  Google Scholar

[21]

D. Furihata and T. Matsuo, Discrete Variational Derivative Method, Chapman & Hall/CRC Numerical Analysis and Scientific Computing. CRC Press, Boca Raton, FL, 2011.  Google Scholar

[22]

J. Fischer, Uniqueness of solutions of the Derrida-Lebowitz-Speer-Spohn equation and quantum drift-diffusion models, Comm. Partial Differential Equations, 38 (2013), 2004-2047.  doi: 10.1080/03605302.2013.823548.  Google Scholar

[23]

L. Giacomelli and F. Otto, Variational formulation for the lubrication approximation of the Hele-Shaw flow, Calc. Var. PDEs, 13 (2001), 377-403.  doi: 10.1007/s005260000077.  Google Scholar

[24]

U. GianazzaG. Savaré and G. Toscani, The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation, Arch. Ration. Mech. Anal., 194 (2009), 133-220.  doi: 10.1007/s00205-008-0186-5.  Google Scholar

[25]

A. E. Hosoi and L. Mahadevan, Peeling, healing and bursting in a lubricated elastic sheet, Phys. Rev. Lett., 93 (2004), 137802. doi: 10.1103/PhysRevLett.93.137802.  Google Scholar

[26]

C. JosserandY. Pomeau and S. Rica, Self-similar singularities in the kinetics of condensation, J. of Low Temp. Physics, 145 (2006), 231-265.  doi: 10.1007/s10909-006-9232-6.  Google Scholar

[27]

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

[28]

A. Jüngel and D. Matthes, The Derrida-Lebowitz-Speer-Spohn equation: existence, non-uniqueness, and decay rates of the solutions, SIAM J. Math. Anal., 39 (2008), 1996-2015.  doi: 10.1137/060676878.  Google Scholar

[29]

A. Jüngel and J. -P. Milišić, A sixth-order nonlinear parabolic equation for quantum systems, SIAM J. Math. Anal., 41 (2009), 1472-1490.  doi: 10.1137/080739021.  Google Scholar

[30]

A. Jüngel and R. Pinnau, Global non-negative solutions of a nonlinear fourth-oder parabolic equation for quantum systems, SIAM J. Math. Anal., 32 (2000), 760-777.  doi: 10.1137/S0036141099360269.  Google Scholar

[31]

A. Jüngel and R. Pinnau, A positivity preserving numerical scheme for a nonlinear fourth-order parabolic equation, SIAM J. Num. Anal., 39 (2001), 385-406.  doi: 10.1137/S0036142900369362.  Google Scholar

[32]

A. Jüngel and I. Violet, First-order entropies for the Derrida-Lebowitz-Speer-Spohn equation, Discrete Cont. Dyn. Sys. B, 8 (2007), 861–877. doi: 10.3934/dcdsb.2007.8.861.  Google Scholar

[33]

J. R. King, The isolation oxidation of silicon: The reaction-controlled case, SIAM J. Appl. Math., 49 (1989), 1064-1080.  doi: 10.1137/0149064.  Google Scholar

[34]

J. R. Lister, G. G. Peng and J. A. Neufeld, Spread of a viscous fluid beneath an elastic sheet, Phys. Rev. Lett., 111 (2013). Google Scholar

[35]

J. Maas and D. Matthes, Long-time behavior of a finite volume discretization for a fourth order diffusion equation, Nonlinearity, 29 (2016), 1992-2023.  doi: 10.1088/0951-7715/29/7/1992.  Google Scholar

[36]

D. Matthes and H. Osberger, A Convergent Lagrangian Discretization for a Nonlinear Fourth-Order Equation, Foundations of Computational Mathematics, 17 (2017), 73-126.  doi: 10.1007/s10208-015-9284-6.  Google Scholar

[37]

T. G. Myers, Thin films with high surface tension, SIAM Rev., 40 (1998), 441-462.  doi: 10.1137/S003614459529284X.  Google Scholar

[38]

A. Novick-Cohen and A. Shishkov, The thin film equation with backwards second order diffusion, Interfaces and Free Boundaries, 12 (2010), 463-496.  doi: 10.4171/IFB/242.  Google Scholar

[39]

A. OronS. H. Davis and S. G. Bankoff, Long-scale evolution of thin liquid films, Rev. Mod. Phys., 69 (1997), 931-980.  doi: 10.1103/RevModPhys.69.931.  Google Scholar

[40]

A. Tarski, A Decision Method for Elementary Algebra and Geometry, University of California Press, Berkeley, CA, 1951.  Google Scholar

[41]

T. P. WitelskiA. J. Bernoff and A. L. Bertozzi, Blowup and dissipation in a critical case unstable thin film equation, Euro. Jnl. of Applied Mathematics, 15 (2004), 223-256.  doi: 10.1017/S0956792504005418.  Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savare, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhäuser Basel, 2008.  Google Scholar

[2]

H. Bae and R. Granero-Belinchón, Global existence and exponential decay to equilibrium for DLSS-Type equations, J. Dyn. Diff. Equat., (2020). doi: 10.1007/s10884-020-09852-5.  Google Scholar

[3]

J. Becker and G. Grün, The thin-film equation: Recent advances and some new perspectives, J. Phys.: Condens. Matter, 17 (2005), 291-307.  doi: 10.1088/0953-8984/17/9/002.  Google Scholar

[4]

F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Diff. Eqs., 83 (1990), 179-206.  doi: 10.1016/0022-0396(90)90074-Y.  Google Scholar

[5]

A. Bertozzi, The mathematics of moving contact lines in thin liquid films, Notices Amer. Math. Soc., 45 (1998), 689-697.   Google Scholar

[6]

P. M. Bleher, J. L. Lebowitz and E. R. Speer, Existence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations, Commun. Pure Appl. Math., 47 (1994), 923–942. doi: 10.1002/cpa.3160470702.  Google Scholar

[7]

C. Bordenave, P. Germain and T. Trogdon, An extension of the Derrida–Lebowitz–Speer–Spohn equation, J. Phys. A: Math. Theor., 48 (2015), 485205. doi: 10.1088/1751-8113/48/48/485205.  Google Scholar

[8]

M. BukalE. 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

[9]

M. Bukal, A. Jüngel and D. Matthes, A multidimensional nonlinear sixth-order quantum diffusion equation, Annales de l'IHP Analyse Non Linéaire, 30 (2013), 337–365. doi: 10.1016/j.anihpc.2012.08.003.  Google Scholar

[10]

M. BurgerL. He and C.-B. Schönlieb, Cahn-Hilliard inpainting and a generalization for grayvalue images, SIAM Journal on Imaging Sciences, 2 (2009), 1129-1167.  doi: 10.1137/080728548.  Google Scholar

[11]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, The Journal of Chemical Physics, 28 (1958), 258-267.  doi: 10.1063/1.1744102.  Google Scholar

[12]

J. A. CarrilloJ. DolbeaultI. Gentil and A. Jüngel, Entropy-Energy inequalities and improved convergence rates for nonlinear parabolic equations, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1027-1050.  doi: 10.3934/dcdsb.2006.6.1027.  Google Scholar

[13]

J. A. CarrilloA. Jüngel and S. Tang, Positive entropic schemes for a nonlinear fourth-order equation, Discrete Contin. Dyn. Syst. B, 3 (2003), 1-20.  doi: 10.3934/dcdsb.2003.3.1.  Google Scholar

[14]

J. A. Carrillo and G. Toscani, Long-Time Asymptotics for Strong Solutions of the Thin Film Equation, Commun. Math. Phys., 225 (2002), 551-571.  doi: 10.1007/s002200100591.  Google Scholar

[15]

X. ChenA. Jüngel and J.-G. Liu, A note on Aubin-Lions-Dubinskii lemmas, Acta Appl. Math., 133 (2014), 33-43.  doi: 10.1007/s10440-013-9858-8.  Google Scholar

[16]

P. ConstantinT. F. DupontR. E. GoldsteinL. P. KadanoffM. J. Shelley and S. -M. Zhou, Droplet breakup in a model of the Hele-Shaw cell, Phys. Rev. E, 47 (1993), 4169-4181.  doi: 10.1103/PhysRevE.47.4169.  Google Scholar

[17]

R. Dal PassoH. Garcke and G. Grün, On a fourth order degenerate parabolic equation: Global entropy estimates and qualitative behaviour of solutions, SIAM J. Math. Anal., 29 (1998), 321-342.  doi: 10.1137/S0036141096306170.  Google Scholar

[18]

P. DegondF. Méhats and C. Ringhofer, Quantum energy-transport and drift-diffusion models, J. Stat. Phys., 118 (2005), 625-667.  doi: 10.1007/s10955-004-8823-3.  Google Scholar

[19]

B. DerridaJ. L. LebowitzE. R. Speer and H. Spohn, Dynamics of an anchored Toom interface, J. Phys. A: Math. Gen., 24 (1991), 4805-4834.  doi: 10.1088/0305-4470/24/20/015.  Google Scholar

[20]

B. DüringD. Matthes and J. P. Milišić, A gradient flow scheme for nonlinear fourth order equations, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 935-959.  doi: 10.3934/dcdsb.2010.14.935.  Google Scholar

[21]

D. Furihata and T. Matsuo, Discrete Variational Derivative Method, Chapman & Hall/CRC Numerical Analysis and Scientific Computing. CRC Press, Boca Raton, FL, 2011.  Google Scholar

[22]

J. Fischer, Uniqueness of solutions of the Derrida-Lebowitz-Speer-Spohn equation and quantum drift-diffusion models, Comm. Partial Differential Equations, 38 (2013), 2004-2047.  doi: 10.1080/03605302.2013.823548.  Google Scholar

[23]

L. Giacomelli and F. Otto, Variational formulation for the lubrication approximation of the Hele-Shaw flow, Calc. Var. PDEs, 13 (2001), 377-403.  doi: 10.1007/s005260000077.  Google Scholar

[24]

U. GianazzaG. Savaré and G. Toscani, The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation, Arch. Ration. Mech. Anal., 194 (2009), 133-220.  doi: 10.1007/s00205-008-0186-5.  Google Scholar

[25]

A. E. Hosoi and L. Mahadevan, Peeling, healing and bursting in a lubricated elastic sheet, Phys. Rev. Lett., 93 (2004), 137802. doi: 10.1103/PhysRevLett.93.137802.  Google Scholar

[26]

C. JosserandY. Pomeau and S. Rica, Self-similar singularities in the kinetics of condensation, J. of Low Temp. Physics, 145 (2006), 231-265.  doi: 10.1007/s10909-006-9232-6.  Google Scholar

[27]

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

[28]

A. Jüngel and D. Matthes, The Derrida-Lebowitz-Speer-Spohn equation: existence, non-uniqueness, and decay rates of the solutions, SIAM J. Math. Anal., 39 (2008), 1996-2015.  doi: 10.1137/060676878.  Google Scholar

[29]

A. Jüngel and J. -P. Milišić, A sixth-order nonlinear parabolic equation for quantum systems, SIAM J. Math. Anal., 41 (2009), 1472-1490.  doi: 10.1137/080739021.  Google Scholar

[30]

A. Jüngel and R. Pinnau, Global non-negative solutions of a nonlinear fourth-oder parabolic equation for quantum systems, SIAM J. Math. Anal., 32 (2000), 760-777.  doi: 10.1137/S0036141099360269.  Google Scholar

[31]

A. Jüngel and R. Pinnau, A positivity preserving numerical scheme for a nonlinear fourth-order parabolic equation, SIAM J. Num. Anal., 39 (2001), 385-406.  doi: 10.1137/S0036142900369362.  Google Scholar

[32]

A. Jüngel and I. Violet, First-order entropies for the Derrida-Lebowitz-Speer-Spohn equation, Discrete Cont. Dyn. Sys. B, 8 (2007), 861–877. doi: 10.3934/dcdsb.2007.8.861.  Google Scholar

[33]

J. R. King, The isolation oxidation of silicon: The reaction-controlled case, SIAM J. Appl. Math., 49 (1989), 1064-1080.  doi: 10.1137/0149064.  Google Scholar

[34]

J. R. Lister, G. G. Peng and J. A. Neufeld, Spread of a viscous fluid beneath an elastic sheet, Phys. Rev. Lett., 111 (2013). Google Scholar

[35]

J. Maas and D. Matthes, Long-time behavior of a finite volume discretization for a fourth order diffusion equation, Nonlinearity, 29 (2016), 1992-2023.  doi: 10.1088/0951-7715/29/7/1992.  Google Scholar

[36]

D. Matthes and H. Osberger, A Convergent Lagrangian Discretization for a Nonlinear Fourth-Order Equation, Foundations of Computational Mathematics, 17 (2017), 73-126.  doi: 10.1007/s10208-015-9284-6.  Google Scholar

[37]

T. G. Myers, Thin films with high surface tension, SIAM Rev., 40 (1998), 441-462.  doi: 10.1137/S003614459529284X.  Google Scholar

[38]

A. Novick-Cohen and A. Shishkov, The thin film equation with backwards second order diffusion, Interfaces and Free Boundaries, 12 (2010), 463-496.  doi: 10.4171/IFB/242.  Google Scholar

[39]

A. OronS. H. Davis and S. G. Bankoff, Long-scale evolution of thin liquid films, Rev. Mod. Phys., 69 (1997), 931-980.  doi: 10.1103/RevModPhys.69.931.  Google Scholar

[40]

A. Tarski, A Decision Method for Elementary Algebra and Geometry, University of California Press, Berkeley, CA, 1951.  Google Scholar

[41]

T. P. WitelskiA. J. Bernoff and A. L. Bertozzi, Blowup and dissipation in a critical case unstable thin film equation, Euro. Jnl. of Applied Mathematics, 15 (2004), 223-256.  doi: 10.1017/S0956792504005418.  Google Scholar

Figure 1.  Numerical evolution of the corrected DLSS equation for unit mass initial datum $ u_0 $ at different time moments: $ t_1 = 5\cdot10^{-6} $, $ t_2 = 4\cdot10^{-5} $, $ t_3 = 2\cdot10^{-4} $, and $ t_4 = 1.5\cdot10^{-3} $
Figure 2.  Errors with respect to time and space discretization parameters. Dashed lines indicate theoretical convergence rates of the numerical scheme
Table 1.  Numerical convergence rates: $ \kappa_t $ in time (left) and $ \kappa_s $ in space (right) calculated according to (67)
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