• Previous Article
    A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains
  • DCDS Home
  • This Issue
  • Next Article
    A constructive approach to robust chaos using invariant manifolds and expanding cones
July  2021, 41(7): 3389-3414. 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  July 2021 Early access  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 and Continuous Dynamical Systems, 2021, 41 (7) : 3389-3414. doi: 10.3934/dcds.2021001
References:
[1]

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

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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).

[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.

[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.

[37]

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

[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.

[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.

[40]

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

[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.

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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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).

[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.

[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.

[37]

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

[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.

[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.

[40]

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

[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.

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)
[1]

Makoto Okumura, Takeshi Fukao, Daisuke Furihata, Shuji Yoshikawa. A second-order accurate structure-preserving scheme for the Cahn-Hilliard equation with a dynamic boundary condition. Communications on Pure and Applied Analysis, 2022, 21 (2) : 355-392. doi: 10.3934/cpaa.2021181

[2]

Yones Esmaeelzade Aghdam, Hamid Safdari, Yaqub Azari, Hossein Jafari, Dumitru Baleanu. Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2025-2039. doi: 10.3934/dcdss.2020402

[3]

José A. Carrillo, Ansgar Jüngel, Shaoqiang Tang. Positive entropic schemes for a nonlinear fourth-order parabolic equation. Discrete and Continuous Dynamical Systems - B, 2003, 3 (1) : 1-20. doi: 10.3934/dcdsb.2003.3.1

[4]

Carlos Banquet, Élder J. Villamizar-Roa. On the management fourth-order Schrödinger-Hartree equation. Evolution Equations and Control Theory, 2020, 9 (3) : 865-889. doi: 10.3934/eect.2020037

[5]

Chunhua Jin, Jingxue Yin, Zejia Wang. Positive periodic solutions to a nonlinear fourth-order differential equation. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1225-1235. doi: 10.3934/cpaa.2008.7.1225

[6]

Toshiko Ogiwara, Danielle Hilhorst, Hiroshi Matano. Convergence and structure theorems for order-preserving dynamical systems with mass conservation. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3883-3907. doi: 10.3934/dcds.2020129

[7]

Makoto Okumura, Daisuke Furihata. A structure-preserving scheme for the Allen–Cahn equation with a dynamic boundary condition. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4927-4960. doi: 10.3934/dcds.2020206

[8]

Gabriele Bonanno, Beatrice Di Bella. Fourth-order hemivariational inequalities. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 729-739. doi: 10.3934/dcdss.2012.5.729

[9]

Marcel Braukhoff, Ansgar Jüngel. Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3335-3355. doi: 10.3934/dcdsb.2020234

[10]

Lili Ju, Xinfeng Liu, Wei Leng. Compact implicit integration factor methods for a family of semilinear fourth-order parabolic equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1667-1687. doi: 10.3934/dcdsb.2014.19.1667

[11]

Zhen-Zhen Tao, Bing Sun. Space-time spectral methods for a fourth-order parabolic optimal control problem in three control constraint cases. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022080

[12]

Jinxing Liu, Xiongrui Wang, Jun Zhou, Huan Zhang. Blow-up phenomena for the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4321-4335. doi: 10.3934/dcdss.2021108

[13]

Pablo Álvarez-Caudevilla, Jonathan D. Evans, Victor A. Galaktionov. Gradient blow-up for a fourth-order quasilinear Boussinesq-type equation. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3913-3938. doi: 10.3934/dcds.2018170

[14]

Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2021, 20 (2) : 651-680. doi: 10.3934/cpaa.2020284

[15]

Benoît Pausader. The focusing energy-critical fourth-order Schrödinger equation with radial data. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1275-1292. doi: 10.3934/dcds.2009.24.1275

[16]

Luca Calatroni, Bertram Düring, Carola-Bibiane Schönlieb. ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 931-957. doi: 10.3934/dcds.2014.34.931

[17]

Wenjun Liu, Zhijing Chen, Zhiyu Tu. New general decay result for a fourth-order Moore-Gibson-Thompson equation with memory. Electronic Research Archive, 2020, 28 (1) : 433-457. doi: 10.3934/era.2020025

[18]

Boling Guo, Jun Wu. Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021205

[19]

Kelin Li, Huafei Di. On the well-posedness and stability for the fourth-order Schrödinger equation with nonlinear derivative term. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4293-4320. doi: 10.3934/dcdss.2021122

[20]

Ying Wen, Jiebao Sun, Zhichang Guo. A new anisotropic fourth-order diffusion equation model based on image features for image denoising. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022004

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (148)
  • HTML views (147)
  • Cited by (1)

Other articles
by authors

[Back to Top]