November  2011, 30(4): 1037-1054. doi: 10.3934/dcds.2011.30.1037

Asymptotics for a generalized Cahn-Hilliard equation with forcing terms

1. 

Department of Applied Mathematics, University of Crete, 714 09 Heraklion, Greece, Greece

2. 

Department of Mathematics, University of Crete, GR-714 09 Heraklion, Crete, Greece

Received  February 2010 Revised  July 2010 Published  May 2011

Motivated by the physical theory of Critical Dynamics the Cahn-Hilliard equation on a bounded space domain is considered and forcing terms of general type are introduced. For such a rescaled equation the limiting inter-face problem is studied and the following are derived: (i) asymptotic results indicating that the forcing terms may slow down the equilibrium locally or globally, (ii) the sharp interface limit problem in the multidimensional case demonstrating a local influence in phase transitions of terms that stem from the chemical potential, while free energy independent terms act on the rest of the domain, (iii) a limiting non-homogeneous linear diffusion equation for the one-dimensional problem in the case of deterministic forcing term that follows the white noise scaling.
Citation: Dimitra Antonopoulou, Georgia Karali, Georgios T. Kossioris. Asymptotics for a generalized Cahn-Hilliard equation with forcing terms. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1037-1054. doi: 10.3934/dcds.2011.30.1037
References:
[1]

N. D. Alikakos, P. W. Bates and X. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model,, Arch. Rat. Mech. Anal., 128 (1994), 165.  doi: 10.1007/BF00375025.  Google Scholar

[2]

N. D. Alikakos, G. Fusco and G. Karali, Ostwald ripening in two dimensions- The rigorous derivation of the equations from Mullins-Sekerka dynamics,, Journ. of Differential Equations, 205 (2004), 1.   Google Scholar

[3]

T. Antal, M. Droz, J. Magnin and Z. Rácz, Formation of liesengang patterns: A spinodal decomposition scenarion,, Phys. Rev. Lett., 83 (1999), 2880.  doi: 10.1103/PhysRevLett.83.2880.  Google Scholar

[4]

G. Belletini, M. S. Gelli, S. Luckhaus and M. Novaga, Deterministic equivalent for the Allen Cahn energy of a scaling law in the Ising model,, Calc. Var., 26 (2006), 429.  doi: 10.1007/s00526-006-0012-6.  Google Scholar

[5]

P. W. Bates and J. Xun, Metastable patterns for the Cahn-Hilliard Equation, Part I,, Journ. of Differential Equations, 111 (1994), 421.   Google Scholar

[6]

P. W. Bates and J. Xun, Metastable patterns for the Cahn-Hilliard Equation, Part II,, Journ. of Differential Equations, 117 (1995), 165.   Google Scholar

[7]

D. Blömker, S. Maier-Paape and T. Wanner, Phase separation in stochastic Cahn-Hilliard models,, in, (2005), 1.   Google Scholar

[8]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy,, J. Chem. Phys., 28 (1958), 258.  doi: 10.1063/1.1744102.  Google Scholar

[9]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system II. Thermodynamic basis,, J. Chem. Phys., 30 (1959), 1121.  doi: 10.1063/1.1730145.  Google Scholar

[10]

G. Cagninalp and P. C. Fife, Dynamics of layered interfaces arising from phase boundaries,, SIAM J. Appl. Math., 48 (1988), 506.  doi: 10.1137/0148029.  Google Scholar

[11]

H. Cook, Brownian motion in spinodal decomposition,, Acta Metallurgica, 18 (1970), 297.  doi: 10.1016/0001-6160(70)90144-6.  Google Scholar

[12]

G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions,", Cambridge University Press, (1992).  doi: 10.1017/CBO9780511666223.  Google Scholar

[13]

G. Da Prato and A. Debussche, Stochastic Cahn-Hilliard equation,, Nonlinear Anal., 26 (1996), 241.  doi: 10.1016/0362-546X(94)00277-O.  Google Scholar

[14]

A. Debussche and L. Zambotti, Conservative stochastic Cahn-Hilliard equation with reflection,, Ann. Probab., 35 (2007), 1706.  doi: 10.1214/009117906000000773.  Google Scholar

[15]

Q. Du and T. Zhang, Numerical approximation of some linear stochastic partial differential equations driven by special additive noises,, SIAM J. Num. Anal., 40 (2002), 1421.  doi: 10.1137/S0036142901387956.  Google Scholar

[16]

C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation,, Arch. Rat. Mech. Anal., 96 (1986), 339.  doi: 10.1007/BF00251803.  Google Scholar

[17]

N. Elezovic and A. Mikelic, On the stochastic Cahn-Hilliard equation,, Nonlinear Anal., 16 (1991), 1169.  doi: 10.1016/0362-546X(91)90204-E.  Google Scholar

[18]

X. Chen, X. Hong and F. Yi, Existence, uniqueness and regularity of solutions of Mullins-Sekerka problem,, Comm. Partial Differential Equations, 21 (1996), 1705.   Google Scholar

[19]

P. C. Fife, Models for phase separation and their mathematics,, El. Journ. Diff. E., 48 (2000), 1.   Google Scholar

[20]

T. Funaki, The scaling limit for a Stochastic PDE and the separation of phases,, Probab. Theory Relat. Fields., 102 (1995), 221.  doi: 10.1007/BF01213390.  Google Scholar

[21]

T. Funaki, Singular limit for stochastic reaction-diffusion equation and generation of random interfaces,, Acta Math. Sinica, 15 (1999), 407.  doi: 10.1007/BF02650735.  Google Scholar

[22]

T. Funaki, Singular limit for reaction-diffusion equation with self-similar Gaussian noise,, in Proceedings of Taniguchi symposium, (2000), 132.   Google Scholar

[23]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance,, Physica D, 92 (1996), 178.  doi: 10.1016/0167-2789(95)00173-5.  Google Scholar

[24]

M. Hildebrand and A. S. Mikhailov, Mesoscopic modeling in the kinetic theory of adsorbates,, J. Phys. Chem., 100 (1996), 19089.  doi: 10.1021/jp961668w.  Google Scholar

[25]

P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena,, J. Rev. Mod. Phys., 49 (1977), 435.  doi: 10.1103/RevModPhys.49.435.  Google Scholar

[26]

G. Kallianpur and J. Xiong, "Stochastic Differential Equations in Infinite Dimensional Spaces,", Institute of Mathematical Statistics, 26 (1995).   Google Scholar

[27]

G. Karali, Phase boundaries motion preserving the volume of each connected component,, Asymptotic Analysis, 49 (2006), 17.   Google Scholar

[28]

G. Karali and M. A. Katsoulakis, The role of multiple microscopic mechanisms in cluster interface evolution,, J. Differential Equations, 235 (2007), 418.   Google Scholar

[29]

M. A. Katsoulakis and D. G. Vlachos, From microscopic interactions to macroscopic laws of cluster evolution,, Phys. Rev. Letters, 84 (2000), 1511.  doi: 10.1103/PhysRevLett.84.1511.  Google Scholar

[30]

K. Kitahara, Y. Oono and D. Jasnow, Phase separation dynamics and external force field,, Mod. Phys. Letters B, 2 (1988), 765.  doi: 10.1142/S0217984988000461.  Google Scholar

[31]

M. Katsoulakis, G. Kossioris and O. Lakkis, Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem,, Interfaces Free Bound, 9 (2007), 1.  doi: 10.4171/IFB/154.  Google Scholar

[32]

G. Kossioris and G. Zouraris, Fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise,, ESAIM: Mathematical Modelling and Numerical Analysis, ().   Google Scholar

[33]

L. D. Landau and E. M. Lifshitz, "Statistical Physics Part 1,", Course of Theoretical Physics, 5 (1994).   Google Scholar

[34]

J. S. Langer, Theory of spinodal decomposition in alloys,, Ann. of Phys., 65 (1971), 53.  doi: 10.1016/0003-4916(71)90162-X.  Google Scholar

[35]

Di Liu, Convergence of the spectral method for stochastic Ginzburg-Landau equation driven by space-time white noise,, Comm. Math. Sci., 1 (2003), 361.   Google Scholar

[36]

B. Øksendal, "Stochastic Differential Equations,", Springer, (2003).   Google Scholar

[37]

R. L. Pego, Front migration in the non-linear Cahn-Hilliard equation,, Proc. R. Soc. Lond. A, 422 (1989), 261.  doi: 10.1098/rspa.1989.0027.  Google Scholar

[38]

J. Printems, On the discretization in time of parabolic stochastic partial differential equations,, Mathematical Modelling and Numerical Analysis, 35 (2001), 1055.  doi: 10.1051/m2an:2001148.  Google Scholar

[39]

T. M. Rogers, K. R. Elder and R. C. Desai, Numerical study of the late stages of spinodal decomposition,, Phys. Rev. B, 37 (1988), 9638.  doi: 10.1103/PhysRevB.37.9638.  Google Scholar

[40]

Y. Yan, Semidiscrete Galerkin approximation for a linear stochastic parabolic partial differential equation driven by an additive noise,, BIT Numerical Mathematics, 44 (2004), 829.  doi: 10.1007/s10543-004-3755-5.  Google Scholar

[41]

Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations,, SIAM J. Numer. Anal., 43 (2005), 1363.  doi: 10.1137/040605278.  Google Scholar

[42]

J. B. Walsh, An introduction to stochastic partial differential equations,, Lecture Notes in Math., 984 (1986), 265.  doi: 10.1007/BFb0074920.  Google Scholar

[43]

Quan-Fang Wang and Shin-ichi Nakagiri, Weak solutions of Cahn-Hilliard equations having forcing terms and optimal control problems,, Mathematical Models in Functional Equations, 1128 (2000), 172.   Google Scholar

show all references

References:
[1]

N. D. Alikakos, P. W. Bates and X. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model,, Arch. Rat. Mech. Anal., 128 (1994), 165.  doi: 10.1007/BF00375025.  Google Scholar

[2]

N. D. Alikakos, G. Fusco and G. Karali, Ostwald ripening in two dimensions- The rigorous derivation of the equations from Mullins-Sekerka dynamics,, Journ. of Differential Equations, 205 (2004), 1.   Google Scholar

[3]

T. Antal, M. Droz, J. Magnin and Z. Rácz, Formation of liesengang patterns: A spinodal decomposition scenarion,, Phys. Rev. Lett., 83 (1999), 2880.  doi: 10.1103/PhysRevLett.83.2880.  Google Scholar

[4]

G. Belletini, M. S. Gelli, S. Luckhaus and M. Novaga, Deterministic equivalent for the Allen Cahn energy of a scaling law in the Ising model,, Calc. Var., 26 (2006), 429.  doi: 10.1007/s00526-006-0012-6.  Google Scholar

[5]

P. W. Bates and J. Xun, Metastable patterns for the Cahn-Hilliard Equation, Part I,, Journ. of Differential Equations, 111 (1994), 421.   Google Scholar

[6]

P. W. Bates and J. Xun, Metastable patterns for the Cahn-Hilliard Equation, Part II,, Journ. of Differential Equations, 117 (1995), 165.   Google Scholar

[7]

D. Blömker, S. Maier-Paape and T. Wanner, Phase separation in stochastic Cahn-Hilliard models,, in, (2005), 1.   Google Scholar

[8]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy,, J. Chem. Phys., 28 (1958), 258.  doi: 10.1063/1.1744102.  Google Scholar

[9]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system II. Thermodynamic basis,, J. Chem. Phys., 30 (1959), 1121.  doi: 10.1063/1.1730145.  Google Scholar

[10]

G. Cagninalp and P. C. Fife, Dynamics of layered interfaces arising from phase boundaries,, SIAM J. Appl. Math., 48 (1988), 506.  doi: 10.1137/0148029.  Google Scholar

[11]

H. Cook, Brownian motion in spinodal decomposition,, Acta Metallurgica, 18 (1970), 297.  doi: 10.1016/0001-6160(70)90144-6.  Google Scholar

[12]

G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions,", Cambridge University Press, (1992).  doi: 10.1017/CBO9780511666223.  Google Scholar

[13]

G. Da Prato and A. Debussche, Stochastic Cahn-Hilliard equation,, Nonlinear Anal., 26 (1996), 241.  doi: 10.1016/0362-546X(94)00277-O.  Google Scholar

[14]

A. Debussche and L. Zambotti, Conservative stochastic Cahn-Hilliard equation with reflection,, Ann. Probab., 35 (2007), 1706.  doi: 10.1214/009117906000000773.  Google Scholar

[15]

Q. Du and T. Zhang, Numerical approximation of some linear stochastic partial differential equations driven by special additive noises,, SIAM J. Num. Anal., 40 (2002), 1421.  doi: 10.1137/S0036142901387956.  Google Scholar

[16]

C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation,, Arch. Rat. Mech. Anal., 96 (1986), 339.  doi: 10.1007/BF00251803.  Google Scholar

[17]

N. Elezovic and A. Mikelic, On the stochastic Cahn-Hilliard equation,, Nonlinear Anal., 16 (1991), 1169.  doi: 10.1016/0362-546X(91)90204-E.  Google Scholar

[18]

X. Chen, X. Hong and F. Yi, Existence, uniqueness and regularity of solutions of Mullins-Sekerka problem,, Comm. Partial Differential Equations, 21 (1996), 1705.   Google Scholar

[19]

P. C. Fife, Models for phase separation and their mathematics,, El. Journ. Diff. E., 48 (2000), 1.   Google Scholar

[20]

T. Funaki, The scaling limit for a Stochastic PDE and the separation of phases,, Probab. Theory Relat. Fields., 102 (1995), 221.  doi: 10.1007/BF01213390.  Google Scholar

[21]

T. Funaki, Singular limit for stochastic reaction-diffusion equation and generation of random interfaces,, Acta Math. Sinica, 15 (1999), 407.  doi: 10.1007/BF02650735.  Google Scholar

[22]

T. Funaki, Singular limit for reaction-diffusion equation with self-similar Gaussian noise,, in Proceedings of Taniguchi symposium, (2000), 132.   Google Scholar

[23]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance,, Physica D, 92 (1996), 178.  doi: 10.1016/0167-2789(95)00173-5.  Google Scholar

[24]

M. Hildebrand and A. S. Mikhailov, Mesoscopic modeling in the kinetic theory of adsorbates,, J. Phys. Chem., 100 (1996), 19089.  doi: 10.1021/jp961668w.  Google Scholar

[25]

P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena,, J. Rev. Mod. Phys., 49 (1977), 435.  doi: 10.1103/RevModPhys.49.435.  Google Scholar

[26]

G. Kallianpur and J. Xiong, "Stochastic Differential Equations in Infinite Dimensional Spaces,", Institute of Mathematical Statistics, 26 (1995).   Google Scholar

[27]

G. Karali, Phase boundaries motion preserving the volume of each connected component,, Asymptotic Analysis, 49 (2006), 17.   Google Scholar

[28]

G. Karali and M. A. Katsoulakis, The role of multiple microscopic mechanisms in cluster interface evolution,, J. Differential Equations, 235 (2007), 418.   Google Scholar

[29]

M. A. Katsoulakis and D. G. Vlachos, From microscopic interactions to macroscopic laws of cluster evolution,, Phys. Rev. Letters, 84 (2000), 1511.  doi: 10.1103/PhysRevLett.84.1511.  Google Scholar

[30]

K. Kitahara, Y. Oono and D. Jasnow, Phase separation dynamics and external force field,, Mod. Phys. Letters B, 2 (1988), 765.  doi: 10.1142/S0217984988000461.  Google Scholar

[31]

M. Katsoulakis, G. Kossioris and O. Lakkis, Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem,, Interfaces Free Bound, 9 (2007), 1.  doi: 10.4171/IFB/154.  Google Scholar

[32]

G. Kossioris and G. Zouraris, Fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise,, ESAIM: Mathematical Modelling and Numerical Analysis, ().   Google Scholar

[33]

L. D. Landau and E. M. Lifshitz, "Statistical Physics Part 1,", Course of Theoretical Physics, 5 (1994).   Google Scholar

[34]

J. S. Langer, Theory of spinodal decomposition in alloys,, Ann. of Phys., 65 (1971), 53.  doi: 10.1016/0003-4916(71)90162-X.  Google Scholar

[35]

Di Liu, Convergence of the spectral method for stochastic Ginzburg-Landau equation driven by space-time white noise,, Comm. Math. Sci., 1 (2003), 361.   Google Scholar

[36]

B. Øksendal, "Stochastic Differential Equations,", Springer, (2003).   Google Scholar

[37]

R. L. Pego, Front migration in the non-linear Cahn-Hilliard equation,, Proc. R. Soc. Lond. A, 422 (1989), 261.  doi: 10.1098/rspa.1989.0027.  Google Scholar

[38]

J. Printems, On the discretization in time of parabolic stochastic partial differential equations,, Mathematical Modelling and Numerical Analysis, 35 (2001), 1055.  doi: 10.1051/m2an:2001148.  Google Scholar

[39]

T. M. Rogers, K. R. Elder and R. C. Desai, Numerical study of the late stages of spinodal decomposition,, Phys. Rev. B, 37 (1988), 9638.  doi: 10.1103/PhysRevB.37.9638.  Google Scholar

[40]

Y. Yan, Semidiscrete Galerkin approximation for a linear stochastic parabolic partial differential equation driven by an additive noise,, BIT Numerical Mathematics, 44 (2004), 829.  doi: 10.1007/s10543-004-3755-5.  Google Scholar

[41]

Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations,, SIAM J. Numer. Anal., 43 (2005), 1363.  doi: 10.1137/040605278.  Google Scholar

[42]

J. B. Walsh, An introduction to stochastic partial differential equations,, Lecture Notes in Math., 984 (1986), 265.  doi: 10.1007/BFb0074920.  Google Scholar

[43]

Quan-Fang Wang and Shin-ichi Nakagiri, Weak solutions of Cahn-Hilliard equations having forcing terms and optimal control problems,, Mathematical Models in Functional Equations, 1128 (2000), 172.   Google Scholar

[1]

Hyung Ju Hwang, Youngmin Oh, Marco Antonio Fontelos. The vanishing surface tension limit for the Hele-Shaw problem. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3479-3514. doi: 10.3934/dcdsb.2016108

[2]

Nataliya Vasylyeva, Vitalii Overko. The Hele-Shaw problem with surface tension in the case of subdiffusion. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1941-1974. doi: 10.3934/cpaa.2016023

[3]

Laurence Cherfils, Alain Miranville, Sergey Zelik. On a generalized Cahn-Hilliard equation with biological applications. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2013-2026. doi: 10.3934/dcdsb.2014.19.2013

[4]

A. Bernardini, J. Bragard, H. Mancini. Synchronization between two Hele-Shaw Cells. Mathematical Biosciences & Engineering, 2004, 1 (2) : 339-346. doi: 10.3934/mbe.2004.1.339

[5]

Francesco Della Porta, Maurizio Grasselli. On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems. Communications on Pure & Applied Analysis, 2016, 15 (2) : 299-317. doi: 10.3934/cpaa.2016.15.299

[6]

Dimitra Antonopoulou, Georgia Karali. Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 31-55. doi: 10.3934/dcdsb.2011.16.31

[7]

Wenbin Chen, Wenqiang Feng, Yuan Liu, Cheng Wang, Steven M. Wise. A second order energy stable scheme for the Cahn-Hilliard-Hele-Shaw equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 149-182. doi: 10.3934/dcdsb.2018090

[8]

Desheng Li, Xuewei Ju. On dynamical behavior of viscous Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2207-2221. doi: 10.3934/dcds.2012.32.2207

[9]

Álvaro Hernández, Michał Kowalczyk. Rotationally symmetric solutions to the Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 801-827. doi: 10.3934/dcds.2017033

[10]

FRANCESCO DELLA PORTA, Maurizio Grasselli. Erratum: "On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems" [Comm. Pure Appl. Anal. 15 (2016), 299--317]. Communications on Pure & Applied Analysis, 2017, 16 (1) : 369-372. doi: 10.3934/cpaa.2017018

[11]

Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127

[12]

Aibo Liu, Changchun Liu. Cauchy problem for a sixth order Cahn-Hilliard type equation with inertial term. Evolution Equations & Control Theory, 2015, 4 (3) : 315-324. doi: 10.3934/eect.2015.4.315

[13]

Riccarda Rossi. On two classes of generalized viscous Cahn-Hilliard equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 405-430. doi: 10.3934/cpaa.2005.4.405

[14]

Annalisa Iuorio, Stefano Melchionna. Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3765-3788. doi: 10.3934/dcds.2018163

[15]

Alain Miranville, Sergey Zelik. The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 275-310. doi: 10.3934/dcds.2010.28.275

[16]

S. Maier-Paape, Ulrich Miller. Connecting continua and curves of equilibria of the Cahn-Hilliard equation on the square. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1137-1153. doi: 10.3934/dcds.2006.15.1137

[17]

Laurence Cherfils, Madalina Petcu, Morgan Pierre. A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1511-1533. doi: 10.3934/dcds.2010.27.1511

[18]

Amy Novick-Cohen, Andrey Shishkov. Upper bounds for coarsening for the degenerate Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 251-272. doi: 10.3934/dcds.2009.25.251

[19]

Gianni Gilardi, A. Miranville, Giulio Schimperna. On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (3) : 881-912. doi: 10.3934/cpaa.2009.8.881

[20]

Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations & Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (7)

[Back to Top]