• Previous Article
    Pointwise gradient estimates for subquadratic elliptic systems with discontinuous coefficients
  • CPAA Home
  • This Issue
  • Next Article
    Ground states for asymptotically periodic fractional Kirchhoff equation with critical Sobolev exponent
November  2019, 18(6): 3161-3179. doi: 10.3934/cpaa.2019142

Time discretization of a nonlinear phase field system in general domains

1. 

Dipartimento di Matematica "F. Casorati", Università di Pavia, Research Associate at the IMATI – C.N.R. Pavia, Via Ferrata 5, 27100 Pavia, Italy

2. 

Department of Mathematics, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan

* Corresponding author

Received  November 2018 Revised  January 2019 Published  May 2019

This paper deals with the nonlinear phase field system
$ \begin{equation*} \begin{cases} \partial_t (\theta +\ell \varphi) - \Delta\theta = f & \mbox{in}\ \Omega\times(0, T), \\ \partial_t \varphi - \Delta\varphi + \xi + \pi(\varphi) = \ell \theta,\ \xi\in\beta(\varphi) & \mbox{in}\ \Omega\times(0, T) \end{cases} \end{equation*} $
in a general domain
$ \Omega\subseteq\mathbb{R}^{d} $
. Here
$ d \in \mathbb{N} $
,
$ T>0 $
,
$ \ell>0 $
,
$ f $
is a source term,
$ \beta $
is a maximal monotone graph and
$ \pi $
is a Lipschitz continuous function. We note that in the above system the nonlinearity
$ \beta+\pi $
replaces the derivative of a potential of double well type. Thus it turns out that the system is a generalization of the Caginalp phase field model and it has been studied by many authors in the case that
$ \Omega $
is a bounded domain. However, for unbounded domains the analysis of the system seems to be at an early stage. In this paper we study the existence of solutions by employing a time discretization scheme and passing to the limit as the time step
$ h $
goes to
$ 0 $
. In the limit procedure we face with the difficulty that the embedding
$ H^1(\Omega) \hookrightarrow L^2(\Omega) $
is not compact in the case of unbounded domains. Moreover, we can prove an interesting error estimate of order
$ h^{1/2} $
for the difference between continuous and discrete solutions.
Citation: Pierluigi Colli, Shunsuke Kurima. Time discretization of a nonlinear phase field system in general domains. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3161-3179. doi: 10.3934/cpaa.2019142
References:
[1]

T. G. AmlerN. D. BotkinK.-H. Hoffmann and K. A. Ruf, Regularity of solutions of a phase field model, Dyn. Partial Differ. Equ., 10 (2013), 353-365. doi: 10.4310/DPDE.2013.v10.n4.a3. Google Scholar

[2]

B. D. Bangola, Global and exponential attractors for a Caginalp type phase-field problem, Cent. Eur. J. Math., 11 (2013), 1651-1676. doi: 10.2478/s11533-013-0258-0. Google Scholar

[3]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach spaces, Noordhoff International Publishing, Leyden, 1976. Google Scholar

[4]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5. Google Scholar

[5]

V. BarbuP. ColliG. GilardiG. Marinoschi and E. Rocca, Sliding mode control for a nonlinear phase-field system, SIAM J. Control Optim., 55 (2017), 2108-2133. doi: 10.1137/15M102424X. Google Scholar

[6]

S. Benzoni-GavageL. ChupinD. Jamet and J. Vovelle, On a phase field model for solid-liquid phase transitions, Discrete Contin. Dyn. Syst., 32 (2012), 1997-2025. doi: 10.3934/dcds.2012.32.1997. Google Scholar

[7]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996. doi: 10.1007/978-1-4612-4048-8. Google Scholar

[8]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827. Google Scholar

[9]

G. CaginalpX. Chen and C. Eck, Numerical tests of a phase field model with second order accuracy, SIAM J. Appl. Math., 68 (2008), 1518-1534. doi: 10.1137/070680965. Google Scholar

[10]

G. Canevari and P. Colli, Convergence properties for a generalization of the Caginalp phase field system, Asymptot. Anal., 82 (2013), 139-162. Google Scholar

[11]

G. Canevari and P. Colli, Solvability and asymptotic analysis of a generalization of the Caginalp phase field system, Commun. Pure Appl. Anal., 11 (2012), 1959-1982. doi: 10.3934/cpaa.2012.11.1959. Google Scholar

[12]

O. CârjăA. Miranville and C. Moroşanu, On the existence, uniqueness and regularity of solutions to the phase-field system with a general regular potential and a general class of nonlinear and non-homogeneous boundary conditions, Nonlinear Anal., 113 (2015), 190-208. doi: 10.1016/j.na.2014.10.003. Google Scholar

[13]

X. ChenG. Caginalp and E. Esenturk, Interface conditions for a phase field model with anisotropic and non-local interactions, Arch. Ration. Mech. Anal., 202 (2011), 349-372. doi: 10.1007/s00205-011-0429-8. Google Scholar

[14]

L. CherfilsS. Gatti and A. Miranville, Long time behavior of the Caginalp system with singular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 11 (2012), 2261-2290. doi: 10.3934/cpaa.2012.11.2261. Google Scholar

[15]

L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115. doi: 10.1007/s10492-009-0008-6. Google Scholar

[16]

P. ColliG. Gilardi and G. Marinoschi, A boundary control problem for a possibly singular phase field system with dynamic boundary conditions, J. Math. Anal. Appl., 434 (2016), 432-463. doi: 10.1016/j.jmaa.2015.09.011. Google Scholar

[17]

P. ColliG. GilardiG. Marinoschi and E. Rocca, Optimal control for a phase field system with a possibly singular potential, Math. Control Relat. Fields, 6 (2016), 95-112. doi: 10.3934/mcrf.2016.6.95. Google Scholar

[18]

P. ColliD. HilhorstF. Issard-Roch and G. Schimperna, Long time convergence for a class of variational phase-field models, Discrete Contin. Dyn. Syst., 25 (2009), 63-81. doi: 10.3934/dcds.2009.25.63. Google Scholar

[19]

M. ContiS. GattiA. Miranville and R. Quintanilla, On a Caginalp phase-field system with two temperatures and memory, Milan J. Math., 85 (2017), 1-27. doi: 10.1007/s00032-017-0263-z. Google Scholar

[20]

M. Conti, S. Gatti and A. Miranville, Attractors for a Caginalp model with a logarithmic potential and coupled dynamic boundary conditions, Anal. Appl. (Singap.), 11 (2013), 1350024, 31 pp. doi: 10.1142/S0219530513500243. Google Scholar

[21]

G. Duvaut, Résolution d'un problème de Stefan (fusion d'un bloc de glace à zéro degré), C. R. Acad. Sci. Paris Sr. A-B, 276 (1973), A1461–A1463. Google Scholar

[22]

C. M. Elliott and S. Zheng, Global existence and stability of solutions to the phase-field equations, in "Free Boundary Problems", Internat. Ser. Numer. Math., 95, 46-58, Birkhäuser Verlag, Basel, (1990). Google Scholar

[23]

M. Frémond, Non-smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9. Google Scholar

[24]

T. FukaoS. Kurima and T. Yokota, Nonlinear diffusion equations as asymptotic limits of Cahn–Hilliard systems on unbounded domains via Cauchy's criterion, Math. Methods Appl. Sci., 41 (2018), 2590-2601. doi: 10.1002/mma.4760. Google Scholar

[25]

M. GrasselliH. Petzeltová and G. Schimperna, Long time behavior of solutions to the Caginalp system with singular potential, Z. Anal. Anwend., 25 (2006), 51-72. doi: 10.4171/ZAA/1277. Google Scholar

[26]

J.W. Jerome, Approximation of Nonlinear Evolution Systems, Mathematics in Science and Engineering 164, Academic Press Inc., Orlando, 1983. Google Scholar

[27]

K.-H. Hoffmann and L. S. Jiang, Optimal control of a phase field model for solidification, Numer. Funct. Anal. Optim., 13 (1992), 11-27. doi: 10.1080/01630569208816458. Google Scholar

[28]

K.-H. HoffmannN. KenmochiM. Kubo and N. Yamazaki, Optimal control problems for models of phase-field type with hysteresis of play operator, Adv. Math. Sci. Appl., 17 (2007), 305-336. Google Scholar

[29]

N. Kenmochi and M. Niezgódka, Evolution systems of nonlinear variational inequalities arising from phase change problems, Nonlinear Anal., 22 (1994), 1163-1180. doi: 10.1016/0362-546X(94)90235-6. Google Scholar

[30]

S. Kurima, Existence and energy estimates of weak solutions for nonlocal Cahn–Hilliard equations on unbounded domains, preprint, arXiv: 1806.06361, (2018).Google Scholar

[31]

S. Kurima, Asymptotic analysis for Cahn–Hilliard type phase field systems related to tumor growth in general domains, Math. Methods Appl. Sci, to appear.Google Scholar

[32]

S. Kurima and T. Yokota, A direct approach to quasilinear parabolic equations on unbounded domains by Brézis's theory for subdifferential operators, Adv. Math. Sci. Appl., 26 (2017), 221-242. Google Scholar

[33]

S. Kurima and T. Yokota, Monotonicity methods for nonlinear diffusion equations and their approximations with error estimates, J. Differential Equations, 263 (2017), 2024-2050. doi: 10.1016/j.jde.2017.03.040. Google Scholar

[34]

A. Miranville, Some mathematical models in phase transition, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 271-306. doi: 10.3934/dcdss.2014.7.271. Google Scholar

[35]

A. Miranville and C. Moroşanu, On the existence, uniqueness and regularity of solutions to the phase-field transition system with non-homogeneous Cauchy-Neumann and nonlinear dynamic boundary conditions, Appl. Math. Model., 40 (2016), 192-207. doi: 10.1016/j.apm.2015.04.039. Google Scholar

[36]

A. Miranville and A. J. Ntsokongo, On anisotropic Caginalp phase-field type models with singular nonlinear terms, J. Appl. Anal. Comput., 8 (2018), 655-674. Google Scholar

[37]

A. Miranville and R. Quintanilla, Spatial decay for several phase-field models, ZAMM Z. Angew. Math. Mech., 93 (2013), 801-810. doi: 10.1002/zamm.201200131. Google Scholar

[38]

A. Miranville and R. Quintanilla, A phase-field model based on a three-phase-lag heat conduction, Appl. Math. Optim., 63 (2011), 133-150. doi: 10.1007/s00245-010-9114-9. Google Scholar

[39]

A. Miranville and R. Quintanilla, Some generalizations of the Caginalp phase-field system, Appl. Anal., 88 (2009), 877-894. doi: 10.1080/00036810903042182. Google Scholar

[40]

T. Miyasita, Global existence and exponential attractor of solutions of Fix-Caginalp equation, Sci. Math. Jpn., 77 (2015), 339-355. Google Scholar

[41]

R. Rossi and G. Savaré, Gradient flows of non convex functionals in Hilbert spaces and applications, ESAIM Control Optim. Calc. Var., 12 (2006), 564-614. doi: 10.1051/cocv:2006013. Google Scholar

[42]

J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360. Google Scholar

[43]

A. Visintin, Models of Phase Transitions, Progress in Nonlinear Differential Equations and their Applications, 28, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4078-5. Google Scholar

show all references

References:
[1]

T. G. AmlerN. D. BotkinK.-H. Hoffmann and K. A. Ruf, Regularity of solutions of a phase field model, Dyn. Partial Differ. Equ., 10 (2013), 353-365. doi: 10.4310/DPDE.2013.v10.n4.a3. Google Scholar

[2]

B. D. Bangola, Global and exponential attractors for a Caginalp type phase-field problem, Cent. Eur. J. Math., 11 (2013), 1651-1676. doi: 10.2478/s11533-013-0258-0. Google Scholar

[3]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach spaces, Noordhoff International Publishing, Leyden, 1976. Google Scholar

[4]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5. Google Scholar

[5]

V. BarbuP. ColliG. GilardiG. Marinoschi and E. Rocca, Sliding mode control for a nonlinear phase-field system, SIAM J. Control Optim., 55 (2017), 2108-2133. doi: 10.1137/15M102424X. Google Scholar

[6]

S. Benzoni-GavageL. ChupinD. Jamet and J. Vovelle, On a phase field model for solid-liquid phase transitions, Discrete Contin. Dyn. Syst., 32 (2012), 1997-2025. doi: 10.3934/dcds.2012.32.1997. Google Scholar

[7]

M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Springer, New York, 1996. doi: 10.1007/978-1-4612-4048-8. Google Scholar

[8]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827. Google Scholar

[9]

G. CaginalpX. Chen and C. Eck, Numerical tests of a phase field model with second order accuracy, SIAM J. Appl. Math., 68 (2008), 1518-1534. doi: 10.1137/070680965. Google Scholar

[10]

G. Canevari and P. Colli, Convergence properties for a generalization of the Caginalp phase field system, Asymptot. Anal., 82 (2013), 139-162. Google Scholar

[11]

G. Canevari and P. Colli, Solvability and asymptotic analysis of a generalization of the Caginalp phase field system, Commun. Pure Appl. Anal., 11 (2012), 1959-1982. doi: 10.3934/cpaa.2012.11.1959. Google Scholar

[12]

O. CârjăA. Miranville and C. Moroşanu, On the existence, uniqueness and regularity of solutions to the phase-field system with a general regular potential and a general class of nonlinear and non-homogeneous boundary conditions, Nonlinear Anal., 113 (2015), 190-208. doi: 10.1016/j.na.2014.10.003. Google Scholar

[13]

X. ChenG. Caginalp and E. Esenturk, Interface conditions for a phase field model with anisotropic and non-local interactions, Arch. Ration. Mech. Anal., 202 (2011), 349-372. doi: 10.1007/s00205-011-0429-8. Google Scholar

[14]

L. CherfilsS. Gatti and A. Miranville, Long time behavior of the Caginalp system with singular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 11 (2012), 2261-2290. doi: 10.3934/cpaa.2012.11.2261. Google Scholar

[15]

L. Cherfils and A. Miranville, On the Caginalp system with dynamic boundary conditions and singular potentials, Appl. Math., 54 (2009), 89-115. doi: 10.1007/s10492-009-0008-6. Google Scholar

[16]

P. ColliG. Gilardi and G. Marinoschi, A boundary control problem for a possibly singular phase field system with dynamic boundary conditions, J. Math. Anal. Appl., 434 (2016), 432-463. doi: 10.1016/j.jmaa.2015.09.011. Google Scholar

[17]

P. ColliG. GilardiG. Marinoschi and E. Rocca, Optimal control for a phase field system with a possibly singular potential, Math. Control Relat. Fields, 6 (2016), 95-112. doi: 10.3934/mcrf.2016.6.95. Google Scholar

[18]

P. ColliD. HilhorstF. Issard-Roch and G. Schimperna, Long time convergence for a class of variational phase-field models, Discrete Contin. Dyn. Syst., 25 (2009), 63-81. doi: 10.3934/dcds.2009.25.63. Google Scholar

[19]

M. ContiS. GattiA. Miranville and R. Quintanilla, On a Caginalp phase-field system with two temperatures and memory, Milan J. Math., 85 (2017), 1-27. doi: 10.1007/s00032-017-0263-z. Google Scholar

[20]

M. Conti, S. Gatti and A. Miranville, Attractors for a Caginalp model with a logarithmic potential and coupled dynamic boundary conditions, Anal. Appl. (Singap.), 11 (2013), 1350024, 31 pp. doi: 10.1142/S0219530513500243. Google Scholar

[21]

G. Duvaut, Résolution d'un problème de Stefan (fusion d'un bloc de glace à zéro degré), C. R. Acad. Sci. Paris Sr. A-B, 276 (1973), A1461–A1463. Google Scholar

[22]

C. M. Elliott and S. Zheng, Global existence and stability of solutions to the phase-field equations, in "Free Boundary Problems", Internat. Ser. Numer. Math., 95, 46-58, Birkhäuser Verlag, Basel, (1990). Google Scholar

[23]

M. Frémond, Non-smooth Thermomechanics, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-04800-9. Google Scholar

[24]

T. FukaoS. Kurima and T. Yokota, Nonlinear diffusion equations as asymptotic limits of Cahn–Hilliard systems on unbounded domains via Cauchy's criterion, Math. Methods Appl. Sci., 41 (2018), 2590-2601. doi: 10.1002/mma.4760. Google Scholar

[25]

M. GrasselliH. Petzeltová and G. Schimperna, Long time behavior of solutions to the Caginalp system with singular potential, Z. Anal. Anwend., 25 (2006), 51-72. doi: 10.4171/ZAA/1277. Google Scholar

[26]

J.W. Jerome, Approximation of Nonlinear Evolution Systems, Mathematics in Science and Engineering 164, Academic Press Inc., Orlando, 1983. Google Scholar

[27]

K.-H. Hoffmann and L. S. Jiang, Optimal control of a phase field model for solidification, Numer. Funct. Anal. Optim., 13 (1992), 11-27. doi: 10.1080/01630569208816458. Google Scholar

[28]

K.-H. HoffmannN. KenmochiM. Kubo and N. Yamazaki, Optimal control problems for models of phase-field type with hysteresis of play operator, Adv. Math. Sci. Appl., 17 (2007), 305-336. Google Scholar

[29]

N. Kenmochi and M. Niezgódka, Evolution systems of nonlinear variational inequalities arising from phase change problems, Nonlinear Anal., 22 (1994), 1163-1180. doi: 10.1016/0362-546X(94)90235-6. Google Scholar

[30]

S. Kurima, Existence and energy estimates of weak solutions for nonlocal Cahn–Hilliard equations on unbounded domains, preprint, arXiv: 1806.06361, (2018).Google Scholar

[31]

S. Kurima, Asymptotic analysis for Cahn–Hilliard type phase field systems related to tumor growth in general domains, Math. Methods Appl. Sci, to appear.Google Scholar

[32]

S. Kurima and T. Yokota, A direct approach to quasilinear parabolic equations on unbounded domains by Brézis's theory for subdifferential operators, Adv. Math. Sci. Appl., 26 (2017), 221-242. Google Scholar

[33]

S. Kurima and T. Yokota, Monotonicity methods for nonlinear diffusion equations and their approximations with error estimates, J. Differential Equations, 263 (2017), 2024-2050. doi: 10.1016/j.jde.2017.03.040. Google Scholar

[34]

A. Miranville, Some mathematical models in phase transition, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 271-306. doi: 10.3934/dcdss.2014.7.271. Google Scholar

[35]

A. Miranville and C. Moroşanu, On the existence, uniqueness and regularity of solutions to the phase-field transition system with non-homogeneous Cauchy-Neumann and nonlinear dynamic boundary conditions, Appl. Math. Model., 40 (2016), 192-207. doi: 10.1016/j.apm.2015.04.039. Google Scholar

[36]

A. Miranville and A. J. Ntsokongo, On anisotropic Caginalp phase-field type models with singular nonlinear terms, J. Appl. Anal. Comput., 8 (2018), 655-674. Google Scholar

[37]

A. Miranville and R. Quintanilla, Spatial decay for several phase-field models, ZAMM Z. Angew. Math. Mech., 93 (2013), 801-810. doi: 10.1002/zamm.201200131. Google Scholar

[38]

A. Miranville and R. Quintanilla, A phase-field model based on a three-phase-lag heat conduction, Appl. Math. Optim., 63 (2011), 133-150. doi: 10.1007/s00245-010-9114-9. Google Scholar

[39]

A. Miranville and R. Quintanilla, Some generalizations of the Caginalp phase-field system, Appl. Anal., 88 (2009), 877-894. doi: 10.1080/00036810903042182. Google Scholar

[40]

T. Miyasita, Global existence and exponential attractor of solutions of Fix-Caginalp equation, Sci. Math. Jpn., 77 (2015), 339-355. Google Scholar

[41]

R. Rossi and G. Savaré, Gradient flows of non convex functionals in Hilbert spaces and applications, ESAIM Control Optim. Calc. Var., 12 (2006), 564-614. doi: 10.1051/cocv:2006013. Google Scholar

[42]

J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360. Google Scholar

[43]

A. Visintin, Models of Phase Transitions, Progress in Nonlinear Differential Equations and their Applications, 28, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4078-5. Google Scholar

[1]

Pierluigi Colli, Gianni Gilardi, Elisabetta Rocca, Jürgen Sprekels. Asymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modelling tumor growth. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 37-54. doi: 10.3934/dcdss.2017002

[2]

M. Motta, C. Sartori. Exit time problems for nonlinear unbounded control systems. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 137-156. doi: 10.3934/dcds.1999.5.137

[3]

Paulo Cesar Carrião, Olimpio Hiroshi Miyagaki. On a class of variational systems in unbounded domains. Conference Publications, 2001, 2001 (Special) : 74-79. doi: 10.3934/proc.2001.2001.74

[4]

Orazio Muscato, Wolfgang Wagner. A stochastic algorithm without time discretization error for the Wigner equation. Kinetic & Related Models, 2019, 12 (1) : 59-77. doi: 10.3934/krm.2019003

[5]

Luca Rossi. Non-existence of positive solutions of fully nonlinear elliptic equations in unbounded domains. Communications on Pure & Applied Analysis, 2008, 7 (1) : 125-141. doi: 10.3934/cpaa.2008.7.125

[6]

Marcone C. Pereira, Ricardo P. Silva. Error estimates for a Neumann problem in highly oscillating thin domains. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 803-817. doi: 10.3934/dcds.2013.33.803

[7]

Elena Bonetti, Elisabetta Rocca. Global existence and long-time behaviour for a singular integro-differential phase-field system. Communications on Pure & Applied Analysis, 2007, 6 (2) : 367-387. doi: 10.3934/cpaa.2007.6.367

[8]

A. Jiménez-Casas. Invariant regions and global existence for a phase field model. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 273-281. doi: 10.3934/dcdss.2008.1.273

[9]

Y. Peng, X. Xiang. Second order nonlinear impulsive time-variant systems with unbounded perturbation and optimal controls. Journal of Industrial & Management Optimization, 2008, 4 (1) : 17-32. doi: 10.3934/jimo.2008.4.17

[10]

Maurizio Grasselli, Giulio Schimperna. Nonlocal phase-field systems with general potentials. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5089-5106. doi: 10.3934/dcds.2013.33.5089

[11]

Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641

[12]

Dalibor Pražák, Jakub Slavík. Attractors and entropy bounds for a nonlinear RDEs with distributed delay in unbounded domains. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1259-1277. doi: 10.3934/dcdsb.2016.21.1259

[13]

Olivier Goubet, Ezzeddine Zahrouni. On a time discretization of a weakly damped forced nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1429-1442. doi: 10.3934/cpaa.2008.7.1429

[14]

T. Colin, Géraldine Ebrard, Gérard Gallice. Semi-discretization in time for nonlinear Zakharov waves equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 263-282. doi: 10.3934/dcdsb.2009.11.263

[15]

Thomas Apel, Mariano Mateos, Johannes Pfefferer, Arnd Rösch. Error estimates for Dirichlet control problems in polygonal domains: Quasi-uniform meshes. Mathematical Control & Related Fields, 2018, 8 (1) : 217-245. doi: 10.3934/mcrf.2018010

[16]

Ulisse Stefanelli. Analysis of a variable time-step discretization for a phase transition model with micro-movements. Communications on Pure & Applied Analysis, 2006, 5 (3) : 659-673. doi: 10.3934/cpaa.2006.5.659

[17]

Pierluigi Colli, Danielle Hilhorst, Françoise Issard-Roch, Giulio Schimperna. Long time convergence for a class of variational phase-field models. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 63-81. doi: 10.3934/dcds.2009.25.63

[18]

Ahmed Bonfoh, Cyril D. Enyi. Large time behavior of a conserved phase-field system. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1077-1105. doi: 10.3934/cpaa.2016.15.1077

[19]

Ken Shirakawa. Stability analysis for phase field systems associated with crystalline-type energies. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 483-504. doi: 10.3934/dcdss.2011.4.483

[20]

S. Gatti, M. Grasselli, V. Pata, M. Squassina. Robust exponential attractors for a family of nonconserved phase-field systems with memory. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 1019-1029. doi: 10.3934/dcds.2005.12.1019

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (39)
  • HTML views (170)
  • Cited by (0)

Other articles
by authors

[Back to Top]