• Previous Article
    Existence and uniqueness of a solution for a class of parabolic equations with two unbounded nonlinearities
  • CPAA Home
  • This Issue
  • Next Article
    Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species
January  2016, 15(1): 219-241. doi: 10.3934/cpaa.2016.15.219

On the Swift-Hohenberg equation with slow and fast dynamics: well-posedness and long-time behavior

1. 

Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi 9, I-20133 Milano, Italy

Received  February 2015 Revised  April 2015 Published  December 2015

We propose a mathematical analysis of the Swift-Hohenberg equation arising from the phase field theory to model the transition from an unstable to a (meta)stable state. We also consider a recent generalization of the original equation, obtained by introducing an inertial term, to predict fast degrees of freedom in the system. We formulate and prove well-posedness results of the concerned models. Afterwards, we analyse the long-time behavior in terms of global and exponential attractors. Finally, by reading the inertial term as a singular perturbation of the Swift-Hohenberg equation, we construct a family of exponential attractors which is Hölder continuous with respect to the perturbative parameter of the system.
Citation: Andrea Giorgini. On the Swift-Hohenberg equation with slow and fast dynamics: well-posedness and long-time behavior. Communications on Pure & Applied Analysis, 2016, 15 (1) : 219-241. doi: 10.3934/cpaa.2016.15.219
References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland, (1992). Google Scholar

[2]

V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equation on $\mathbbR^3$,, \emph{Discrete Contin. Dynam. Systems}, 7 (2001), 719. doi: 10.3934/dcds.2001.7.719. Google Scholar

[3]

M. Cross and H. Greenside, Pattern Formation and Dynamics in Nonequilibrium Systems,, Cambridge University Press, (2009). Google Scholar

[4]

D. Danilov, P. Galenko and V. Lebedev, Phase-field-crystal and Swift-Hohenberg equations with fast dynamics,, \emph{Phys. Rev. E}, 79 (2009), 1. doi: 10.1103/PhysRevE.79.051110. Google Scholar

[5]

J. Duan, V. J. Ervin, H. Gao and G. Lin, Asymptotic dynamical difference between the nonlocal and local Swift-Hohenberg models,, \emph{J. Math. Phys.}, 41 (2000), 2077. doi: 10.1063/1.533228. Google Scholar

[6]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 135 (2005), 703. doi: 10.1017/S030821050000408X. Google Scholar

[7]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $R^3$,, \emph{C.R. Acad. Sci. Paris. S\'er. I}, 330 (2000), 713. doi: 10.1016/S0764-4442(00)00259-7. Google Scholar

[8]

K. R. Elder, M. Grant and J. Viñals, Dynamic scaling and quasi-ordered states in the 2-dimensional Swift-Hohenberg equation,, \emph{Phys. Rev. A}, 46 (1992), 7618. Google Scholar

[9]

A. B. Ezersky, M. I. Rabinovich and P. D. Weidman, The Dynamics of Patterns,, World Scientific Publishing, (2000). doi: 10.1142/9789812813350. Google Scholar

[10]

J. García-Ojalvo, A. Hernández-Machado and J. M. Sancho, Effects of external noise on the Swift-Hohenberg equation,, \emph{Phys. Rev. Lett.}, 71 (1992), 1542. Google Scholar

[11]

M. Grasselli and H. Wu, Well-posedness and longtime behavior for the modified phase-field crystal equation,, \emph{Math. Models Methods Appl. Sci.}, 24 (2014), 2743. doi: 10.1142/S0218202514500365. Google Scholar

[12]

M. Grasselli and H. Wu, Robust exponential attractors for the modified phase-field crystal equation,, \emph{Discrete Contin. Dyn. Syst. Ser. A}, 35 (2015), 2539. doi: 10.3934/dcds.2015.35.2539. Google Scholar

[13]

J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation,, \emph{J. Differential Equations}, 73 (1988), 197. doi: 10.1016/0022-0396(88)90104-0. Google Scholar

[14]

P. C. Hohenberg and J. Swift, Hydrodynamic fluctuations at the convective instability,, \emph{Phys. Rev. A}, 15 (1977), 319. Google Scholar

[15]

P. C. Hohenberg and J. Swift, Effects of additive noise at the onset of rayleigh-Bénard convection,, \emph{Phys. Rev. A}, 46 (1992), 4773. Google Scholar

[16]

J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires,, Dunod, (1969). Google Scholar

[17]

T. Ma, L. Y. Song and Y. D. Zhang, Global attractor of a modified Swift-Hohenberg equation in Hk spaces,, \emph{Nonlinear Anal.}, 72 (2010), 183. doi: 10.1016/j.na.2009.06.103. Google Scholar

[18]

A. Miranville, V. Pata and S. Zelik, Exponential attractors for singurarly perturbed damped wave equations: a simple construction,, \emph{Asymptot. Anal.}, 53 (2007), 1. Google Scholar

[19]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, \emph{Handbook of Differential Equations: Evolutionary Equations}, 4 (2008), 103. doi: 10.1016/S1874-5717(08)00003-0. Google Scholar

[20]

S. H. Park and J. Y. Park, Pullback attractor for a non-autonomous modified Swift-Hohenberg equation,, \emph{Comput. Math. Appl.}, 67 (2014), 542. doi: 10.1016/j.camwa.2013.11.011. Google Scholar

[21]

L. A. Peletier and J. F. Williams, Some canonical bifurcations in the Swift-Hohenberg equation,, \emph{SIAM J. Appl. Dyn. Sys.}, 6 (2007), 208. doi: 10.1137/050647232. Google Scholar

[22]

L. M. Pismen, Patterns and Interfaces in Dissipative Dynamics,, Springer-Verlag, (2006). Google Scholar

[23]

R. Temam, Infinite-Dimensional Dynamical System in Mechanics and Physics,, Second Edition, (1997). doi: 10.1007/978-1-4612-0645-3. Google Scholar

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland, (1992). Google Scholar

[2]

V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equation on $\mathbbR^3$,, \emph{Discrete Contin. Dynam. Systems}, 7 (2001), 719. doi: 10.3934/dcds.2001.7.719. Google Scholar

[3]

M. Cross and H. Greenside, Pattern Formation and Dynamics in Nonequilibrium Systems,, Cambridge University Press, (2009). Google Scholar

[4]

D. Danilov, P. Galenko and V. Lebedev, Phase-field-crystal and Swift-Hohenberg equations with fast dynamics,, \emph{Phys. Rev. E}, 79 (2009), 1. doi: 10.1103/PhysRevE.79.051110. Google Scholar

[5]

J. Duan, V. J. Ervin, H. Gao and G. Lin, Asymptotic dynamical difference between the nonlocal and local Swift-Hohenberg models,, \emph{J. Math. Phys.}, 41 (2000), 2077. doi: 10.1063/1.533228. Google Scholar

[6]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 135 (2005), 703. doi: 10.1017/S030821050000408X. Google Scholar

[7]

M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $R^3$,, \emph{C.R. Acad. Sci. Paris. S\'er. I}, 330 (2000), 713. doi: 10.1016/S0764-4442(00)00259-7. Google Scholar

[8]

K. R. Elder, M. Grant and J. Viñals, Dynamic scaling and quasi-ordered states in the 2-dimensional Swift-Hohenberg equation,, \emph{Phys. Rev. A}, 46 (1992), 7618. Google Scholar

[9]

A. B. Ezersky, M. I. Rabinovich and P. D. Weidman, The Dynamics of Patterns,, World Scientific Publishing, (2000). doi: 10.1142/9789812813350. Google Scholar

[10]

J. García-Ojalvo, A. Hernández-Machado and J. M. Sancho, Effects of external noise on the Swift-Hohenberg equation,, \emph{Phys. Rev. Lett.}, 71 (1992), 1542. Google Scholar

[11]

M. Grasselli and H. Wu, Well-posedness and longtime behavior for the modified phase-field crystal equation,, \emph{Math. Models Methods Appl. Sci.}, 24 (2014), 2743. doi: 10.1142/S0218202514500365. Google Scholar

[12]

M. Grasselli and H. Wu, Robust exponential attractors for the modified phase-field crystal equation,, \emph{Discrete Contin. Dyn. Syst. Ser. A}, 35 (2015), 2539. doi: 10.3934/dcds.2015.35.2539. Google Scholar

[13]

J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation,, \emph{J. Differential Equations}, 73 (1988), 197. doi: 10.1016/0022-0396(88)90104-0. Google Scholar

[14]

P. C. Hohenberg and J. Swift, Hydrodynamic fluctuations at the convective instability,, \emph{Phys. Rev. A}, 15 (1977), 319. Google Scholar

[15]

P. C. Hohenberg and J. Swift, Effects of additive noise at the onset of rayleigh-Bénard convection,, \emph{Phys. Rev. A}, 46 (1992), 4773. Google Scholar

[16]

J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires,, Dunod, (1969). Google Scholar

[17]

T. Ma, L. Y. Song and Y. D. Zhang, Global attractor of a modified Swift-Hohenberg equation in Hk spaces,, \emph{Nonlinear Anal.}, 72 (2010), 183. doi: 10.1016/j.na.2009.06.103. Google Scholar

[18]

A. Miranville, V. Pata and S. Zelik, Exponential attractors for singurarly perturbed damped wave equations: a simple construction,, \emph{Asymptot. Anal.}, 53 (2007), 1. Google Scholar

[19]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, \emph{Handbook of Differential Equations: Evolutionary Equations}, 4 (2008), 103. doi: 10.1016/S1874-5717(08)00003-0. Google Scholar

[20]

S. H. Park and J. Y. Park, Pullback attractor for a non-autonomous modified Swift-Hohenberg equation,, \emph{Comput. Math. Appl.}, 67 (2014), 542. doi: 10.1016/j.camwa.2013.11.011. Google Scholar

[21]

L. A. Peletier and J. F. Williams, Some canonical bifurcations in the Swift-Hohenberg equation,, \emph{SIAM J. Appl. Dyn. Sys.}, 6 (2007), 208. doi: 10.1137/050647232. Google Scholar

[22]

L. M. Pismen, Patterns and Interfaces in Dissipative Dynamics,, Springer-Verlag, (2006). Google Scholar

[23]

R. Temam, Infinite-Dimensional Dynamical System in Mechanics and Physics,, Second Edition, (1997). doi: 10.1007/978-1-4612-0645-3. Google Scholar

[1]

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

[2]

J. Burke, Edgar Knobloch. Multipulse states in the Swift-Hohenberg equation. Conference Publications, 2009, 2009 (Special) : 109-117. doi: 10.3934/proc.2009.2009.109

[3]

Jongmin Han, Masoud Yari. Dynamic bifurcation of the complex Swift-Hohenberg equation. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 875-891. doi: 10.3934/dcdsb.2009.11.875

[4]

Peng Gao. Averaging principles for the Swift-Hohenberg equation. Communications on Pure & Applied Analysis, 2020, 19 (1) : 293-310. doi: 10.3934/cpaa.2020016

[5]

Maurizio Grasselli, Hao Wu. Robust exponential attractors for the modified phase-field crystal equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2539-2564. doi: 10.3934/dcds.2015.35.2539

[6]

Stefan Meyer, Mathias Wilke. Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces. Evolution Equations & Control Theory, 2013, 2 (2) : 365-378. doi: 10.3934/eect.2013.2.365

[7]

Kei Matsuura, Mitsuharu Otani. Exponential attractors for a quasilinear parabolic equation. Conference Publications, 2007, 2007 (Special) : 713-720. doi: 10.3934/proc.2007.2007.713

[8]

Yuncherl Choi, Taeyoung Ha, Jongmin Han, Doo Seok Lee. Bifurcation and final patterns of a modified Swift-Hohenberg equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2543-2567. doi: 10.3934/dcdsb.2017087

[9]

Ling-Jun Wang. The dynamics of small amplitude solutions of the Swift-Hohenberg equation on a large interval. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1129-1156. doi: 10.3934/cpaa.2012.11.1129

[10]

Yanfeng Guo, Jinqiao Duan, Donglong Li. Approximation of random invariant manifolds for a stochastic Swift-Hohenberg equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1701-1715. doi: 10.3934/dcdss.2016071

[11]

Shengfu Deng. Periodic solutions and homoclinic solutions for a Swift-Hohenberg equation with dispersion. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1647-1662. doi: 10.3934/dcdss.2016068

[12]

John Burke, Edgar Knobloch. Normal form for spatial dynamics in the Swift-Hohenberg equation. Conference Publications, 2007, 2007 (Special) : 170-180. doi: 10.3934/proc.2007.2007.170

[13]

Toshiyuki Ogawa, Takashi Okuda. Bifurcation analysis to Swift-Hohenberg equation with Steklov type boundary conditions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 273-297. doi: 10.3934/dcds.2009.25.273

[14]

Tarek Saanouni. Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (1) : 273-291. doi: 10.3934/cpaa.2014.13.273

[15]

Cecilia Cavaterra, M. Grasselli. Robust exponential attractors for population dynamics models with infinite time delay. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1051-1076. doi: 10.3934/dcdsb.2006.6.1051

[16]

Peter E. Kloeden, José Real, Chunyou Sun. Robust exponential attractors for non-autonomous equations with memory. Communications on Pure & Applied Analysis, 2011, 10 (3) : 885-915. doi: 10.3934/cpaa.2011.10.885

[17]

Michele Coti Zelati. Global and exponential attractors for the singularly perturbed extensible beam. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 1041-1060. doi: 10.3934/dcds.2009.25.1041

[18]

Masoud Yari. Attractor bifurcation and final patterns of the n-dimensional and generalized Swift-Hohenberg equations. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 441-456. doi: 10.3934/dcdsb.2007.7.441

[19]

José A. Langa, Alain Miranville, José Real. Pullback exponential attractors. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1329-1357. doi: 10.3934/dcds.2010.26.1329

[20]

Jongmin Han, Chun-Hsiung Hsia. Dynamical bifurcation of the two dimensional Swift-Hohenberg equation with odd periodic condition. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2431-2449. doi: 10.3934/dcdsb.2012.17.2431

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]