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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 and 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, Amsterdam, 1992. [2] V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equation on $\mathbb{R}^3$, Discrete Contin. Dynam. Systems, 7 (2001), 719-735. doi: 10.3934/dcds.2001.7.719. [3] M. Cross and H. Greenside, Pattern Formation and Dynamics in Nonequilibrium Systems, Cambridge University Press, New York, 2009. [4] D. Danilov, P. Galenko and V. Lebedev, Phase-field-crystal and Swift-Hohenberg equations with fast dynamics, Phys. Rev. E, 79 (2009), 1-15. doi: 10.1103/PhysRevE.79.051110. [5] J. Duan, V. J. Ervin, H. Gao and G. Lin, Asymptotic dynamical difference between the nonlocal and local Swift-Hohenberg models, J. Math. Phys., 41 (2000), 2077-2089. doi: 10.1063/1.533228. [6] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 703-730. doi: 10.1017/S030821050000408X. [7] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $R^3$, C.R. Acad. Sci. Paris. Sér. I, 330 (2000), 713-718. doi: 10.1016/S0764-4442(00)00259-7. [8] K. R. Elder, M. Grant and J. Viñals, Dynamic scaling and quasi-ordered states in the 2-dimensional Swift-Hohenberg equation, Phys. Rev. A, 46 (1992), 7618-7629. [9] A. B. Ezersky, M. I. Rabinovich and P. D. Weidman, The Dynamics of Patterns, World Scientific Publishing, River Edge, 2000. doi: 10.1142/9789812813350. [10] J. García-Ojalvo, A. Hernández-Machado and J. M. Sancho, Effects of external noise on the Swift-Hohenberg equation, Phys. Rev. Lett., 71 (1992), 1542-1546. [11] M. Grasselli and H. Wu, Well-posedness and longtime behavior for the modified phase-field crystal equation, Math. Models Methods Appl. Sci., 24 (2014), 2743-2783. doi: 10.1142/S0218202514500365. [12] M. Grasselli and H. Wu, Robust exponential attractors for the modified phase-field crystal equation, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 2539-2564. doi: 10.3934/dcds.2015.35.2539. [13] J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Differential Equations, 73 (1988), 197-214. doi: 10.1016/0022-0396(88)90104-0. [14] P. C. Hohenberg and J. Swift, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 (1977), 319-328. [15] P. C. Hohenberg and J. Swift, Effects of additive noise at the onset of rayleigh-Bénard convection, Phys. Rev. A, 46 (1992), 4773-4785. [16] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969. [17] T. Ma, L. Y. Song and Y. D. Zhang, Global attractor of a modified Swift-Hohenberg equation in Hk spaces, Nonlinear Anal., 72 (2010), 183-191. doi: 10.1016/j.na.2009.06.103. [18] A. Miranville, V. Pata and S. Zelik, Exponential attractors for singurarly perturbed damped wave equations: a simple construction, Asymptot. Anal., 53 (2007), 1-12. [19] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handbook of Differential Equations: Evolutionary Equations, 4 (2008) 103-200. doi: 10.1016/S1874-5717(08)00003-0. [20] S. H. Park and J. Y. Park, Pullback attractor for a non-autonomous modified Swift-Hohenberg equation, Comput. Math. Appl., 67 (2014), 542-548. doi: 10.1016/j.camwa.2013.11.011. [21] L. A. Peletier and J. F. Williams, Some canonical bifurcations in the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Sys., 6 (2007) 208-235. doi: 10.1137/050647232. [22] L. M. Pismen, Patterns and Interfaces in Dissipative Dynamics, Springer-Verlag, Berlin, 2006. [23] R. Temam, Infinite-Dimensional Dynamical System in Mechanics and Physics, Second Edition, Applied Mathematical Sciences, Vol. 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

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##### References:
 [1] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. [2] V. Belleri and V. Pata, Attractors for semilinear strongly damped wave equation on $\mathbb{R}^3$, Discrete Contin. Dynam. Systems, 7 (2001), 719-735. doi: 10.3934/dcds.2001.7.719. [3] M. Cross and H. Greenside, Pattern Formation and Dynamics in Nonequilibrium Systems, Cambridge University Press, New York, 2009. [4] D. Danilov, P. Galenko and V. Lebedev, Phase-field-crystal and Swift-Hohenberg equations with fast dynamics, Phys. Rev. E, 79 (2009), 1-15. doi: 10.1103/PhysRevE.79.051110. [5] J. Duan, V. J. Ervin, H. Gao and G. Lin, Asymptotic dynamical difference between the nonlocal and local Swift-Hohenberg models, J. Math. Phys., 41 (2000), 2077-2089. doi: 10.1063/1.533228. [6] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 703-730. doi: 10.1017/S030821050000408X. [7] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $R^3$, C.R. Acad. Sci. Paris. Sér. I, 330 (2000), 713-718. doi: 10.1016/S0764-4442(00)00259-7. [8] K. R. Elder, M. Grant and J. Viñals, Dynamic scaling and quasi-ordered states in the 2-dimensional Swift-Hohenberg equation, Phys. Rev. A, 46 (1992), 7618-7629. [9] A. B. Ezersky, M. I. Rabinovich and P. D. Weidman, The Dynamics of Patterns, World Scientific Publishing, River Edge, 2000. doi: 10.1142/9789812813350. [10] J. García-Ojalvo, A. Hernández-Machado and J. M. Sancho, Effects of external noise on the Swift-Hohenberg equation, Phys. Rev. Lett., 71 (1992), 1542-1546. [11] M. Grasselli and H. Wu, Well-posedness and longtime behavior for the modified phase-field crystal equation, Math. Models Methods Appl. Sci., 24 (2014), 2743-2783. doi: 10.1142/S0218202514500365. [12] M. Grasselli and H. Wu, Robust exponential attractors for the modified phase-field crystal equation, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 2539-2564. doi: 10.3934/dcds.2015.35.2539. [13] J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation, J. Differential Equations, 73 (1988), 197-214. doi: 10.1016/0022-0396(88)90104-0. [14] P. C. Hohenberg and J. Swift, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 (1977), 319-328. [15] P. C. Hohenberg and J. Swift, Effects of additive noise at the onset of rayleigh-Bénard convection, Phys. Rev. A, 46 (1992), 4773-4785. [16] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969. [17] T. Ma, L. Y. Song and Y. D. Zhang, Global attractor of a modified Swift-Hohenberg equation in Hk spaces, Nonlinear Anal., 72 (2010), 183-191. doi: 10.1016/j.na.2009.06.103. [18] A. Miranville, V. Pata and S. Zelik, Exponential attractors for singurarly perturbed damped wave equations: a simple construction, Asymptot. Anal., 53 (2007), 1-12. [19] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handbook of Differential Equations: Evolutionary Equations, 4 (2008) 103-200. doi: 10.1016/S1874-5717(08)00003-0. [20] S. H. Park and J. Y. Park, Pullback attractor for a non-autonomous modified Swift-Hohenberg equation, Comput. Math. Appl., 67 (2014), 542-548. doi: 10.1016/j.camwa.2013.11.011. [21] L. A. Peletier and J. F. Williams, Some canonical bifurcations in the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Sys., 6 (2007) 208-235. doi: 10.1137/050647232. [22] L. M. Pismen, Patterns and Interfaces in Dissipative Dynamics, Springer-Verlag, Berlin, 2006. [23] R. Temam, Infinite-Dimensional Dynamical System in Mechanics and Physics, Second Edition, Applied Mathematical Sciences, Vol. 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.
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