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October  2014, 34(10): 3921-3944. doi: 10.3934/dcds.2014.34.3921

Estimates on the distance of inertial manifolds

José M. Arrieta 1, and  Esperanza Santamaría 1,

1. 

Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain, Spain

Received  March 2013 Revised  May 2013 Published  April 2014

In this paper we obtain estimates on the distance of inertial manifolds for dynamical systems generated by evolutionary parabolic type equations. We consider the situation where the systems are defined in different phase spaces and we estimate the distance in terms of the distance of the resolvent operators of the corresponding elliptic operators and the distance of the nonlinearities of the equations.
Keywords: abstract parabolic equations, perturbations, Inertial manifolds, estimates, thin domains..
Mathematics Subject Classification: Primary: 35B42, 35K9.
Citation: José M. Arrieta, Esperanza Santamaría. Estimates on the distance of inertial manifolds. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 3921-3944. doi: 10.3934/dcds.2014.34.3921
References:
[1]

J. M. Arrieta and A. N. Carvalho, Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain, Journal of Differential Equations, 199 (2004), 143-178. doi: 10.1016/j.jde.2003.09.004.

[2]

J. M. Arrieta, A. N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains I. continuity of the set of equilibria, Journal of Differential Equations, 231 (2006). doi: 10.1016/j.jde.2006.06.002.

[3]

J. M. Arrieta, A. N. Carvalho and G. Lozada-Cruz, Dynamics in Dumbbell Domains III. Continuity of attractors, Journal of Differential Equations, 247 (2009), 225-259. doi: 10.1016/j.jde.2008.12.014.

[4]

J. M. Arrieta and E. Santamaría, Distance of attractors for thin domains, in preparation.

[5]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.

[6]

P. W. Bates, K. Lu and C. Zeng, Existence and persistence of invariant manifolds for semiflows in banach space, Mem. Am. Math. Soc., 135 (1998). doi: 10.1090/memo/0645.

[7]

A. N. Carvalho, J. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical-Systems, Applied Mathematical Sciences, 182, Springer, 2012. doi: 10.1007/978-1-4614-4581-4.

[8]

A. N. Carvalho and S. Piskarev, A general approximation scheme for attractors of abstract parabolic problems, Numerical Functional Analysis and Optimization, 27 (2006), 785-829. doi: 10.1080/01630560600882723.

[9]

S.-N. Chow, X.-B. Lin and K. Lu, Smooth invariant foliations in infinite dimensional spaces, Journal of Differential Equations, 94 (1991), 266-291. doi: 10.1016/0022-0396(91)90093-O.

[10]

S.-N. Chow, K. Lu and G. R. Sell, Smoothness of inertial manifolds, Journal of Mathematical Analysis and Applications, 169 (1992), 283-312. doi: 10.1016/0022-247X(92)90115-T.

[11]

J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, London Mathematical Society Lecture Note Series, 278, Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511526404.

[12]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, 1988.

[13]

J. K. Hale and G. Raugel, Reaction-Diffusion Equation on Thin Domains, J. Math. Pures et Appl., 71 (1992), 33-95.

[14]

D. B. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.

[15]

D. A. Jones, A. M. Stuart and E. S. Titi, Persistence of invariant sets for dissipative evolution equations, Journal of Mathematical Analysis and Applications, 219 (1998), 479-502. doi: 10.1006/jmaa.1997.5847.

[16]

P. S. Ngiamsunthorn, Invariant manifolds for parabolic equations under perturbation of the domain, Nonlinear Analysis TMA, 80 (2013), 23-48. doi: 10.1016/j.na.2012.12.001.

[17]

G. Raugel, Dynamics of partial differential equations on thin domains, in Dynamical Systems (Montecatini Terme, 1994), Lecture Notes in Math., 1609, Springer, Berlin, 1995, 208-315. doi: 10.1007/BFb0095241.

[18]

J. C. Robinson, Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2001.

[19]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer, 2002.

[20]

N. Varchon, Domain perturbation and invariant manifolds, J. Evol. Equ., 12 (2012), 547-569. doi: 10.1007/s00028-012-0144-4.

show all references

References:
[1]

J. M. Arrieta and A. N. Carvalho, Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain, Journal of Differential Equations, 199 (2004), 143-178. doi: 10.1016/j.jde.2003.09.004.

[2]

J. M. Arrieta, A. N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains I. continuity of the set of equilibria, Journal of Differential Equations, 231 (2006). doi: 10.1016/j.jde.2006.06.002.

[3]

J. M. Arrieta, A. N. Carvalho and G. Lozada-Cruz, Dynamics in Dumbbell Domains III. Continuity of attractors, Journal of Differential Equations, 247 (2009), 225-259. doi: 10.1016/j.jde.2008.12.014.

[4]

J. M. Arrieta and E. Santamaría, Distance of attractors for thin domains, in preparation.

[5]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.

[6]

P. W. Bates, K. Lu and C. Zeng, Existence and persistence of invariant manifolds for semiflows in banach space, Mem. Am. Math. Soc., 135 (1998). doi: 10.1090/memo/0645.

[7]

A. N. Carvalho, J. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical-Systems, Applied Mathematical Sciences, 182, Springer, 2012. doi: 10.1007/978-1-4614-4581-4.

[8]

A. N. Carvalho and S. Piskarev, A general approximation scheme for attractors of abstract parabolic problems, Numerical Functional Analysis and Optimization, 27 (2006), 785-829. doi: 10.1080/01630560600882723.

[9]

S.-N. Chow, X.-B. Lin and K. Lu, Smooth invariant foliations in infinite dimensional spaces, Journal of Differential Equations, 94 (1991), 266-291. doi: 10.1016/0022-0396(91)90093-O.

[10]

S.-N. Chow, K. Lu and G. R. Sell, Smoothness of inertial manifolds, Journal of Mathematical Analysis and Applications, 169 (1992), 283-312. doi: 10.1016/0022-247X(92)90115-T.

[11]

J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, London Mathematical Society Lecture Note Series, 278, Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511526404.

[12]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, 1988.

[13]

J. K. Hale and G. Raugel, Reaction-Diffusion Equation on Thin Domains, J. Math. Pures et Appl., 71 (1992), 33-95.

[14]

D. B. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.

[15]

D. A. Jones, A. M. Stuart and E. S. Titi, Persistence of invariant sets for dissipative evolution equations, Journal of Mathematical Analysis and Applications, 219 (1998), 479-502. doi: 10.1006/jmaa.1997.5847.

[16]

P. S. Ngiamsunthorn, Invariant manifolds for parabolic equations under perturbation of the domain, Nonlinear Analysis TMA, 80 (2013), 23-48. doi: 10.1016/j.na.2012.12.001.

[17]

G. Raugel, Dynamics of partial differential equations on thin domains, in Dynamical Systems (Montecatini Terme, 1994), Lecture Notes in Math., 1609, Springer, Berlin, 1995, 208-315. doi: 10.1007/BFb0095241.

[18]

J. C. Robinson, Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2001.

[19]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer, 2002.

[20]

N. Varchon, Domain perturbation and invariant manifolds, J. Evol. Equ., 12 (2012), 547-569. doi: 10.1007/s00028-012-0144-4.

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