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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 & 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[13]

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

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[13]

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

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

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