May  2015, 20(3): 781-810. doi: 10.3934/dcdsb.2015.20.781

Strong uniform attractors for non-autonomous dissipative PDEs with non translation-compact external forces

1. 

Department of Mathematics, University of Surrey, Guildford, GU2 7XH

Received  April 2014 Revised  August 2014 Published  January 2015

We give a comprehensive study of strong uniform attractors of nonautonomous dissipative systems for the case where the external forces are not translation compact. We introduce several new classes of external forces that are not translation compact, but nevertheless allow the attraction in a strong topology of the phase space to be verified and discuss in a more detailed way the class of so-called normal external forces introduced before. We also develop a unified approach to verify the asymptotic compactness for such systems based on the energy method and apply it to a number of equations of mathematical physics including the Navier-Stokes equations, damped wave equations and reaction-diffusing equations in unbounded domains.
Citation: Sergey Zelik. Strong uniform attractors for non-autonomous dissipative PDEs with non translation-compact external forces. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 781-810. doi: 10.3934/dcdsb.2015.20.781
References:
[1]

C. Anh and N. Quang, Uniform attractors for nonautonomous parabolic equations involving weighted p-Laplacian operators,, Ann. Polon. Math., 98 (2010), 251. doi: 10.4064/ap98-3-5. Google Scholar

[2]

A. Babin and M. Vishik, Attractors of Evolutionary Equations,, North Holland, (1992). Google Scholar

[3]

J. Ball, Global attractors for damped semilinear wave equations,, Discrete Contin. Dyn. Syst., 10 (2004), 31. doi: 10.3934/dcds.2004.10.31. Google Scholar

[4]

A. Carvalho, J. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems,, Applied Mathematical Sciences, (2013). doi: 10.1007/978-1-4614-4581-4. Google Scholar

[5]

V. Chepyzhov, On uniform attractors of dynamic processes and nonautonomous equations of mathematical physics,, Russian Math. Surveys, 68 (2013), 349. Google Scholar

[6]

V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society Colloquium Publications, (2002). Google Scholar

[7]

V. Chepyzhov and M. Vishik, Attractors of nonautonomous dynamical systems and their dimension,, J. Math. Pures Appl., 73 (1994), 279. Google Scholar

[8]

V. Chepyzhov and M. Vishik, Attractors of non-autonomous evolution equations with translation-compact symbols,, in Partial Differential Operators and Mathematical Physics (Holzhau, (1994), 49. Google Scholar

[9]

H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Theory Related Fields, 100 (1994), 365. doi: 10.1007/BF01193705. Google Scholar

[10]

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

[11]

V. Kalantarov, A. Savostianov and S. Zelik, Attractors for damped quintic wave equations in bounded domains,, submitted., (). Google Scholar

[12]

S. Lu, H. Wu and C. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces,, Discrete Contin. Dyn. Syst., 13 (2005), 701. doi: 10.3934/dcds.2005.13.701. Google Scholar

[13]

S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces,, J. Differential Equations, 230 (2006), 196. doi: 10.1016/j.jde.2006.07.009. Google Scholar

[14]

S. Lu., Attractors for nonautonomous reaction-diffusion systems with symbols without strong translation compactness,, Asymptot. Anal., 54 (2007), 197. Google Scholar

[15]

S. Ma, C. Zhong and H. Song, Attractors for nonautonomous 2D Navier-Stokes equations with less regular symbols,, Nonlinear Anal., 71 (2009), 4215. doi: 10.1016/j.na.2009.02.107. Google Scholar

[16]

S. Ma, X. Cheng and H. Li, Attractors for non-autonomous wave equations with a new class of external forces,, J. Math. Anal. Appl., 337 (2008), 808. doi: 10.1016/j.jmaa.2007.03.108. Google Scholar

[17]

S. Ma and C. Zhong, The attractors for weakly damped non-autonomous hyperbolic equations with a new class of external forces,, Discrete Contin. Dyn. Syst., 18 (2007), 53. doi: 10.3934/dcds.2007.18.53. Google Scholar

[18]

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

[19]

I. Moise, R. Rosa and X. Wang, Attractors for non-compact semigroups via energy equations,, Nonlinearity, 11 (1998), 1369. doi: 10.1088/0951-7715/11/5/012. Google Scholar

[20]

I. Moise, R. Rosa and X. Wang, Attractors for noncompact nonautonomous systems via energy equations,, Discrete Contin. Dyn. Syst., 10 (2004), 473. doi: 10.3934/dcds.2004.10.473. Google Scholar

[21]

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

[22]

A. Robertson and W. Robertson, Topological Vector Spaces,, Reprint of the second edition, (1980). Google Scholar

[23]

Y. Xie, K. Zhu and C. Sun, The existence of uniform attractors for non-autonomous reaction-diffusion equations on the whole space,, J. Math. Phys., 53 (2012). doi: 10.1063/1.4746693. Google Scholar

show all references

References:
[1]

C. Anh and N. Quang, Uniform attractors for nonautonomous parabolic equations involving weighted p-Laplacian operators,, Ann. Polon. Math., 98 (2010), 251. doi: 10.4064/ap98-3-5. Google Scholar

[2]

A. Babin and M. Vishik, Attractors of Evolutionary Equations,, North Holland, (1992). Google Scholar

[3]

J. Ball, Global attractors for damped semilinear wave equations,, Discrete Contin. Dyn. Syst., 10 (2004), 31. doi: 10.3934/dcds.2004.10.31. Google Scholar

[4]

A. Carvalho, J. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems,, Applied Mathematical Sciences, (2013). doi: 10.1007/978-1-4614-4581-4. Google Scholar

[5]

V. Chepyzhov, On uniform attractors of dynamic processes and nonautonomous equations of mathematical physics,, Russian Math. Surveys, 68 (2013), 349. Google Scholar

[6]

V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society Colloquium Publications, (2002). Google Scholar

[7]

V. Chepyzhov and M. Vishik, Attractors of nonautonomous dynamical systems and their dimension,, J. Math. Pures Appl., 73 (1994), 279. Google Scholar

[8]

V. Chepyzhov and M. Vishik, Attractors of non-autonomous evolution equations with translation-compact symbols,, in Partial Differential Operators and Mathematical Physics (Holzhau, (1994), 49. Google Scholar

[9]

H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Theory Related Fields, 100 (1994), 365. doi: 10.1007/BF01193705. Google Scholar

[10]

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

[11]

V. Kalantarov, A. Savostianov and S. Zelik, Attractors for damped quintic wave equations in bounded domains,, submitted., (). Google Scholar

[12]

S. Lu, H. Wu and C. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces,, Discrete Contin. Dyn. Syst., 13 (2005), 701. doi: 10.3934/dcds.2005.13.701. Google Scholar

[13]

S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces,, J. Differential Equations, 230 (2006), 196. doi: 10.1016/j.jde.2006.07.009. Google Scholar

[14]

S. Lu., Attractors for nonautonomous reaction-diffusion systems with symbols without strong translation compactness,, Asymptot. Anal., 54 (2007), 197. Google Scholar

[15]

S. Ma, C. Zhong and H. Song, Attractors for nonautonomous 2D Navier-Stokes equations with less regular symbols,, Nonlinear Anal., 71 (2009), 4215. doi: 10.1016/j.na.2009.02.107. Google Scholar

[16]

S. Ma, X. Cheng and H. Li, Attractors for non-autonomous wave equations with a new class of external forces,, J. Math. Anal. Appl., 337 (2008), 808. doi: 10.1016/j.jmaa.2007.03.108. Google Scholar

[17]

S. Ma and C. Zhong, The attractors for weakly damped non-autonomous hyperbolic equations with a new class of external forces,, Discrete Contin. Dyn. Syst., 18 (2007), 53. doi: 10.3934/dcds.2007.18.53. Google Scholar

[18]

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

[19]

I. Moise, R. Rosa and X. Wang, Attractors for non-compact semigroups via energy equations,, Nonlinearity, 11 (1998), 1369. doi: 10.1088/0951-7715/11/5/012. Google Scholar

[20]

I. Moise, R. Rosa and X. Wang, Attractors for noncompact nonautonomous systems via energy equations,, Discrete Contin. Dyn. Syst., 10 (2004), 473. doi: 10.3934/dcds.2004.10.473. Google Scholar

[21]

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

[22]

A. Robertson and W. Robertson, Topological Vector Spaces,, Reprint of the second edition, (1980). Google Scholar

[23]

Y. Xie, K. Zhu and C. Sun, The existence of uniform attractors for non-autonomous reaction-diffusion equations on the whole space,, J. Math. Phys., 53 (2012). doi: 10.1063/1.4746693. Google Scholar

[1]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087

[2]

Michael Zgurovsky, Mark Gluzman, Nataliia Gorban, Pavlo Kasyanov, Liliia Paliichuk, Olha Khomenko. Uniform global attractors for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 2053-2065. doi: 10.3934/dcdsb.2017120

[3]

Ahmed Y. Abdallah, Rania T. Wannan. Second order non-autonomous lattice systems and their uniform attractors. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1827-1846. doi: 10.3934/cpaa.2019085

[4]

Vladimir V. Chepyzhov. Trajectory attractors for non-autonomous dissipative 2d Euler equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 811-832. doi: 10.3934/dcdsb.2015.20.811

[5]

Gaocheng Yue. Attractors for non-autonomous reaction-diffusion equations with fractional diffusion in locally uniform spaces. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1645-1671. doi: 10.3934/dcdsb.2017079

[6]

Chunyou Sun, Daomin Cao, Jinqiao Duan. Non-autonomous wave dynamics with memory --- asymptotic regularity and uniform attractor. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 743-761. doi: 10.3934/dcdsb.2008.9.743

[7]

Grzegorz Łukaszewicz, James C. Robinson. Invariant measures for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4211-4222. doi: 10.3934/dcds.2014.34.4211

[8]

Flank D. M. Bezerra, Vera L. Carbone, Marcelo J. D. Nascimento, Karina Schiabel. Pullback attractors for a class of non-autonomous thermoelastic plate systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3553-3571. doi: 10.3934/dcdsb.2017214

[9]

Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17

[10]

Bixiang Wang. Random attractors for non-autonomous stochastic wave equations with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 269-300. doi: 10.3934/dcds.2014.34.269

[11]

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

[12]

Peter E. Kloeden, Jacson Simsen. Pullback attractors for non-autonomous evolution equations with spatially variable exponents. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2543-2557. doi: 10.3934/cpaa.2014.13.2543

[13]

Zhijian Yang, Yanan Li. Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4899-4912. doi: 10.3934/dcdsb.2019036

[14]

Alexandre N. Carvalho, José A. Langa, James C. Robinson. Non-autonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 703-747. doi: 10.3934/dcdsb.2015.20.703

[15]

David Cheban, Cristiana Mammana. Continuous dependence of attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 499-515. doi: 10.3934/dcds.2007.18.499

[16]

Noriaki Yamazaki. Global attractors for non-autonomous multivalued dynamical systems associated with double obstacle problems. Conference Publications, 2003, 2003 (Special) : 935-944. doi: 10.3934/proc.2003.2003.935

[17]

Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice systems with random coupled coefficients. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2221-2245. doi: 10.3934/cpaa.2016035

[18]

Abiti Adili, Bixiang Wang. Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 643-666. doi: 10.3934/dcdsb.2013.18.643

[19]

Abiti Adili, Bixiang Wang. Random attractors for non-autonomous stochastic FitzHugh-Nagumo systems with multiplicative noise. Conference Publications, 2013, 2013 (special) : 1-10. doi: 10.3934/proc.2013.2013.1

[20]

Xiaolin Jia, Caidi Zhao, Juan Cao. Uniform attractor of the non-autonomous discrete Selkov model. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 229-248. doi: 10.3934/dcds.2014.34.229

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]