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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

Sergey Zelik 1,

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.
Keywords: Non-autonomous equations, energy method., dissipative systems, asymptotic compactness, uniform attractors.
Mathematics Subject Classification: 35B40, 35B4.
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), 251-271. doi: 10.4064/ap98-3-5.  Google Scholar

[2]

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

[3]

J. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. 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, 182, Springer, New York, 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), 349-382.  Google Scholar

[6]

V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002.  Google Scholar

[7]

V. Chepyzhov and M. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333.  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), Oper. Theory Adv. Appl., 78, Birkhäuser, Basel, 1995, 49-60.  Google Scholar

[9]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. 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), 703-730. 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), 701-719. 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), 196-212. 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), 197-210.  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), 4215-4222. 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), 808-820. 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), 53-70. 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, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008, 103-200. 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), 1369-1393. 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), 473-496. doi: 10.3934/dcds.2004.10.473.  Google Scholar

[21]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 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, Cambridge Tracts in Mathematics, 53, Cambridge University Press, Cambridge-New York, 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), 082703, 11 pp. 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), 251-271. doi: 10.4064/ap98-3-5.  Google Scholar

[2]

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

[3]

J. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. 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, 182, Springer, New York, 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), 349-382.  Google Scholar

[6]

V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002.  Google Scholar

[7]

V. Chepyzhov and M. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333.  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), Oper. Theory Adv. Appl., 78, Birkhäuser, Basel, 1995, 49-60.  Google Scholar

[9]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. 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), 703-730. 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), 701-719. 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), 196-212. 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), 197-210.  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), 4215-4222. 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), 808-820. 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), 53-70. 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, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008, 103-200. 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), 1369-1393. 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), 473-496. doi: 10.3934/dcds.2004.10.473.  Google Scholar

[21]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 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, Cambridge Tracts in Mathematics, 53, Cambridge University Press, Cambridge-New York, 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), 082703, 11 pp. doi: 10.1063/1.4746693.  Google Scholar

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