-
Previous Article
Attractors of multivalued semi-flows generated by solutions of optimal control problems
- DCDS-B Home
- This Issue
-
Next Article
Forward attraction of pullback attractors and synchronizing behavior of gradient-like systems with nonautonomous perturbations
On relation between attractors for single and multivalued semiflows for a certain class of PDEs
1. | Faculty of Mathematics and Computer Sciences, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland |
2. | Faculty of Mathematics, Informatics, and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland |
3. | Faculty of Mathematics and Computer Sciences, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland |
Sometimes it is not possible to prove the uniqueness of the weak solutions for problems of mathematical physics, but it is possible to bootstrap their regularity to the regularity of strong solutions which are unique. In this paper we formulate an abstract setting for such class of problems and we provide the conditions under which the global attractors for both strong and weak solutions coincide and the fractal dimension of the common attractor is finite. We present two problems belonging to this class: planar Rayleigh–Bénard flow of thermomicropolar fluid and surface quasigeostrophic equation on torus.
References:
[1] |
J. M. Arrieta, A. Rodríguez–Bernal and J. Valero,
Dynamics of a reaction diffusion equation with a discontinuous nonlinearity, Int. J. Bifurcat. Chaos, 16 (2006), 2965-2984.
doi: 10.1142/S0218127406016586. |
[2] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North Holland, Amsterdam, London, New York, Tokyo, 1992. |
[3] |
F. Balibrea, T. Caraballo, P. E. Kloeden and J. Valero,
Recent developments in dynamical systems: Three perspectives, Int. J. Bifurcat. Chaos, 20 (2010), 2591-2636.
doi: 10.1142/S0218127410027246. |
[4] |
J. M. Ball,
Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, Nonlinear Sci., 7 (1997), 475-502.
doi: 10.1007/s003329900037. |
[5] |
L. A. Cafarelli and A. Vasseur,
Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[6] |
T. Caraballo, P. Marín-Rubio and J. C. Robinson,
A comparison between two theories for multi-valued semiflows and their asymptotic behaviour, Set-Valued Anal., 11 (2003), 297-322.
doi: 10.1023/A:1024422619616. |
[7] |
V. V. Chepyzhov and M. I. Vishik,
Trajectory attractors for evolution equations, CR. Acad. Sci. I-Math., 321 (1995), 1309-1314.
|
[8] |
A. Cheskidov and M. Dai,
The existence of a global attractor for the forced critical surface quasi-geostrophic equation in $L^2$, Journal of Mathematical Fluid Mechanics, 20 (2018), 213-225.
doi: 10.1007/s00021-017-0324-7. |
[9] |
A. Cheskidov and C. Foiaş,
On global attractors of the 3D Navier Stokes equations, Journal of Differential Equations, 231 (2006), 714-754.
doi: 10.1016/j.jde.2006.08.021. |
[10] |
J. W. Cholewa and T. Dłotko, Bi-spaces global attractors in abstract parabolic equations, Banach Center Publications, PWN, 60 2003, 13–26.
doi: 10.4064/bc60-0-1. |
[11] |
J. W. Cholewa, R. Czaja and G. Mola,
Remarks on the fractal dimension of bi-space global and exponential attractors, Bollettino dell'Unione Matematica Italiana, 1 (2008), 121-145.
|
[12] |
I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, American Mathematical Society, 195 2008, viii+183 pp.
doi: 10.1090/memo/0912. |
[13] |
P. Constantin, M. Coti Zelati and V. Vicol,
Uniformly attracting limit sets for the critically dissipative SQG equation, Nonlinearity, 29 (2016), 298-318.
doi: 10.1088/0951-7715/29/2/298. |
[14] |
P. Constantin, A. J. Majda and E. Tabak,
Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar, Nonlinearity, 7 (1994), 1495-1533.
doi: 10.1088/0951-7715/7/6/001. |
[15] |
P. Constantin, A. Tarfulea and V. Vicol,
Long time dynamics of forced critical SQG, Communications in Mathematical Physics, 335 (2015), 93-141.
doi: 10.1007/s00220-014-2129-3. |
[16] |
P. Constantin and V. Vicol,
Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geometric and Functional Analysis, 22 (2012), 1289-1321.
doi: 10.1007/s00039-012-0172-9. |
[17] |
M. Coti Zelati and P. Kalita,
Minimality properties of set-valued processes and their pullback attractors, SIAM Journal on Mathematical Analysis, 47 (2015), 1530-1561.
doi: 10.1137/140978995. |
[18] |
M. Coti Zelati and P. Kalita,
Smooth attractors for weak solutions of the SQG equation with critical dissipation, Discrete and Continuous Dynamical Systems - Series B, 55 (2017), 1857-1873.
doi: 10.3934/dcdsb.2017110. |
[19] |
A. Eden, C. Foiaş, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, John Wiley & Sons/Masson, Chichester, New York, Brisbane, Toronto, Singapore/Paris, Milan, Barcelona, 1994. |
[20] |
A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1–18. |
[21] |
A. C. Eringen, Theory of thermomicrofluids, J. Math. Anal. Appl., 38 (1972), 480–496. Google Scholar |
[22] |
P. Kalita, J. A. Langa and G. Łukaszewicz, Micropolar meets Newtonian. The Rayleigh–Bénard problem, Physica D: Nonlinear Phenomena, accepted for publication
doi: 10.1016/j.physd.2018.12.004. |
[23] |
P. Kalita, G. Łukaszewicz and J. Siemianowski, Rayleigh–Bénard problem for thermomicropolar Fluids, Topological Methods in Nonlinear Analysis, accepted for publication
doi: 10.12775/TMNA.2018.012. |
[24] |
O. V. Kapustyan and J. Valero, Comparison between trajectory and global attractors for evolution systems without uniqueness of solutions, Int. J. Bifurcat. Chaos, 20 (2010), 2723–2734.
doi: 10.1142/S0218127410027313. |
[25] |
A. Kiselev and F. Nazarov, A variation on a theme of Cafarelli and Vasseur, Journal of Mathematical Sciences, 166 (2010), 31–39.
doi: 10.1007/s10958-010-9842-z. |
[26] |
G. Łukaszewicz, Micropolar Fluids. Theory and Applications, Birkhäuser Boston, Inc., Boston, MA, 1999.
doi: 10.1007/978-1-4612-0641-5. |
[27] |
G. Łukaszewicz, Long time behavior of 2D micropolar fluid flows, Mathematical and Computer Modelling, 34 (2001), 487–509.
doi: 10.1016/S0895-7177(01)00078-4. |
[28] |
G. Łukaszewicz, Asymptotic behavior of micropolar fluid flows, International Journal of Engineering Science, 41 (2003), 259–269.
doi: 10.1016/S0020-7225(02)00208-2. |
[29] |
V. S. Melnik and J. Valero, On attractors of multivalued semiflows and differential inclusions, Set-Valued Anal., 6 (1998), 83–111.
doi: 10.1023/A:1008608431399. |
[30] |
V. S. Melnik and J. Valero, Addendum to ''On Attractors of Multivalued Semiflows and Differential Inclusions'' [Set-Valued Anal. 6 (1998), 83–111], Set-Valued Anal., 16 (2008), 507–509.
doi: 10.1007/s11228-007-0066-4. |
[31] |
L. E. Payne and B. Straughan, Critical Rayleigh numbers for oscillatory and nonlinear convection in an isotropic thermomicropolar fluid, International Journal of Engineering Science, 27 (1989), 827–836.
doi: 10.1016/0020-7225(89)90049-9. |
[32] |
S. G. Resnick, Dynamical Problems in Non-Linear Advective Partial Differential Equations, ProQuest LLC, Ann Arbor, MI, Ph.D. Thesis, 1995, The University of Chicago. |
[33] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, UK, 2001.
doi: 10.1007/978-94-010-0732-0. |
[34] |
B. Straughan, The Energy Method, Stability, and Nonlinear Convection, Springer, 2004.
doi: 10.1007/978-0-387-21740-6. |
[35] |
A. Tarasińska, Global attractor for heat convection problem in a micropolar fluid, Mathematical Methods in the Applied Sciences, 29 (2006), 1215–1236.
doi: 10.1002/mma.720. |
[36] |
A. Tarasińska, Pullback attractor for heat convection problem in a micropolar fluid, Nonlinear Analysis: Real World Applications, 11 (2010), 1458–1471.
doi: 10.1016/j.nonrwa.2009.03.003. |
[37] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Second Edition, Springer–Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[38] |
R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis, SIAM, Philadelphia, Pennsylvania, 1983. |
[39] |
J. Valero, Finite and infinite dimensional attractors of multi-valued reaction diffusion equations, Acta Math. Hungary, 88 (2000), 239–258.
doi: 10.1023/A:1006769315268. |
show all references
References:
[1] |
J. M. Arrieta, A. Rodríguez–Bernal and J. Valero,
Dynamics of a reaction diffusion equation with a discontinuous nonlinearity, Int. J. Bifurcat. Chaos, 16 (2006), 2965-2984.
doi: 10.1142/S0218127406016586. |
[2] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North Holland, Amsterdam, London, New York, Tokyo, 1992. |
[3] |
F. Balibrea, T. Caraballo, P. E. Kloeden and J. Valero,
Recent developments in dynamical systems: Three perspectives, Int. J. Bifurcat. Chaos, 20 (2010), 2591-2636.
doi: 10.1142/S0218127410027246. |
[4] |
J. M. Ball,
Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, Nonlinear Sci., 7 (1997), 475-502.
doi: 10.1007/s003329900037. |
[5] |
L. A. Cafarelli and A. Vasseur,
Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[6] |
T. Caraballo, P. Marín-Rubio and J. C. Robinson,
A comparison between two theories for multi-valued semiflows and their asymptotic behaviour, Set-Valued Anal., 11 (2003), 297-322.
doi: 10.1023/A:1024422619616. |
[7] |
V. V. Chepyzhov and M. I. Vishik,
Trajectory attractors for evolution equations, CR. Acad. Sci. I-Math., 321 (1995), 1309-1314.
|
[8] |
A. Cheskidov and M. Dai,
The existence of a global attractor for the forced critical surface quasi-geostrophic equation in $L^2$, Journal of Mathematical Fluid Mechanics, 20 (2018), 213-225.
doi: 10.1007/s00021-017-0324-7. |
[9] |
A. Cheskidov and C. Foiaş,
On global attractors of the 3D Navier Stokes equations, Journal of Differential Equations, 231 (2006), 714-754.
doi: 10.1016/j.jde.2006.08.021. |
[10] |
J. W. Cholewa and T. Dłotko, Bi-spaces global attractors in abstract parabolic equations, Banach Center Publications, PWN, 60 2003, 13–26.
doi: 10.4064/bc60-0-1. |
[11] |
J. W. Cholewa, R. Czaja and G. Mola,
Remarks on the fractal dimension of bi-space global and exponential attractors, Bollettino dell'Unione Matematica Italiana, 1 (2008), 121-145.
|
[12] |
I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, American Mathematical Society, 195 2008, viii+183 pp.
doi: 10.1090/memo/0912. |
[13] |
P. Constantin, M. Coti Zelati and V. Vicol,
Uniformly attracting limit sets for the critically dissipative SQG equation, Nonlinearity, 29 (2016), 298-318.
doi: 10.1088/0951-7715/29/2/298. |
[14] |
P. Constantin, A. J. Majda and E. Tabak,
Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar, Nonlinearity, 7 (1994), 1495-1533.
doi: 10.1088/0951-7715/7/6/001. |
[15] |
P. Constantin, A. Tarfulea and V. Vicol,
Long time dynamics of forced critical SQG, Communications in Mathematical Physics, 335 (2015), 93-141.
doi: 10.1007/s00220-014-2129-3. |
[16] |
P. Constantin and V. Vicol,
Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geometric and Functional Analysis, 22 (2012), 1289-1321.
doi: 10.1007/s00039-012-0172-9. |
[17] |
M. Coti Zelati and P. Kalita,
Minimality properties of set-valued processes and their pullback attractors, SIAM Journal on Mathematical Analysis, 47 (2015), 1530-1561.
doi: 10.1137/140978995. |
[18] |
M. Coti Zelati and P. Kalita,
Smooth attractors for weak solutions of the SQG equation with critical dissipation, Discrete and Continuous Dynamical Systems - Series B, 55 (2017), 1857-1873.
doi: 10.3934/dcdsb.2017110. |
[19] |
A. Eden, C. Foiaş, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, John Wiley & Sons/Masson, Chichester, New York, Brisbane, Toronto, Singapore/Paris, Milan, Barcelona, 1994. |
[20] |
A. C. Eringen, Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1–18. |
[21] |
A. C. Eringen, Theory of thermomicrofluids, J. Math. Anal. Appl., 38 (1972), 480–496. Google Scholar |
[22] |
P. Kalita, J. A. Langa and G. Łukaszewicz, Micropolar meets Newtonian. The Rayleigh–Bénard problem, Physica D: Nonlinear Phenomena, accepted for publication
doi: 10.1016/j.physd.2018.12.004. |
[23] |
P. Kalita, G. Łukaszewicz and J. Siemianowski, Rayleigh–Bénard problem for thermomicropolar Fluids, Topological Methods in Nonlinear Analysis, accepted for publication
doi: 10.12775/TMNA.2018.012. |
[24] |
O. V. Kapustyan and J. Valero, Comparison between trajectory and global attractors for evolution systems without uniqueness of solutions, Int. J. Bifurcat. Chaos, 20 (2010), 2723–2734.
doi: 10.1142/S0218127410027313. |
[25] |
A. Kiselev and F. Nazarov, A variation on a theme of Cafarelli and Vasseur, Journal of Mathematical Sciences, 166 (2010), 31–39.
doi: 10.1007/s10958-010-9842-z. |
[26] |
G. Łukaszewicz, Micropolar Fluids. Theory and Applications, Birkhäuser Boston, Inc., Boston, MA, 1999.
doi: 10.1007/978-1-4612-0641-5. |
[27] |
G. Łukaszewicz, Long time behavior of 2D micropolar fluid flows, Mathematical and Computer Modelling, 34 (2001), 487–509.
doi: 10.1016/S0895-7177(01)00078-4. |
[28] |
G. Łukaszewicz, Asymptotic behavior of micropolar fluid flows, International Journal of Engineering Science, 41 (2003), 259–269.
doi: 10.1016/S0020-7225(02)00208-2. |
[29] |
V. S. Melnik and J. Valero, On attractors of multivalued semiflows and differential inclusions, Set-Valued Anal., 6 (1998), 83–111.
doi: 10.1023/A:1008608431399. |
[30] |
V. S. Melnik and J. Valero, Addendum to ''On Attractors of Multivalued Semiflows and Differential Inclusions'' [Set-Valued Anal. 6 (1998), 83–111], Set-Valued Anal., 16 (2008), 507–509.
doi: 10.1007/s11228-007-0066-4. |
[31] |
L. E. Payne and B. Straughan, Critical Rayleigh numbers for oscillatory and nonlinear convection in an isotropic thermomicropolar fluid, International Journal of Engineering Science, 27 (1989), 827–836.
doi: 10.1016/0020-7225(89)90049-9. |
[32] |
S. G. Resnick, Dynamical Problems in Non-Linear Advective Partial Differential Equations, ProQuest LLC, Ann Arbor, MI, Ph.D. Thesis, 1995, The University of Chicago. |
[33] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, UK, 2001.
doi: 10.1007/978-94-010-0732-0. |
[34] |
B. Straughan, The Energy Method, Stability, and Nonlinear Convection, Springer, 2004.
doi: 10.1007/978-0-387-21740-6. |
[35] |
A. Tarasińska, Global attractor for heat convection problem in a micropolar fluid, Mathematical Methods in the Applied Sciences, 29 (2006), 1215–1236.
doi: 10.1002/mma.720. |
[36] |
A. Tarasińska, Pullback attractor for heat convection problem in a micropolar fluid, Nonlinear Analysis: Real World Applications, 11 (2010), 1458–1471.
doi: 10.1016/j.nonrwa.2009.03.003. |
[37] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Second Edition, Springer–Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[38] |
R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis, SIAM, Philadelphia, Pennsylvania, 1983. |
[39] |
J. Valero, Finite and infinite dimensional attractors of multi-valued reaction diffusion equations, Acta Math. Hungary, 88 (2000), 239–258.
doi: 10.1023/A:1006769315268. |
[1] |
Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020345 |
[2] |
Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021015 |
[3] |
Fang Li, Bo You. On the dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021024 |
[4] |
Wenjun Liu, Hefeng Zhuang. Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 907-942. doi: 10.3934/dcdsb.2020147 |
[5] |
Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270 |
[6] |
Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299 |
[7] |
Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081 |
[8] |
Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243 |
[9] |
Zheng Han, Daoyuan Fang. Almost global existence for the Klein-Gordon equation with the Kirchhoff-type nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020287 |
[10] |
Yu Zhou, Xinfeng Dong, Yongzhuang Wei, Fengrong Zhang. A note on the Signal-to-noise ratio of $ (n, m) $-functions. Advances in Mathematics of Communications, 2020 doi: 10.3934/amc.2020117 |
[11] |
Hongwei Liu, Jingge Liu. On $ \sigma $-self-orthogonal constacyclic codes over $ \mathbb F_{p^m}+u\mathbb F_{p^m} $. Advances in Mathematics of Communications, 2020 doi: 10.3934/amc.2020127 |
[12] |
Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073 |
[13] |
Lisa Hernandez Lucas. Properties of sets of Subspaces with Constant Intersection Dimension. Advances in Mathematics of Communications, 2021, 15 (1) : 191-206. doi: 10.3934/amc.2020052 |
[14] |
Xin Zhong. Singularity formation to the nonhomogeneous magneto-micropolar fluid equations. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021021 |
[15] |
Zsolt Saffer, Miklós Telek, Gábor Horváth. Analysis of Markov-modulated fluid polling systems with gated discipline. Journal of Industrial & Management Optimization, 2021, 17 (2) : 575-599. doi: 10.3934/jimo.2019124 |
[16] |
Hai Q. Dinh, Bac T. Nguyen, Paravee Maneejuk. Constacyclic codes of length $ 8p^s $ over $ \mathbb F_{p^m} + u\mathbb F_{p^m} $. Advances in Mathematics of Communications, 2020 doi: 10.3934/amc.2020123 |
[17] |
João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138 |
[18] |
Sabira El Khalfaoui, Gábor P. Nagy. On the dimension of the subfield subcodes of 1-point Hermitian codes. Advances in Mathematics of Communications, 2021, 15 (2) : 219-226. doi: 10.3934/amc.2020054 |
[19] |
Chandra Shekhar, Amit Kumar, Shreekant Varshney, Sherif Ibrahim Ammar. $ \bf{M/G/1} $ fault-tolerant machining system with imperfection. Journal of Industrial & Management Optimization, 2021, 17 (1) : 1-28. doi: 10.3934/jimo.2019096 |
[20] |
Saadoun Mahmoudi, Karim Samei. Codes over $ \frak m $-adic completion rings. Advances in Mathematics of Communications, 2020 doi: 10.3934/amc.2020122 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]