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March  2019, 24(3): 1199-1227. doi: 10.3934/dcdsb.2019012

On relation between attractors for single and multivalued semiflows for a certain class of PDEs

Piotr Kalita 1, , Grzegorz Łukaszewicz 2, and  Jakub Siemianowski 3,

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

Received  January 2018 Revised  May 2018 Published  March 2019 Early access  January 2019

Fund Project: Work of P.K.G.L,and J.S.was supported by National Science Center (NCN) of Poland under project No.DEC-2017/25/B/ST1/00302,work of P.K.and J.S.was partially supported by NCN of Poland under project No.UMO-2016/22/A/ST1/00077.

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.

Keywords: m-semiflow, global attractor, fractal dimension, thermomicropolar fluid, quasigeostrophic equation.
Mathematics Subject Classification: Primary: 35B41, 35B65, 76D03, 80M35.
Citation: Piotr Kalita, Grzegorz Łukaszewicz, Jakub Siemianowski. On relation between attractors for single and multivalued semiflows for a certain class of PDEs. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1199-1227. doi: 10.3934/dcdsb.2019012
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.

[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.

[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.

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