Topological Methods in Nonlinear Analysis

Rayleigh-Bénard problem for thermomicropolar fluids

Piotr Kalita, Grzegorz Łukaszewicz, and Jakub Siemianowski

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Abstract

The two-dimensional Rayleigh-Bénard problem for a thermomicropolar fluids model is considered. The existence of suitable weak solutions which may not be unique, and the existence of the unique strong solution are proved. The global attractor for the m-semiflow associated with weak solutions and the global attractor for semiflow associated with strong solutions are shown to be equal.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 52, Number 2 (2018), 477-514.

Dates
First available in Project Euclid: 25 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1532484286

Digital Object Identifier
doi:10.12775/TMNA.2018.012

Mathematical Reviews number (MathSciNet)
MR3915648

Zentralblatt MATH identifier
07051677

Citation

Kalita, Piotr; Łukaszewicz, Grzegorz; Siemianowski, Jakub. Rayleigh-Bénard problem for thermomicropolar fluids. Topol. Methods Nonlinear Anal. 52 (2018), no. 2, 477--514. doi:10.12775/TMNA.2018.012. https://projecteuclid.org/euclid.tmna/1532484286


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References

  • R.A. Adams and J.J. Fournier, Sobolev Spaces, Pure and Applied Mathematics, Elsevier/Academic Press, Amsterdam, 2003.
  • M. Boukrouche, G. Łukaszewicz and J. Real, On pullback attractors for a class of two-dimensional turbulent shear flows, Internat. J. Engrg. Sci. 44 (2006), 830–844.
  • H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.
  • L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. Math. 171 (2010) 1903–1930.
  • V.V. Chepyzhov, M. Conti and V. Pata, A minimal approach to the theory of global attractors, Discrete Contin. Dyn. Syst. 32 (2012), 2079–2088.
  • A. Cheskidov and M. Dai, The existence of a global attractor for the forced critical surface quasi-geostrophic equation in $L^2$, J. Math. Fluid Mech. (2017), DOI: 10.1007/s00021-017-0324-7.
  • P. Constantin, M. Coti Zelati and V. Vicol, Uniformly attracting limit sets for the critically dissipative SQG equation, Nonlinearity 29 (2016), 298–318.
  • P. Constantin and Ch. Doering, Heat transfer in convective turbulence, Nonlinearity 9 (1996), 1049–1060.
  • M. Coti Zelati, On the theory of global attractors and Lyapunov functionals, Set-Valued Var. Anal. 21 (2013), 127–149.
  • M. Coti Zelati and P. Kalita, Minimality properties of set-valued processes and their pullback attractors, SIAM J. Math. Anal. 47 (2015), 1530–1561.
  • M. Coti Zelati and P. Kalita, Smooth attractors for weak solutions of the SQG equation with critical dissipation, Discrete Contin. Dyn. Syst. Ser. B 22 (2017), 1957–1873.
  • T. Dłotko and C.Y. Sun, \rom2D Quasi-geostrophic equation; sub-critical and critical cases, Nonlinear Anal. 150 (2017), 38–60.
  • Ch. Doering and P.Constantin, Variational bounds on energy dissipation in incompressible flows. \romIII. Convection, Phys. Rev. E 53 (1996), 5957–5981.
  • A.C. Eringen, Theory of micropolar fluids, J. Math. Mech. 16 (1966), 1–18.
  • A.C. Eringen, Theory of thermomicrofluids, J. Math. Anal. Appl. 38 (1972), 480–496.
  • A.C. Eringen, Microcontinuum Field Theories \romII: Fluent Media, Springer, New York, 2001.
  • L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 1998.
  • P. Kalita, J.A. Langa and G. Łukaszewicz, Micropolar meets Newtonian. The Rayleigh–Bénard problem. (submitted)
  • P. Kalita and G. Łukaszewicz, Global attractors for multivalued semiflows with weak continuity properties, Nonlinear Anal. 101 (2014), 124–143.
  • P. Kalita and G. Łukaszewicz, Navier–Stokes Equations. An Introduction with Applications, Springer, Cham, 2016.
  • J.-L. Lions and E. Magenes, Problèmes aux Limites Non Homogènes et applications, Vol. 1, Dunod, Paris, 1968.
  • G. Łukaszewicz, Micropolar Fluids. Theory and Applications, Birkhäuser, Boston, MA, 1999.
  • G. Łukaszewicz, Long time behavior of \rom2D micropolar fluid flows, Math. Comput. Modelling 34 (2001), 487–509,
  • V.S. Melnik and J. Valero, On attractors of multivalued semiflows and differential inclusions, Set-Valued Anal. 6 (1998), 83–111.
  • L. E. Payne and B. Straughan, Critical Rayleigh numbers for oscillatory and nonlinear convection in an isotropic thermomicropolar fluid, Int. J. Engng. Sci. 27 (1989), 827–836.
  • J.C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, 2001.
  • B. Rummler, The eigenfunctions of the Stokes operator in special domains. II, Z. Angew. Math. Mech. 77 (1997), 669–675.
  • J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4) 146 (1987), 65–96.
  • A. Tarasińska, Global attractor for heat convection problem in a micropolar fluid, Math. Methods Appl. Sci. 29 (2006), 1215–1236.
  • A. Tarasińska, Pullback attractor for heat convection problem in a micropolar fluid, Nonlinear Anal. Real World Appl. 11 (2010), 1458–1471.
  • R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997.
  • R. Temam, Navier–Stokes equations. Theory and numerical analysis, North-Holland, Amsterdam, 1977.
  • X. Wang, Bound on vertical heat transport at large Prandtl number, Phys. D 237 (2008), 854–858.
  • C. Zhao, W. Sun, and C.H. Hsu, Pullback dynamical behaviors of the non-autonomous micropolar fluid flows, Dyn. Partial Differ. Equ. 12 (2015), 265–288.