The Oberbeck–Boussinesq approximation and Rayleigh–Bénard convection revisited

Eduard Feireisl; Elisabetta Rocca; Giulio Schimperna

doi:10.3934/dcds.2024032

Article Contents

2024, Volume 44, Issue 8: 2387-2402. Doi: 10.3934/dcds.2024032

This issue Previous Article On semidiscrete models dominated by the heat, wave and Laplace equations Next Article Periodic dynamics of a mosquito population suppression model based on Wolbachia-infected males

The Oberbeck–Boussinesq approximation and Rayleigh–Bénard convection revisited

1.
Institute of Mathematics of the Academy of Sciences of the Czech Republic, Zitna 25, CZ-115 67 Praha 1, Czech Republic
2.
Dipartimento di Matematica, Università degli studi di Pavia and IMATI-C.N.R., via Ferrata, 5-27100 Pavia, Italy

^*Corresponding author: Elisabetta Rocca
^*Corresponding author: Elisabetta Rocca

Received: February 2024

Early access: March 19, 2024

Published online: March 19, 2024

The first author is supported byThe work of E.F. was partially supported by the Czech Sciences Foundation (GAČR), Grant Agreement 24-11034S. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO:67985840. The work of E.R. and G.S. has been partially supported by the MIUR-PRIN Grant 2020F3NCPX "Mathematics for industry 4.0 (Math4I4)". The present paper also benefits from the support of the GNAMPA (Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica).

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

We consider the Oberbeck–Boussinesq approximation driven by an inhomogeneous temperature distribution on the boundary of a bounded fluid domain. The relevant boundary conditions are perturbed by a non–local term arising in the incompressible limit of the Navier–Stokes–Fourier system. The long time behaviour of the resulting initial/boundary value problem is investigated.

Keywords:

Mathematics Subject Classification: Primary: 76D05, 35B50, 35B40; Secondary: 45K05.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	A. Abbatiello, M. Bulíček and Kaplický, On the exponential decay in time of solutions to ageneralized Navier-Stokes-Fourier system, J. Differential Equations, 379 (2024), 762-793. doi: 10.1016/j.jde.2023.10.036.
[2]	A. Abbatiello and E. Feireisl, The Oberbeck-Boussinesq system with non-local boundary conditions, Quart. Appl. Math., 81 (2023), 297-306.
[3]	P. Bella, E. Feireisl and F. Oschmann, Rigorous derivation of the Oberbeck-Boussinesq approximation revealing unexpected term, Comm. Math. Phys., 403 (2023), 1245-1273. doi: 10.1007/s00220-023-04823-5.
[4]	B. Birnir and N. Svanstedt, Existence theory and strong attractors for the Rayleigh-Bénard problem with a large aspect ratio, Discrete Contin. Dyn. Syst., 10 (2004), 53-74. doi: 10.3934/dcds.2004.10.53.
[5]	E. Bodenschatz, W. Pesch and G. Ahlers, Recent developments in Rayleigh-Bénard convection, Annu. Rev. Fluid Mech., 32 (2000), 709-778. doi: 10.1146/annurev.fluid.32.1.709.
[6]	M. Cabral, R. Rosa and R. Temam, Existence and dimension of the attractor for the Bénard problem on channel-like domains, Discrete Contin. Dyn. Syst., 10 (2004), 89-116. doi: 10.3934/dcds.2004.10.89.
[7]	R. Denk, M. Hieber and J. Prüss, Optimal ${L}^p-{L}^q$-estimates for parabolic boundary value problems with inhomogenous data, Math. Z., 257 (2007), 193-224. doi: 10.1007/s00209-007-0120-9.
[8]	R. E. Ecke and O. Shishkina, Turbulent rotating Rayleigh-Bénard convection, Annu. Rev. Fluid Mech., 55 (2023), 603-638.
[9]	F. Fanelli and E. Feireisl, Thermally driven fluid convection in the incompressible limit regime, 2023, arXiv: 2310.03881.
[10]	F. Fanelli, E. Feireisl and M. Hofmanová, Ergodic theory for energetically open compressible fluid flows, Phys. D, 423 (2021), Paper No. 132914, 25 pp. doi: 10.1016/j.physd.2021.132914.
[11]	E. Feireisl and A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, 2$^{nd}$ edition, Advances in Mathematical Fluid Mechanics. Birkhäuser/Springer, Cham, 2017.
[12]	C. Foias, O. Manley and R. Temam, Attractors for the Bénard problem: Existence and physical bounds on their fractal dimension, Nonlinear Anal., 11 (1987), 939-967. doi: 10.1016/0362-546X(87)90061-7.
[13]	J. Málek and J. Nečas, A finite-dimensional attractor for the three dimensional flow of incompressible fluid, J. Differential. Equations, 127 (1996), 498-518. doi: 10.1006/jdeq.1996.0080.
[14]	J. Málek and D. Pražák, Large time behavior via the method of l-trajectories, J. Differential. Equations, 181 (2002), 243-279. doi: 10.1006/jdeq.2001.4087.
[15]	G. R. Sell, Global attractors for the three-dimensional Navier-Stokes equations, J. Dynamics Differential Equations, 8 (1996), 1-33. doi: 10.1007/BF02218613.
[16]	X. Wang, A note on long time behavior of solutions to the Boussinesq system at large Prandtl number, Contemp. Math. Amer. Math. Soc., 371 (2005), 315-323. doi: 10.1090/conm/371/06862.