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On the Boussinesq system with nonlinear thermal diffusion. (English) Zbl 0897.35065
The authors deal with the Boussinesq system of hydrodynamics \[ u_t+ (u\cdot \nabla)u -\text{div} \bigl(\mu (\theta) D(u)\bigr) +\nabla p=F (\theta), \quad \text{div} u=0, \] \[ \theta_t +u\cdot \nabla \theta- \Delta\varphi (\theta)=0. \] which is appropriately transformed to allow the use of projection technique used for Navier-Stokes equations. Under suitable conditions they show the existence of solutions in a domain \(\Omega \subset \mathbb{R}^N\), and some time interval \([0,T]\). The solution is searched in the space \(L^\infty (0,T;L^2 (\Omega)) \cap L^2(0,t; W_\sigma^{1,2} (\Omega))\). An interative procedure is proposed to decouple \(u\) and \(\theta\).

35Q35 PDEs in connection with fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
80A20 Heat and mass transfer, heat flow (MSC2010)
Full Text: DOI
[1] Antontsev, A.; Diaz, J.I., Space and time localization in the flow of two inmiscible fluids through a porous medium: energy methods applied to systems, Nonlinear anal., theory, meth. & app., Vol 16, No 4, 299-313, (1991) · Zbl 0716.76066
[2] Antontsev A. & Diaz J.I., Energy Methods for Free Boundary Problems in Continuum Mechanics. To appear in Birkhauser. · Zbl 0855.00021
[3] Benilan, P., Equations d’evolution dans un space de Banach quelconque et applications, ()
[4] Boussinesq, J., ()
[5] Casas, E., The Navier-Stokes equations coupled with the heat equation: analysis and control, Control and cybernetics, Vol 23, No 4, 605-620, (1994) · Zbl 0901.49003
[6] Diaz, J.I.; Galiano, G., Topological methods in nonlinear analysis, (), To appear in · Zbl 0897.35065
[7] Diaz, J.I.; Veron, L., Local vanishing properties of solutions to elliptic and parabolic equations, Trans. am. math. soc., 290, 787-814, (1985) · Zbl 0579.35003
[8] Diaz, J.I.; Vrabie, I.I., Compactness of the Green operator of nonlinear diffusion equations: applications to Boussinesq type systems in fluid dynamics, (), 399-416 · Zbl 0841.35048
[9] Foias, C.; Manley, O.; Temam, R., Attractors for the benard problem: existence and physical bounds on their fractal dimension, Nonlinear analysis, Vol 11, No 8, 939-967, (1987) · Zbl 0646.76098
[10] Gontscharowa, O., Solvability of the nonstationary problem for the free convection equation with the temperature dependent viscosity, Dinamika sploshnoi sredy, (1990), No 96
[11] Gontscharowa, O., About the uniqueness of the solution of the two-dimensional nonstationary problem for the equations of free convection with viscousity depending on temperature, Red sib.mat.j. no 260, V92, (1990)
[12] Joseph, D.D., Stability of fluid motions I and II. Springer tracts in natural philosophy, Vol 28, (1976), Berlin
[13] Ladyzhenskaya, O.A., ()
[14] Ladyzhenskaya, O.A.; Solonnikov, V.A.; Ural’ceva, N.N., Linear and quasilinear equations of parabolic type, () · Zbl 0164.12302
[15] Milhaljan, J.M., A rigorous exposition of the Boussinesq approximations applicable to a thin layer of fluid, Astrophysics J., Vol 136, 1126, (1962)
[16] Oberbeck, A., Uber die Wärmeleitung der flüssigkeiten bei der berücksichtigung der strömungen infolge von temperaturdifferenzen, Annalen der physik und chemie, 7, 271, (1879) · JFM 11.0787.01
[17] Rodrigues, J.F., Weak solutions for thermoconvective flows of Boussinesq-Stefan type, (), 93-116, Harlow · Zbl 0794.76087
[18] Rulla, J., Weak solutions to Stefan problems with prescribed convection, SIAM J. math. anal., Vol 18, No 6, 1784-1800, (1987) · Zbl 0642.35080
[19] Temam, R., ()
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