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On the equations of free convection in an absolutely heat-conducting fluid. (English. Russian original) Zbl 0808.76089

Sib. Math. J. 34, No. 5, 988-998 (1993); translation from Sib. Mat. Zh. 34, No. 5, 218-229 (1993).
We study the Oberbeck-Boussinesq asymptotic model of free convection in the limit case when the thermal conductivity coefficient \(\delta\) tends to \(\infty\) at a constant Rayleigh number. We discuss questions of solvability for initial-boundary value problems. The situation here is the same as that for the system of Navier-Stokes equations: in the two- dimensional case global solvability is provable for both viscous and inviscid fluids, and in the three-dimensional case the existence of smooth solutions can be proved only for a small time, and global existence theorems can be established only for weak solutions and only in the case of a viscous fluid.
Finally, we deal with the growth of solutions in time for an inviscid fluid. The kinetic energy (as well as enstrophy – that is, the integral of the squared vorticity – in the case of a horizontal strip) is monotone increasing, which results in a slow explosion (passing to infinity at infinite time) or in transition to some state in which the temperature is balanced and the fluid particles move only horizontally. The same happens as the time tends to \(-\infty\).

MSC:

76R10 Free convection
80A20 Heat and mass transfer, heat flow (MSC2010)
35Q35 PDEs in connection with fluid mechanics
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[1] L. D. Landau and E. M. Lifshits, Hydrodynamics [in Russian], Nauka, Moscow (1986). · Zbl 0664.76001
[2] G. Z. Gershuni and E. M. Zhukhovitskiî, Convective Stability of an Incompressible Fluid [in Russian], Nauka, Moscow (1972).
[3] V. I. Yudovich, ?On free convection equations in Oberbeck-Boussinesq approximation,? submitted to VINITI, 1990, No. 6225-B90.
[4] V. I. Yudovich, ?Asymptotic behavior of limit cycles for the Lorenz system at large Rayleigh numbers,? submitted to VINITI, 1978, No. 2611-78.
[5] M. R. Ukhovskiî and V. I. Yudovich, ?On the equations of stationary convection,? Prikl. Mat. Mekh.,27, No. 2, 295-300 (1963).
[6] V. I. Yudovich, A Linearization Method in Hydrodynamic Stability Theory [in Russian], Izdat. Rostov. Univ., Rostov-on-Don (1984). · Zbl 0553.76038
[7] O. A. Ladyzhenskaya, Mathematical Problems of the Dynamics of a Viscous Incompressible Fluid [in Russian], Fizmatgiz, Moscow (1961). · Zbl 0131.09402
[8] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis [Russian translation], Mir, Moscow (1981). · Zbl 0529.35002
[9] N. D. Kopachevskiî, S. G. Kreîn, and Kan Zuî Ngo, Operator Methods in Linear Hydrodynamics: Evolution and Spectral Problems [in Russian], Nauka, Moscow (1989). · Zbl 0681.76001
[10] N. M. Gyunter, ?On the fundamental problem of hydrodynamics,? Izv. Fiz.-Mat. Inst. Steklov.,2, 1-168 (1926).
[11] N. M. Gyunter, ?On the motion of a fluid encapsulated in a given moving vessel,? Izv. Akad. Nauk SSSR Ser. Fiz.-Mat., 1323-1348, 1503-1532 (1926); 621-656, 735-756, 1139-1162 (1927); 9-30 (1928).
[12] L. Lichtenstein, Grundlagen der Hydromechanik, Berlin (1929). · JFM 55.1124.01
[13] V. I. Yudovich, ?On the problem of nonstationary flows of a perfect incompressible fluid through a fixed domain,? Dokl. Akad. Nauk SSSR,146, No. 3, 561-564 (1962).
[14] V. I. Yudovich, ?Nonstationary flows of a perfect incompressible fluid,? Zh. Vychisl. Mat. i Mat. Fiz.,3, No. 6, 1032-1036 (1963).
[15] V. I. Yudovich, ?A two-dimensional nonstationary problem of the flow of a perfect incompressible fluid through a fixed domain,? Mat. Sb.,64, No. 4, 562-588 (1964).
[16] W. Wolibner, ?Un theoreme sur l’existence des mouvement plan d’un fluid parfait, homogene, incompressible, pendant un temps infinitement long prehensible,? Math. Z., Bd. 37, No. 5, 698-726 (1933). · Zbl 0008.06901 · doi:10.1007/BF01474610
[17] T. Kato, ?Nonstationary flows of viscous and ideal fluids in ?3,? J. Funct. Anal.,9, 296-305 (1972). · Zbl 0229.76018 · doi:10.1016/0022-1236(72)90003-1
[18] D. G. Ebin and J. Marsden, ?Groups of diffeomorphisms and the motion of an incornpressible fluid,? Ann. Math.,92, 102-163 (1970). · Zbl 0211.57401 · doi:10.2307/1970699
[19] E. Lorenz, ?Deterministic nonperiodic flow,? J. Atmospheric Sci.,20, 130-141 (1963). · Zbl 1417.37129 · doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
[20] V. I. Yudovich, ?On a finite-dimensional model of free convection,? Izv. Akad. Nauk SSSR Ser. Fiz. Atmosfer. i Okeana,26, No. 12, 1323-1333 (1991).
[21] V. I. Yudovich, ?Free convection and ramification,? Prikl. Mat. Mekh.,31, No. 1, 101-111 (1967). · Zbl 0173.28803
[22] V. I. Yudovich, ?An example of loss of stability with the generation of a secondary flow of a fluid in a closed vessel,? Mat. Sb.,74, No. 4, 565-579 (1967).
[23] M. M. Vaînberg and V. A. Trenogin, Ramification Theory in Solving Nonlinear Equations [in Russian], Nauka, Moscow (1972).
[24] M. I. Vishik and L. A. Lyusternik, ?Solution of some perturbation problems in the case of matrices and selfadjoint and not selfadjoint differential equations,? Uspekhi Mat. Nauk.,15, No. 3, 3-78 (1960). · Zbl 0096.08702
[25] V. I. Yudovich, ?Stability of convection flows,? Prikl. Mat. Mekh.,31, No. 2, 272-281 (1967).
[26] V. V. Nemytskiî and V. V. Stepanov, Qualitive Theory of Differential Equations [in Russian], Gos. Izdat. Tekhn. Teoret. Lit., Moscow-Leningrad (1949).
[27] E. A. Barbashin, Introduction to Stability Theory [in Russian], Nauka, Moscow (1967). · Zbl 0198.19703
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