Geodesic
Linear stability of equilibria of a fluid that is a nonconductor of heat
Sbornik. Mathematics, Tome 82 (1995) no. 1, pp. 117-134 
V. I. Yudovich. Linear stability of equilibria of a fluid that is a nonconductor of heat. Sbornik. Mathematics, Tome 82 (1995) no. 1, pp. 117-134. http://geodesic.mathdoc.fr/item/SM_1995_82_1_a5/
@article{SM_1995_82_1_a5,
     author = {V. I. Yudovich},
     title = {Linear stability of equilibria of a~fluid that is a~nonconductor of heat},
     journal = {Sbornik. Mathematics},
     pages = {117--134},
     year = {1995},
     volume = {82},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1995_82_1_a5/}
}
TY  - JOUR
AU  - V. I. Yudovich
TI  - Linear stability of equilibria of a fluid that is a nonconductor of heat
JO  - Sbornik. Mathematics
PY  - 1995
SP  - 117
EP  - 134
VL  - 82
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1995_82_1_a5/
LA  - en
ID  - SM_1995_82_1_a5
ER  - 
%0 Journal Article
%A V. I. Yudovich
%T Linear stability of equilibria of a fluid that is a nonconductor of heat
%J Sbornik. Mathematics
%D 1995
%P 117-134
%V 82
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1995_82_1_a5/
%G en
%F SM_1995_82_1_a5

Voir la notice de l'article provenant de la source Math-Net.Ru

Convective stability is studied in the linear approximation of equilibria of a strongly viscous fluid that is a nonconductor of heat where the fluid fills a bounded domain in a gravitational field. The corresponding system consists of the heat equation with transport in a velocity field and the steady-state Stokes system for the velocity and pressure. The latter includes an Archimedean force proportional to the temperature. It is proved that equilibria for which the temperature strictly increases upward are stable in $L_2$ with respect to the temperature and in $W_2^2$ with respect to the velocity. Here, however, perturbations may die out arbitrarily slowly (Banach–Steinhaus stability). Under rough violation of the condition of monotonicity of the temperature the equilibrium is unstable. Some critical cases of stability are also considered.

[1] Yudovich V. I., Ob uravneniyakh svobodnoi konvektsii v priblizhenii Oberbeka–Bussineska, Rukopis dep. v VINITI, No 6225-V90, 1990

[2] Yudovich V. I., O konvektsii v silno vyazkoi zhidkosti. 1: globalnaya razreshimost, dissipativnost, pervyi perekhod, Rukopis dep. v VINITI. No 826-V91, 1991

[3] Yudovich V. I., O konvektsii v silno vyazkoi zhidkosti. 2: polzuschie techeniya neteploprovodnoi zhidkosti, Rukopis dep. v VINITI. No 2964-V92, 1992 | MR

[4] Yudovich V. I., Metod linearizatsii v gidrodinamicheskoi teorii ustoichivosti, Izd-vo RGU, Rostov n/D

[5] Yudovich V. I., “O potere gladkosti reshenii uravnenii Eilera so vremenem”, Dinamika sploshnoi sredy, no. 16, In-t gidrodinamiki SO AN SSSR, 1974, 71–78 | MR

[6] Ladyzhenskaya O. A., Matematicheskie voprosy dinamiki vyazkoi neszhimaemoi zhidkosti, Gos. izd-vo fiz.-mat. literatury, M., 1961

[7] Sobolev S. L., “Ob odnoi teoreme funktsionalnogo analiza”, Matem. sbornik, 4:3 (1938), 471–497 | Zbl

[8] Yudovich V. I., “O nekotorykh otsenkakh, svyazannykh s integralnymi operatorami i resheniyami ellipticheskikh uravnenii”, DAN SSSR, 138:4 (1961), 805–808 | MR | Zbl

[9] Pokhozhaev S. I., “O teoreme vlozheniya Soboleva v sluchae $pl=n$”, Doklady nauchno-tekhnicheskoi konferentsii Moskovskogo energeticheskogo instituta, Izd-vo MEI, M., 1965, 158–170

[10] Naimark M. A., Normirovannye koltsa, GITTL, M., 1956, 488 pp.

[11] Banach S., Steinhaus H., “Sur le principe de la condensation des singularites”, Fund. Math., 9 (1927), 51–57

[12] Sobolev S. L., Nekotorye prilozheniya funktsionalnogo analiza v matematicheskoi fizike, Izd-vo LGU, L., 1950

[13] Natanson I. I., Teoriya funktsii veschestvennoi peremennoi, Nauka, M., 1974, 480 pp. | MR