Geodesic
Asymptotics of the elements of attractors corresponding to singularly perturbed parabolic equations
Sbornik. Mathematics, Tome 74 (1993) no. 2, pp. 513-529 
M. I. Vishik; M. Yu. Skvortsov. Asymptotics of the elements of attractors corresponding to singularly perturbed parabolic equations. Sbornik. Mathematics, Tome 74 (1993) no. 2, pp. 513-529. http://geodesic.mathdoc.fr/item/SM_1993_74_2_a12/
@article{SM_1993_74_2_a12,
     author = {M. I. Vishik and M. Yu. Skvortsov},
     title = {Asymptotics of the elements of attractors corresponding to singularly perturbed parabolic equations},
     journal = {Sbornik. Mathematics},
     pages = {513--529},
     year = {1993},
     volume = {74},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1993_74_2_a12/}
}
TY  - JOUR
AU  - M. I. Vishik
AU  - M. Yu. Skvortsov
TI  - Asymptotics of the elements of attractors corresponding to singularly perturbed parabolic equations
JO  - Sbornik. Mathematics
PY  - 1993
SP  - 513
EP  - 529
VL  - 74
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1993_74_2_a12/
LA  - en
ID  - SM_1993_74_2_a12
ER  - 
%0 Journal Article
%A M. I. Vishik
%A M. Yu. Skvortsov
%T Asymptotics of the elements of attractors corresponding to singularly perturbed parabolic equations
%J Sbornik. Mathematics
%D 1993
%P 513-529
%V 74
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1993_74_2_a12/
%G en
%F SM_1993_74_2_a12

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

In a domain $\Omega^n\Subset\mathbf R^n$ we consider the first boundary value problem for a quasilinear parabolic fourth-order equation with a small parameter in the highest derivatives, which degenerates for $\varepsilon=0$ into a second order equation. It is well known that the semigroup corresponding to this problem has an attractor, that is, an invariant attracting set in the phase space. In this paper we investigate the structure of this attractor by means of an asymptotic expansion in $\varepsilon$. The dominant term of the asymptotics is the solution of a second-order equation. The asymptotic expansion also contains boundary layer functions, which are responsible for the deterioration of the differential properties of the elements of the attractor near the boundary. The asymptotics constructed in this way (with an estimate of the remainder) enable us to study the differential properties of attractors and their behavior as $\varepsilon\to0$ in any interior subdomain $\Omega'$, $\overline\Omega'\subset\Omega$. For simplicity, the investigation is carried out in the case when $\Omega$ is a bounded cylindrical domain. The generalization to $\Omega\Subset\mathbf R^n$ does not present any difficulties.

[1] Vishik M. I., Lyusternik L. A., “Regulyarnoe vyrozhdenie i pogranichnyi sloi dlya lineinykh differentsialnykh uravnenii s malym parametrom”, UMN, 12:5 (1957), 3–122 | MR | Zbl

[2] Vishik M. I., Lyusternik L. A., “Reshenie nekotorykh zadach o vozmuscheniyakh v sluchae matrits i samosopryazhennykh i nesamosopryazhennykh differentsialnykh uravnenii”, UMN, 15:3 (1960), 3–80 | MR | Zbl

[3] Trenogin V. A., “Razvitie i prilozheniya metoda Lyusternika–Vishika”, UMN, 25:4 (1970), 123–158 | MR

[4] Babin A. V., Vishik M. I., Attraktory evolyutsionnykh uravnenii, Nauka, M., 1989 | MR | Zbl

[5] Skvortsov M. Yu., “Maksimalnyi attraktor polugruppy, sootvetstvuyuschei pervoi kraevoi zadache dlya singulyarno vozmuschennogo parabolicheskogo uravneniya”, UMN, 42:2 (1987), 243–244 | MR | Zbl

[6] Skvortsov M. Yu., Attraktory singulyarno vozmuschennykh dinamicheskikh sistem, Dep. VINITI No 8603-V86

[7] Skvortsov M. Yu., “Attraktory singulyarno vozmuschennykh dinamicheskikh sistem”, UMN, 42:4 (1987), 154

[8] Lions Zh.-L., Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972 | MR

[9] Besov O. V., Ilin V. P., Nikolskii S. M., Integralnye predstavleniya funktsii i teoremy vlozheniya, Nauka, M., 1975 | MR | Zbl

[10] Nikolskii S. M., Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, M., 1969 | MR

[11] Agranovich M. S., Vishik M. I., “Ellipticheskie zadachi s parametrom i parabolicheskie zadachi obschego vida”, UMN, 19:3 (1964), 53–161