$G$-convergence of periodic parabolic operators with a small parameter by the time derivative
In this paper, we consider a sequence $\mathcal{P}^k$ of divergent parabolic operators of the second order, which are periodic in time with period $T = \text{const}$, and a sequence $\mathcal{P}^k_{\psi}$ of shifts of these operators by an arbitrary periodic vector function $ \psi \in X = \{L^2((0,...
Збережено в:
| Дата: | 1993 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
1993
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/5840 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512060181315584 |
|---|---|
| author | Sidenko, N. R. Сиденко, Н. Р. Сиденко, Н. Р. |
| author_facet | Sidenko, N. R. Сиденко, Н. Р. Сиденко, Н. Р. |
| author_sort | Sidenko, N. R. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-19T09:18:55Z |
| description | In this paper, we consider a sequence $\mathcal{P}^k$ of divergent parabolic operators of the second order, which are periodic in time with period $T = \text{const}$, and a sequence $\mathcal{P}^k_{\psi}$ of shifts of these operators by an arbitrary periodic vector function $ \psi \in X = \{L^2((0, T) \times \Omega)\}^n$ where $\Omega$ is a bounded Lipschitz domain in the space $\mathbb{R}^n$.
The compactness of the family $\{P_{Ψ^k} ¦ Ψ \in X, k \in ℕ\}$ in $k$ with respect to strong $G$-convergence, the convergence of arbitrary solutions of the equations with the operator $\mathcal{P}^k_{\psi}$, and the local character of the strong $G$-convergence in $Ω$ are proved under the assumptions that the matrix of coefficients of $L^2$ is uniformly elliptic and bounded and that their time derivatives are uniformly bounded in the space $L^2(Ω; L^2(0,T))$. |
| first_indexed | 2026-03-24T03:22:47Z |
| format | Article |
| fulltext |
0075
0076
0077
0078
0079
0080
0081
0082
0083
0084
0085
0086
0087
0088
|
| id | umjimathkievua-article-5840 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:22:47Z |
| publishDate | 1993 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/20/036deca76e778655c195e1890c057720.pdf |
| spelling | umjimathkievua-article-58402020-03-19T09:18:55Z $G$-convergence of periodic parabolic operators with a small parameter by the time derivative $G$-сходимость параболических периодических операторов с малым параметром при производной по времени Sidenko, N. R. Сиденко, Н. Р. Сиденко, Н. Р. In this paper, we consider a sequence $\mathcal{P}^k$ of divergent parabolic operators of the second order, which are periodic in time with period $T = \text{const}$, and a sequence $\mathcal{P}^k_{\psi}$ of shifts of these operators by an arbitrary periodic vector function $ \psi \in X = \{L^2((0, T) \times \Omega)\}^n$ where $\Omega$ is a bounded Lipschitz domain in the space $\mathbb{R}^n$. The compactness of the family $\{P_{Ψ^k} ¦ Ψ \in X, k \in ℕ\}$ in $k$ with respect to strong $G$-convergence, the convergence of arbitrary solutions of the equations with the operator $\mathcal{P}^k_{\psi}$, and the local character of the strong $G$-convergence in $Ω$ are proved under the assumptions that the matrix of coefficients of $L^2$ is uniformly elliptic and bounded and that their time derivatives are uniformly bounded in the space $L^2(Ω; L^2(0,T))$. Рассматривается последовательность $\mathcal{P}^k$ периодических по времени с периодом $T = \text{const}$ параболических дивергентных операторов второго порядка и их сдвигов $\mathcal{P}^k_{\psi}$ на произвольную периодическую вектор-функцию $\psi \in X = \{L^2((0, T) \times \Omega)\}^n$, где $\Omega$ - ограниченная Липшицева область в $\mathbb{R}^n$. При условиях равномерной эллиптичности и ограниченности матрицы коэффициентов $\mathcal{P}^k$ и равномерной ограниченности их временной производной в пространстве $L^{\infty}(\Omega; L^2(0, t))$ доказаны утверждения о компактности по $k$ семейства $\{\mathcal{P}^k_{\psi} | \psi \in X, k \in \mathbb{N}\}$ относительно сильной $G$-сходимостн, сходимости произвольных решений уравнений с оператором, локальности сильной $G$-сходимости в $\Omega$. Institute of Mathematics, NAS of Ukraine 1993-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/5840 Ukrains’kyi Matematychnyi Zhurnal; Vol. 45 No. 4 (1993); 525–538 Український математичний журнал; Том 45 № 4 (1993); 525–538 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/5840/8363 https://umj.imath.kiev.ua/index.php/umj/article/view/5840/8364 Copyright (c) 1993 Sidenko N. R. |
| spellingShingle | Sidenko, N. R. Сиденко, Н. Р. Сиденко, Н. Р. $G$-convergence of periodic parabolic operators with a small parameter by the time derivative |
| title | $G$-convergence of periodic parabolic operators with a small parameter by the time derivative |
| title_alt | $G$-сходимость параболических периодических операторов с малым параметром при производной по времени |
| title_full | $G$-convergence of periodic parabolic operators with a small parameter by the time derivative |
| title_fullStr | $G$-convergence of periodic parabolic operators with a small parameter by the time derivative |
| title_full_unstemmed | $G$-convergence of periodic parabolic operators with a small parameter by the time derivative |
| title_short | $G$-convergence of periodic parabolic operators with a small parameter by the time derivative |
| title_sort | $g$-convergence of periodic parabolic operators with a small parameter by the time derivative |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/5840 |
| work_keys_str_mv | AT sidenkonr gconvergenceofperiodicparabolicoperatorswithasmallparameterbythetimederivative AT sidenkonr gconvergenceofperiodicparabolicoperatorswithasmallparameterbythetimederivative AT sidenkonr gconvergenceofperiodicparabolicoperatorswithasmallparameterbythetimederivative AT sidenkonr gshodimostʹparaboličeskihperiodičeskihoperatorovsmalymparametrompriproizvodnojpovremeni AT sidenkonr gshodimostʹparaboličeskihperiodičeskihoperatorovsmalymparametrompriproizvodnojpovremeni AT sidenkonr gshodimostʹparaboličeskihperiodičeskihoperatorovsmalymparametrompriproizvodnojpovremeni |