| |
 |
UNIVERSITY OF BUCHAREST
FACULTY OF PHYSICS Guest
2024-11-22 1:32 |
 |
|
|
|
Conference: Bucharest University Faculty of Physics 2003 Meeting
Section: Theoretical Physics and Applied Mathematics Seminar
Title:
Despre omogenizarea unor probleme parabolice cu
conditii dinamice pe frontiera
Authors:
Claudia Timofte
Affiliation:
Catedra de Matematici, Facultatea de Fizica, Universitatea Bucuresti,
C.P. MG-11, Bucuresti, Magurele
E-mail: claudiatimofte@hotmail.com
E-mail
Keywords:
Abstract:
In acest articol vom analiza comportamentul asimptotic al
solutiei unei probleme parabolice, cu conditii dinamice pe
frontiera. Astfel de probleme apar in mod natural in
multe modele matematice din teoria transferului caldurii
intr-un solid in contact cu un fluid aflat n miscare, in modelarea curgerilor partial saturate in
medii poroase, in probleme de difuzie in medii poroase
(vezi [1]-[3]).
Fie $\varepsilon$ un parametru mic ce ia valori \^{\i}ntr-un \c
sir de numere tinz\^{a}nd c\u atre $0$ \c si fie $\Omega \subset
{\rm I\! R}^{N}$ un domeniu m\u arginit. Acesta va fi perforat
\^{\i}n mod periodic, cu perfora\c tii identice, de ordinul lui
$\varepsilon $. \^{I}n domeniul perforat $\Omega^{\varepsilon}$
vom considera ecua\c tia c\u aldurii, cu o condi\c tie Dirichlet
pe frontiera exterioar\u a \c si o condi\c tie dinamic\u a pe
suprafa\c ta perfora\c tiilor. A\c sa cum vom vedea, la limit\u a,
c\^{a}nd $\varepsilon \rightarrow 0$, vom ob\c tine o ecua\c tie
cu coeficien\c ti constan\c ti, dar cu un termen nou, datorat
influen\c tei condi\c tiei dinamice neomogene impuse pe suprafa\c
ta perfora\c tiilor.
Mai precis, vom studia comportamentul asimptotic al solu\c tiei
$u^{\varepsilon }$ a urm\u atoarei pro\-bleme:
\begin{equation}
\left\{
\begin{array}{l}
\displaystyle\frac{\partial u^{\varepsilon }}{\partial t}-\Delta
u^{\varepsilon
}=f(t,x),\quad \textrm{ \^{\i}n }\ \Omega ^{\varepsilon }\times (0,T), \\\\
\displaystyle \frac{\partial u^{\varepsilon }}{\partial
n}+\varepsilon \frac{\partial u^{\varepsilon }}{\partial
t}=\varepsilon g(t,x),\quad
\textrm{ pe } S^{\varepsilon }\times (0,T),\\\\
u^{\varepsilon }(0,x)=u^{0}(x),\quad \textrm{ \^{\i}n }\ \Omega
^{\varepsilon },\, u^{\varepsilon }(0,x)=v^{0}(x),\ \textrm{ pe }\
S^{\varepsilon },\\\\
u^{\varepsilon }=0,\quad \textrm{ pe }\partial \Omega \times
(0,T).
\end{array}
\right.
\end{equation}
\noindent Aici, $f\in L^{2}(0,T;L^{2}(\Omega )),$ $g\in
L^{2}(0,T;H_{0}^{1}(\Omega )),$ $u^{0}\in L^{2}(\Omega ),$
$v^{0}\in L^{2}(S^{\varepsilon }),$ $[0,T]$ este intervalul de
timp pe care vom lucra \c si $S^{\varepsilon }$ este frontiera
perfora\c tiilor.
\medskip
\noindent Principalul rezultat de convergen\c t\u a din acest
articol este urm\u atorul (vezi [3]):
\medskip
\textbf{Teorema 1. } Fie $u^{\varepsilon }$ solu\c tia unic\u a a
problemei
(1). Atunci, exist\u a o extensie $ \stackrel{\sim }{u}^{\varepsilon }$ a solu\c tiei $%
u^{\varepsilon }$ la $\Omega \times (0,T)$ astfel \^{\i}nc\^{a}t
\[
\stackrel{\sim }{u} ^{\varepsilon }\rightarrow u \quad
\textrm{tare \^{\i}n } L^{2}(0,T; L^{2}(\Omega),
\]
\noindent unde $u$ este solu\c tia unic\u a a urm\u atorului
sistem:
\begin{equation}
\left\{
\begin{array}{l}
\displaystyle (\frac{\left| Y^{*}\right| }{\left| Y\right|
}+\frac{\left|
\partial F\right| }{\left| Y\right| })\frac{\partial u}{\partial
t}-\nabla (Q\nabla u)=\frac{\left| Y^{*}\right| }{\left| Y\right|
}f+\frac{\left| \partial F\right| }{\left| Y\right| }g,\quad
\textrm{\^{\i}n }\Omega \times (0,T), \\\\
u=0,\quad \textrm{pe } \partial \Omega \times (0,T),\\\\
u(0,x)=u^{0}(x),\quad \textrm{ \^{\i}n }\Omega.
\end{array}
\right.
\end{equation}
\noindent Aici $Q=((q_{ij}))$ este matricea omogenizat\u a, ale
c\u arei elemente sunt date prin:
\smallskip
\begin{equation}
q_{ij}=\frac{\left| Y^{*}\right| }{\left| Y\right| }\delta _{ij}-\frac{1}{%
\left| Y\right| }\int_{Y^{*}}\frac{\partial \eta _{j}}{\partial
y_{i}}dy,
\end{equation}
\smallskip
\noindent \^{\i}n func\c tie de solu\c tiile $\eta _{j}$ ale
sistemului:
\begin{equation}
\left\{
\begin{array}{l}
-\Delta \eta_{j}=0,\quad \textrm{\^{\i}n }Y^{\star},\\\\
\partial (\eta _{j}-y_{j})/ \partial n=0,\quad \textrm{pe }\partial F,\\\\
\eta _{j}\ \textrm{ este }Y-\textrm{periodic\u a}.
\end{array}
\right.
\end{equation}
\bigskip
\noindent \^{I}n formulele anterioare, $F$ este perfora\c tia
elementar\u a, $Y$ este celula elementar\u a de periodicitate \c
si $Y^{\star}=Y\setminus {\overline F}$.
\bigskip
Deci, la limit\u a, c\^{a}nd $\varepsilon \rightarrow 0,$ ob\c
tinem o ecua\c tie neomogen\u a cu coeficien\c ti constan\c ti, cu
o condi\c tie Dirichlet pe frontier\u a, cu un termen suplimentar
datorat contribu\c tiei p\u ar\c tii dinamice a condi\c tiei
noastre pe suprafa\c ta perfora\c tiilor.
\bigskip
\bigskip
[1] \, \textsc{S.N. Antontsev, A.V. Kazhikhov, Monakhov%
}, \textit{Boundary Value Problems in Mechanics of Nonhomogeneous
Fluids}, North-Holland, Amsterdam, 1990.
\medskip
[2] \, \textsc{I. Bejenaru, J. I. D\`{\i }az, I. I. Vrabie},
\emph{An abstract approximate controllability result and
applications to elliptic and parabolic systems with dynamic
boundary conditions}, Electronic Journal of Differential Equation,
\textbf{48} (2001), 1-19.
\medskip
[3] \, \textsc{C. Timofte}, \emph{Modele matematice pentru studiul
asimptotic al mediilor neomogene}, Editura Didactic\u a \c si
Pedagogic\u a, 2003.
|
|
|
|