[Author]
Meshkova, Yu. M.
[Title]
Homogenization with the corrector for periodic parabolic systems in the
$L_2(\mathbb{R}^d)$-norm
[AMS Subj-class]
35B27 Homogenization; equations in media with periodic structure
35K45 Initial value problems for second-order parabolic systems
[Keywords]
periodic differential operators, parabolic systems, homogenization,
operator error estimates
[Abstract]
In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider a self-adjoint matrix
elliptic second order differential operator $\mathcal{B}_\varepsilon$,
$0<\varepsilon \leqslant 1$. The principal part of the operator is given
in a factorized form, the operator contains first and zero order terms.
The operator $\mathcal{B}_\varepsilon$ is positive definite, its coefficients
are periodic and depend on $\mathbf{x}/\varepsilon$. We study the behavior of
the operator exponential $e^{-\mathcal{B}_\varepsilon t}$, $t\geqslant 0$, in
the small period limit. Approximation in the $(L_2\rightarrow L_2)$-operator
norm with error estimate of order $O(\varepsilon ^2)$ is obtained. In this
approximation, the corrector is taken into account. The results are applied
to homogenization of the solutions for the parabolic Cauchy problem.
[Comments]
LaTeX, Russian, 69 pp.
Postal address:
Chebyshev Laboratory, St. Petersburg State University, 14th Line V.O., 29B,
Saint Petersburg 199178 Russia.
[Contact e-mail]
y.meshkova@spbu.ru