论文标题
$ L_2（\ Mathbb {r}^d）$的定期抛物线系统的均质化，并考虑了校正器
Homogenization of periodic parabolic systems in the $L_2(\mathbb{R}^d)$-norm with the corrector taken into account
论文作者
论文摘要
在$ l_2（\ Mathbb {r}^d; \ Mathbb {C}^n）$中，考虑一个自动化矩阵第二阶零件椭圆形差异操作员$ \ MATHCAL {B} _ \ VAREPSILON $，$ 0 <\ VAREPSILON \ vARepsilon \ leqslant 1 $。操作员的主要部分以分解形式给出，操作员包含第一和零订单项。运算符$ \ Mathcal {B} _ \ Varepsilon $是正定的，其系数是周期性的，并且取决于$ \ Mathbf {x}/\ Varepsilon $。我们研究了操作员指数$ e^{ - \ Mathcal {b} _ \ Varepsilon T} $，$ t \ geqslant 0 $的少量行为。获得$（l_2 \ rightarrow l_2）$ - 运算符的近似值，并获得了订单$ o（\ varepsilon ^2）$的错误估算。在此近似值中考虑了校正器。该结果应用于抛物线系统库奇问题的溶液的均质化。
In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, consider a self-adjoint matrix second order elliptic differential operator $\mathcal{B}_\varepsilon$, $0<\varepsilon \leqslant 1$. The principal part of the operator is given in a factorised 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 behaviour in the small period limit of the operator exponential $e^{-\mathcal{B}_\varepsilon t}$, $t\geqslant 0$. The approximation in the $(L_2\rightarrow L_2)$-operator norm with error estimate of order $O(\varepsilon ^2)$ is obtained. The corrector is taken into account in this approximation. The results are applied to homogenization of the solutions for the Cauchy problem for parabolic systems.
