Homogenization of periodic parabolic systems in the 𝐿₂(ℝ^{𝕕})-norm with the corrector taken into account,St. Petersburg Mathematical Journal

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Homogenization of periodic parabolic systems in the 𝐿₂(ℝ^{𝕕})-norm with the corrector taken into account
St. Petersburg Mathematical Journal ( IF 0.7 ) Pub Date : 2020-06-11 , DOI: 10.1090/spmj/1619
Yu. M. Meshkova

Abstract:In $L_2(\mathbb{R}^d;\mathbb{C}^n)$ , consider a selfadjoint matrix second order elliptic differential operator $\mathcal {B}_\varepsilon$ , $0<\varepsilon \leq 1$ . 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 $\mathbf {x}/\varepsilon$ . The behavior in the small period limit is studied for the operator exponential $e^{-\mathcal {B}_\varepsilon t}$ , $t\geq 0$ . 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.

中文翻译：

考虑校正器的𝐿_2（ℝ^ {𝕕}）范数中周期抛物线系统的均质化

摘要：，考虑自伴矩阵二阶椭圆微分算子，。运算符的主要部分以因子分解形式给出，该运算符包含一阶和零阶项。算子是正定的，其系数是周期性的并取决于。对于算子指数，研究了在小周期极限内的行为。获得-operator范式中具有阶次误差估计的近似值。在该近似中考虑校正器。将结果应用于抛物型系统柯西问题的解的均质化。 $L_2（\ mathbb {R} ^ d; \ mathbb {C} ^ n）$ $\ mathcal {B} _ \ varepsilon$ $0 <\ varepsilon \ leq 1$ $\ mathcal {B} _ \ varepsilon$ $\ mathbf {x} / \ varepsilon$ $e ^ {-\数学{B} _ \ varepsilon t}$ $t \ geq 0$ $（L_2 \ rightarrow L_2）$ $O（\ varepsilon ^ 2）$

更新日期：2020-06-11

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