arXiv subjects

Yu. M. Meshkova

Publications and source records attributed to Yu. M. Meshkova.

4 recordsLinked to original sources

arXiv2020

Homogenization of periodic parabolic systems in the $L_2(\mathbb{R}^d)$-norm with the corrector taken into account

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.

math.AP↗

arXiv2018

Homogenization of the first initial boundary-value problem for parabolic systems: operator error estimates

Let $\mathcal{O}\subset\mathbb{R}^d$ be a bounded domain of class $C^{1,1}$. In $L_2(\mathcal{O};\mathbb{C}^n)$, we consider a selfadjoint matrix second order elliptic differential operator $B_{D,\varepsilon}$, $0<\varepsilon\leqslant1$, with the Dirichlet boundary condition. The principal part of the operator is given in a factorized form. The operator involves first and zero order terms. The operator $B_{D,\varepsilon}$ is positive definite; its coefficients are periodic and depend on $\mathbf{x}/\varepsilon$. We study the behavior of the operator exponential $e^{-B_{D,\varepsilon}t}$, $t>0$, as $\varepsilon\rightarrow 0$. We obtain approximations for the exponential $e^{-B_{D,\varepsilon}t}$ in the operator norm on $L_2(\mathcal{O};\mathbb{C}^n)$ and in the norm of operators acting from $L_2(\mathcal{O};\mathbb{C}^n)$ to the Sobolev space $H^1(\mathcal{O};\mathbb{C}^n)$. The results are applied to homogenization of solutions of the first initial boundary-value problem for parabolic systems.

math.AP↗

arXiv2015

Two-parametric error estimates in homogenization of second order elliptic systems in $\mathbb{R}^d$ including lower order terms

In $L_2({\mathbb R}^d;{\mathbb C}^n)$, we consider a selfadjoint operator ${\mathcal B}_\varepsilon$, $0< \varepsilon \leqslant 1$, given by the differential expression $b({\mathbf D})^* g({\mathbf x}/\varepsilon)b({\mathbf D}) + \sum_{j=1}^d (a_j({\mathbf x}/\varepsilon) D_j +D_j a_j({\mathbf x}/\varepsilon)^*) + Q({\mathbf x}/\varepsilon)$, where $b({\mathbf D}) = \sum_{l=1}^d b_l D_l$ is the first order differential operator, and $g, a_j, Q$ are matrix-valued functions in ${\mathbb R}^d$ periodic with respect to some lattice $Γ$. It is assumed that $g$ is bounded and positive definite, while $a_j$ and $Q$ are, in general, unbounded. We study the generalized resolvent $({\mathcal B}_\varepsilon - ζQ_0({\mathbf x}/\varepsilon))^{-1}$, where $Q_0$ is a $Γ$-periodic, bounded and positive definite matrix-valued function, and $ζ$ is a complex-valued parameter. Approximations for the generalized resolvent in the $(L_2 \to L_2)$- and $(L_2 \to H^1)$-norms with two-parametric error estimates (with respect to the parameters $\varepsilon$ and $ζ$) are obtained.

math.AP↗

arXiv2015

Homogenization of initial boundary value problems for parabolic systems with periodic coefficients

Let $\mathcal{O} \subset \mathbb{R}^d$ be a bounded domain of class $C^{1,1}$. In the Hilbert space $L_2(\mathcal{O};\mathbb{C}^n)$, we consider matrix elliptic second order differential operators $\mathcal{A}_{D,\varepsilon}$ and $\mathcal{A}_{N,\varepsilon}$ with the Dirichlet or Neumann boundary condition on $\partial \mathcal{O}$, respectively. Here $\varepsilon>0$ is the small parameter. The coefficients of the operators are periodic and depend on $\mathbf{x}/\varepsilon$. The behavior of the operator $e^{-\mathcal{A}_{†,\varepsilon}t}$, $†=D,N$, for small $\varepsilon$ is studied. It is shown that, for fixed $t>0$, the operator $e^{-\mathcal{A}_{†,\varepsilon}t}$ converges in the $L_2$-operator norm to $e^{-\mathcal{A}_†^0 t}$, as $\varepsilon \to 0$. Here $\mathcal{A}_†^0$ is the effective operator with constant coefficients. For the norm of the difference of the operators $e^{-\mathcal{A}_{†,\varepsilon}t}$ and $e^{-\mathcal{A}_†^0 t}$ a sharp order estimate (of order $O(\varepsilon)$) is obtained. Also, we find approximation for the exponential $e^{-\mathcal{A}_{†,\varepsilon}t}$ in the $(L_2\rightarrow H^1)$-norm with error estimate of order $O(\varepsilon ^{1/2})$; in this approximation, a corrector is taken into account. The results are applied to homogenization of solutions of initial boundary value problems for parabolic systems.

math.AP↗