Div-curl lemma revisited: Applications in electromagnetism

Marián Slodička; Ján Jr. Buša

Kybernetika (2010)

  • Volume: 46, Issue: 2, page 328-340
  • ISSN: 0023-5954

Abstract

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Two new time-dependent versions of div-curl results in a bounded domain Ω ⊂ ℝ 3 are presented. We study a limit of the product v k w k , where the sequences v k and w k belong to Ł 2 ( Ω ) . In Theorem 2.1 we assume that ∇ × v k is bounded in the L p -norm and ∇ · w k is controlled in the L r -norm. In Theorem 2.2 we suppose that ∇ × w k is bounded in the L p -norm and ∇ · w k is controlled in the L r -norm. The time derivative of w k is bounded in both cases in the norm of - 1 ( Ω ) . The convergence (in the sense of distributions) of v k w k to the product v w of weak limits of v k and w k is shown.

How to cite

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Slodička, Marián, and Buša, Ján Jr.. "Div-curl lemma revisited: Applications in electromagnetism." Kybernetika 46.2 (2010): 328-340. <http://eudml.org/doc/196724>.

@article{Slodička2010,
abstract = {Two new time-dependent versions of div-curl results in a bounded domain $\Omega \subset \mathbb \{R\}^3$ are presented. We study a limit of the product $\{v\}_k\{w\}_k$, where the sequences $\{v\}_k$ and $\{w\}_k$ belong to $Ł_\{2\}(\Omega )$. In Theorem 2.1 we assume that $\nabla \times \{v\}_k$ is bounded in the $L_p$-norm and $\nabla \cdot \{w\}_k$ is controlled in the $L_r$-norm. In Theorem 2.2 we suppose that $\nabla \times \{w\}_k$ is bounded in the $L_p$-norm and $\nabla \cdot \{w\}_k$ is controlled in the $L_r$-norm. The time derivative of $\{w\}_k$ is bounded in both cases in the norm of $\{-1\}(\Omega )$. The convergence (in the sense of distributions) of $\{v\}_k\{w\}_k$ to the product $\{v\}\{w\}$ of weak limits of $\{v\}_k$ and $\{w\}_k$ is shown.},
author = {Slodička, Marián, Buša, Ján Jr.},
journal = {Kybernetika},
keywords = {compensated compactness; convergence; vector fields; convergence; compensated compactness; vector fields},
language = {eng},
number = {2},
pages = {328-340},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Div-curl lemma revisited: Applications in electromagnetism},
url = {http://eudml.org/doc/196724},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Slodička, Marián
AU - Buša, Ján Jr.
TI - Div-curl lemma revisited: Applications in electromagnetism
JO - Kybernetika
PY - 2010
PB - Institute of Information Theory and Automation AS CR
VL - 46
IS - 2
SP - 328
EP - 340
AB - Two new time-dependent versions of div-curl results in a bounded domain $\Omega \subset \mathbb {R}^3$ are presented. We study a limit of the product ${v}_k{w}_k$, where the sequences ${v}_k$ and ${w}_k$ belong to $Ł_{2}(\Omega )$. In Theorem 2.1 we assume that $\nabla \times {v}_k$ is bounded in the $L_p$-norm and $\nabla \cdot {w}_k$ is controlled in the $L_r$-norm. In Theorem 2.2 we suppose that $\nabla \times {w}_k$ is bounded in the $L_p$-norm and $\nabla \cdot {w}_k$ is controlled in the $L_r$-norm. The time derivative of ${w}_k$ is bounded in both cases in the norm of ${-1}(\Omega )$. The convergence (in the sense of distributions) of ${v}_k{w}_k$ to the product ${v}{w}$ of weak limits of ${v}_k$ and ${w}_k$ is shown.
LA - eng
KW - compensated compactness; convergence; vector fields; convergence; compensated compactness; vector fields
UR - http://eudml.org/doc/196724
ER -

References

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