The div–curl lemma for sequences whose divergence and curl are compact in $ {W}^{-1,1}$

Conti, Sergio; Dolzmann, Georg; Müller, Stefan

doi:10.1016/j.crma.2010.11.013

Partial Differential Equations/Mathematical Problems in Mechanics

The div–curl lemma for sequences whose divergence and curl are compact in

W^{- 1, 1}

[Le lemme div–rot pour les suites dont la divergence et la boucle sont bornées dans

W^{- 1, 1}

]

Présenté par : John M. Ball

Conti, Sergio ¹ ; Dolzmann, Georg ² ; Müller, Stefan ^1,³

¹ Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
² Universität Regensburg, 93040 Regensburg, Germany
³ Hausdorff Center for Mathematics, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany

Comptes Rendus. Mathématique, Tome 349 (2011) no. 3-4, pp. 175-178.

Résumé
Abstract

On montre que $u_{k} \cdot v_{k}$ converge faiblement vers $u \cdot v$ si $u_{k} ⇀ u$ faiblement dans $L^{p}$ , $v_{k} ⇀ v$ faiblement dans $L^{q}$ , les séquences $div u_{k}$ et $rot v_{k}$ sont compactes dans l'espace dual de $W_{0}^{1, \infty}$ et $u_{k} \cdot v_{k}$ est équi-intégrable, pour $p, q \in (1, \infty)$ , $1 / p + 1 / q = 1$ . En effet, on n'utilise que l'équi-intégrabilité du produit scalaire $u_{k} \cdot v_{k}$ , et non pas celle de chacune des suites.

It is shown that $u_{k} \cdot v_{k}$ converges weakly to $u \cdot v$ if $u_{k} ⇀ u$ weakly in $L^{p}$ and $v_{k} ⇀ v$ weakly in $L^{q}$ with $p, q \in (1, \infty)$ , $1 / p + 1 / q = 1$ , under the additional assumptions that the sequences $div u_{k}$ and $curl v_{k}$ are compact in the dual space of $W_{0}^{1, \infty}$ and that $u_{k} \cdot v_{k}$ is equi-integrable. The main point is that we only require equi-integrability of the scalar product $u_{k} \cdot v_{k}$ and not of the individual sequences.

Reçu le : 2009-07-12
Accepté le : 2010-03-23
Publié le : 2011-01-03

DOI : 10.1016/j.crma.2010.11.013

Affiliations des auteurs :

Conti, Sergio ¹ ; Dolzmann, Georg ² ; Müller, Stefan ^{1, 3}

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     title = {The div{\textendash}curl lemma for sequences whose divergence and curl are compact in $ {W}^{-1,1}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {175--178},
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Conti, Sergio; Dolzmann, Georg; Müller, Stefan. The div–curl lemma for sequences whose divergence and curl are compact in $ {W}^{-1,1}$. Comptes Rendus. Mathématique, Tome 349 (2011) no. 3-4, pp. 175-178. doi : 10.1016/j.crma.2010.11.013. http://www.numdam.org/articles/10.1016/j.crma.2010.11.013/

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