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

TL;DR

This paper extends the div-curl lemma to sequences with divergence and curl that are compact in a negative Sobolev space, requiring only equi-integrability of the scalar product, not the individual sequences.

Contribution

It establishes a div-curl lemma for sequences with divergence and curl compactness in W^{-1,1}, requiring only equi-integrability of the product, not the sequences themselves.

Findings

Weak convergence of products under divergence and curl compactness.

Only the scalar product needs to be equi-integrable.

Results apply in the setting of W^{-1,1} spaces.

Abstract

It is shown that $u_k \cdot v_k$ converges weakly to $u\cdot v$ if $u_k\weakto u$ weakly in $L^p$ and $v_k\weakly v$ 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 $W^{1,\infty}_0$ 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.