Higher regularity of divergence free Sobolev vector fields

Question

Consider a vector field $v$ defined in $U \subset \mathbb{R} ^n$ such that $ v \in W^{1, p} (U)$ with $U$ boudned and as musch regular as you wish. I know Sobolev embeddings, and that in general $v$ is continuous (Hölder in fact) if $p>n$, and that there are counterexamples in the limit case $p=n$. However I was wondering: if we also assumed that $div (u)=0$, do we have a little more in terms of regularity? And how much more?