For $u:\mathbb{R}^3 \to \mathbb{R}^3$, if $u \in W^{n,p}(\mathbb{R}^3)$, then what is the regularity of $\nabla \cdot u$?
From the definition of a Sobolev space I think it follows immediately that $$ u \in W^{n,p}(\mathbb{R}^3) \implies \nabla ^k u \in W^{n-k,p}(\mathbb{R}^3) $$ assuming $n>k$.
The question is whether $$ u \in W^{n,p}(\mathbb{R}^3) \implies \nabla \cdot u \in W^{n,p}(\mathbb{R}^3) $$
In this case $u$ is a vector function and therefore $\nabla u$ is the Jacobian. Therefore we know that given that $u$ is in some Sobolev space, the Jacobian is also bounded with respect to the same norm.
In the case of $p=1$ or $p=\infty$ the operator/matrix norm reduces to a simple sum of the absolute value of the elements, e.g. for $p=1$ $$ \|\nabla u \| = \max_j \sum_i \left|\frac{\partial u_i}{\partial x_j}\right| $$ from which one can see that the divergence is then also bounded. Similarly for $p=\infty$.
In case $p=2$ the p-norm is bounded by the Frobenius norm which results in the same answer.
What about the general case?