Interpretation of nabla followed by a dot (which is not meant as divergence)

Question

I am studying a paper and the authors use $\nabla \cdot u$ for a real valued function $u\colon \mathbb{R}^n \to \mathbb{R}$ and I am quite confused how to interpret that, even after long searches in books and the internet. They use this construct also for other real valued functions at other positions in the paper, therefore I think its not a mistake. Can anyone see what is meant by that(I have no background in physics, maybe there is such a notation in physics)? Since $u$ is real valued I hardly can interpret it as divergence.

Answer 1

What they maybe could mean if $u({\bf x}) \in \mathbb R$, ${\bf x} \in \mathbb R^N$:

$$\nabla \cdot u = \sum_{k=1}^N \frac{\partial u}{\partial x_k}$$

i.e. expanding the scalar $u$ to all of the input dimensions and take divergence of the resulting field.

In other words:

$$\nabla \cdot (u {\bf 1}_N)$$ Where ${\bf 1}_N$ is the column vector of $N$ ones.

Answer 2

Thanks for the additional comments first. If one looks close in their paper they say divergence to $\nabla u$ without dot. But anyway, I think I found an answer, at least satisfactory enough for me. I dont know what the authors exactly mean but after reading several other papers which are cited by the authors, I think they maybe could mean $\nabla \cdot u := |\nabla u|^2$ which is the same as $\sum\limits_{i = 1}^n (\partial_{x_i} u)^2$. I never saw this notation before so I am not sure if they mean this but anyhow it works just fine for me.

Answer 3

I think there is a typo on page 10 after eq. 84 article where $\nabla \varphi$ should be $\nabla \cdot\varphi$ and in fact in the preprint at page 19 we find "$\nabla \cdot\varphi$ represents the divergence of $\varphi$". They only use $\nabla \cdot\varphi$, never $\nabla \varphi$. So I suppose the use as definition $\nabla \cdot\varphi=\sum_{\nu}\partial_\nu u$.