I have the following question. Let $b_1, b_2$ two suitably regular vector fields such that $\operatorname{div}b_1=0$ and $\langle b_1, b_2\rangle=0$. What can we say about $\operatorname{div}b_2$? Is it true that it is equal to zero?
According to me, we cannot say that it is equal to zero but I cannot find a counterexample.
We could have $b_1=0$, and then $b_2$ can be whatever it wants.
For a less trivial example, you could pick $b_1$ to be a vector field that "rotates" around the origin of the Euclidean plane (at each point $x$, we have $\langle b_1(\vec x), \vec x\rangle=0$, and $\|\vec x\|=\|\vec y\|\implies \|b_1(\vec x)\|=\|b_1(\vec y)\|$) such as $$ b_1(\vec x)=\begin{bmatrix}0&-1\\1&0\end{bmatrix}\vec x $$ Now let $b_2(\vec x)$ be parallel to $\vec x$ and make sure it has non-zero divergence, such as $b_2(\vec x)=\vec x$.