For single div--curl system, i.e.
$$\nabla\cdot {\bf u} = f,\quad \nabla\times {\bf u} = {\bf b}, \tag1$$
the theorem 3.5 in this paper (
Junichi Aramaki, L^p Theory for the div-curl System, Int. Journal of Math. Analysis, Vol. 8, 2014, no. 6, 259 - 271. http://dx.doi.org/10.12988/ijma.2014.4112 ) says that : ${\rm div}\ {\bf b} =0$ is one of sufficient conditions of existence on the single div--curl system (Eqs. (1)).
Proposition A: It is well known that sufficient conditions are independent. In the single div-curl system, it is that “$\nabla \cdot {\bf b}=0$” is independent of Eqs. (1).
Maxwell equations without sources read:
$$\nabla\times{\bf E}=-\frac{\partial {\bf B}}{\partial t} \tag{2a}$$
$$\nabla\times{\bf B}=\frac{\partial {\bf E}}{\partial t} \tag{2b}$$
$$\nabla\cdot{\bf E}=0 \tag{2c}$$
$$\nabla\cdot{\bf B}=0. \tag{2d}$$
Taking divergence of Eqs (2a) and (2b), it gives $$\frac{\partial}{\partial t}{\bf \nabla\cdot B}=0,\qquad \frac{\partial}{\partial t} {\bf \nabla\cdot E}=0. \tag3$$ Generally (in many books), from Eqs (3), if ${\rm div} {\bf B}=0$ and ${\rm div} {\bf E}=0$ are satisfied at initial time, then they will be hold in all time. Maxwell divergence equations, which are not independent, can be seen as initial conditons of Maxwell curl ones.
The question is:
As we all know, Maxwell equations are double div--curl systems. That corresponding to “$\nabla\cdot{\bf b}=0$” (of Eqs. (1)) is Eqs. (3) of Maxwell equations. If the above Proposition A can be “transferred” to Maxwell equations, Eqs. (3) are independent of Maxwell equations. There is no other way to make Eqs.(3) hold, rather than to make Eqs. (2c) and (2d) hold in all time-space.
Therefore, Eqs. (2c)&(2d) cannot be seen as initial conditoins of Eqs. (2a)&(2b), and they are independent of Maxwell curl ones.
Is it right?
Please see arxiv:1002.0892 (https://arxiv.org/abs/1002.0892 ) for more details.