Is this "Transfer" correct?

Question

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.

Answer 1

I think that the argument being posed here has the following problem. Equation $(3)$ is not used as an initial condition for solving Maxwell's equations. As posed, equation $(3)$ assumes that you are away from any sources. More appropriately, you need to assume a source. From Faraday's law and Gauss law for the $\mathbf{B}$ field, this reads instead, $$\frac{\partial}{\partial t}\nabla\cdot{\mathbf{B}} = \frac{\partial\rho_m}{\partial{t}} = -\nabla\cdot\mathbf{M}$$ Which is a continuity equation for magnetic charge. Similarly, from Ampere's law and Gauss law, $$\frac{\partial}{\partial t}\nabla\cdot{\mathbf{D}} = \frac{\partial{\rho}}{\partial{t}} = -\nabla\cdot\mathbf{J}$$ Which is a continuity equation for electric charge.

Rather, $(3)$ is more properly read that that we also have initial conditions that satisfy the continuity equations:

$$\nabla\cdot\mathbf{M} = -\frac{\partial{\rho_m}}{\partial{t}}\tag{4a}$$ $$\nabla\cdot\mathbf{J} = -\frac{\partial\rho}{\partial{t}} \tag{4b}$$

If $\frac{\partial{\rho_m}}{\partial{t}}=0$ and $\frac{\partial{\rho}}{\partial{t}}=0$, then the initial condition is that $\nabla\cdot\mathbf{M} = 0$, and $\nabla\cdot\mathbf{J} = 0$. This must hold across boundary interfaces, which means if there is a steady $\mathbf{M}$ and $\mathbf{J}$, they most be normal continuous across all boundaries.

Perhaps you are confusing this with statements surrounding the Yee-algorithm used to solve Maxwell's equations where $(3)$ is commonly posed as an ancillary condition. The Yee-algorithm only discretizes the curl equations. In order to satisfy Gauss's laws in a source free medium, it is stated that $(3)$ must have an initial condition of 0. However, what should also be stated is that the primary initial condition is that the field intensities must satisfy the proper initial conditions enforced by the curl equations. Commonly the fields start at $0$, which clearly satisfies all initial condition constraints. Another starting point is a static field, which must satisfy the static curl equations, which require $\mathbf{E}$ and $\mathbf{H}$ to be tangential continuous across all source free boundaries. Also, flux densities should satisfy $\textrm{(2c)}$ and $\textrm{(2d)}$. And in conducting medium, $\textrm{(4a)}$ and $\textrm{(4b)}$ for conducting medium, $\mathbf{J}$ and $\mathbf{m}$ must satisfy (4a) and (4b), assuming $\frac{\partial}{\partial{t}} = 0$. If all this holds true, then by default $(3)$ must be true.

With this said, $(3)$ is not an initial condition used for solving Maxwell's equations. It is a consequence of properly posed initial conditions.