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I would be interested in a detailed description of Maxwell's equations from a mathematical point of view, that is as first-order partial differential equations. Taking the equations in SI units in vacuum, $$ \begin{align} \nabla \cdot \mathbf{E}(\mathbf{r}, t) &= \frac{\rho(\mathbf{r}, t)}{\epsilon_0} \tag{1} \label{eq 1}\\ \nabla \cdot \mathbf{B}(\mathbf{r}, t) &= 0 \tag{2} \label{eq 2} \\ \frac{\partial \mathbf{B}(\mathbf{r}, t)}{\partial t} &= -\nabla \times \mathbf{E}(\mathbf{r}, t) \tag{3} \label{eq 3} \\ \frac{\partial \mathbf{E}(\mathbf{r}, t)}{\partial t} &= \frac{1}{\epsilon_0 \mu_0} \nabla \times \mathbf{B}(\mathbf{r}, t) - \frac{1}{\epsilon_0} \mathbf{j}(\mathbf{r}, t) \tag{4} \label{eq 4} \end{align} $$ where $\mathbf{r} = (x, y, z)$. As a first step, I will assume the functions $\rho(\mathbf{r}, t)$ and $\mathbf{j}(\mathbf{r}, t)$ to be prescribed, but arbitrary functions not affected by the electromagnetic field. Ignoring equations \eqref{eq 1} and \eqref{eq 2} for the moment, the remaining two equations form a system of six linear first-order coupled partial differential equations for six unknown functions $E_i(\mathbf{r}, t)$ and $B_i(\mathbf{r}, t)$ of four independent variables $x, y, z, t$.

First of all, is there a general theorem for partial differential equations that guarantees the existence of a solution for $\mathbf{E}$ and $\mathbf{B}$ for arbitrary $\mathbf{j}(\mathbf{r}, t)$? Secondly, if solutions for given $\mathbf{j}$ exist, what data is needed to uniquely determine a solution? In particular, is the specification of $\mathbf{E}(\mathbf{r}, t_0)$ and $\mathbf{B}(\mathbf{r}, t_0)$ for all $\mathbf{r}$ at some time $t_0$ enough?

Now add equations \eqref{eq 1} and \eqref{eq 2} . The system of equations now consists of eight linear first-order coupled partial differential equations for the six unknown functions. What about the existence and uniqueness of solutions now? Starting from \eqref{eq 3} and \eqref{eq 4}, it seems to me that adding \eqref{eq 1} and \eqref{eq 2} amounts to adding constraints on the possible solutions of \eqref{eq 3} and \eqref{eq 4}. Is there a guarantee that these constraints, for arbitrary $\rho(\mathbf{r}, t)$, are compatible with general solutions to \eqref{eq 3} and \eqref{eq 4}? Does adding these constraints reduce the number of functions which have to be solved for so that a reduced system of differential equations can be written down?

Essentially, I am interested in what a mathematician would have to say if I gave him these coupled equations for six unknown functions $u_{1,...,6}(\mathbf{r}, t)$ (the components of the fields), without any physical interpretation.

Ideally, I would be interested in answers to these questions, with as much mathematical rigour as possible, that only involve the fields $\mathbf{E}$ and $\mathbf{B}$. A description in terms of scalar or vector potentials would also be interesting, but only in addition to a treatment in terms of the fields themselves.

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  • $\begingroup$ Not meaning to be annoying, but what's the motivation for considering equations (3) and (4) by themselves? I've grown up considering all four equations together. $\endgroup$
    – Kenny Wong
    Jul 20, 2020 at 22:22
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    $\begingroup$ Also, what's the motivation for insisting that the discussion be framed directly in terms of $\mathbf E $ and $\mathbf B$? Are you aware that, if we frame the discussion in terms of the scalar and vector potentials $\phi$ and $\mathbf A$ in Lorenz gauge, then the Maxwell's equations reduce to the wave equation, for which there is plenty of theory, and for which a solution can be written down in terms of retarded potentials? en.wikipedia.org/wiki/Retarded_potential $\endgroup$
    – Kenny Wong
    Jul 20, 2020 at 22:24
  • $\begingroup$ @Kenny Wong : trying to understand the Maxwell equations as an initial value problem for E and B is a good exercise before moving to do the same for the more complex Einstein equations (writing things as an initial value problem is fundamental when you have to run a simulation on a computer: you set the t=0 field configuration and then evolve in time step by step). With this is mind it is clear that (1) and (2) are conditions that you have to impose when you construct the initial t=0 field configurations for E and B (1 and 2 do not contain time derivatives). $\endgroup$
    – Quillo
    Jul 20, 2020 at 23:17
  • $\begingroup$ The remaining (2) and (3) are the dynamical equations for the E and B fields (6 scalars). The four variables are x y z t, so you need 24 boundary conditions to ensure uniqueness. Note that, if E(t=0) and B(t=0) satisfy (1) and (2), then evolving E and B with (2) and (3) guarantees that the constraint (1) and (2) are satisfied at all times (easy to see if you consider that there is a constraint between the density and the current imposed by the continuity equation). To summarize: you set the t=0 initial configurations of E and B that satisfy (1) and (2) and the boundary conditions. Then evolve. $\endgroup$
    – Quillo
    Jul 20, 2020 at 23:22
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    $\begingroup$ @Quantum : the continuity equation is implied by Maxwell equations, you can derive it using only the Maxwell equations. It is very easy: derive in time (1) and compare with the divergence of (4). $\endgroup$
    – Quillo
    Jul 22, 2020 at 10:45

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This is a partial answer. The answer to your second question ("Secondly ...") can be found on Page 647 of Courant and Hilbert, Methods of Mathematical Physics Volume II. The answer is yes, for a given current and initial values, solutions to (3) and (4) are unique. A proof is given there, that when the current is zero, solutions are unique. This answers your question because if you have two solutions with the same current and initial values, their difference satisfies the vacuum equations with zero initial values. That is what is proved to be zero in the book.

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  • $\begingroup$ +1, very nice and useful reference, thank you! $\endgroup$
    – Quillo
    Jul 24, 2020 at 18:49

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