Is there a proof of the uniqueness of the solutions of Maxwell equations with non-fixed charges? I know that if $\rho(t,\vec x)$, $\vec J(t,\vec x)$, $\vec E(0,\vec x)$ and $\vec B(0,\vec x)$ are given beforehand, with $\nabla \cdot \vec E(0,\vec x) = 4\pi\rho(0,\vec x)$ and $\nabla \cdot \vec B(0,\vec x) = 0$, and $\rho(t,\vec x)$, $\vec J(t,\vec x)$ satisfying the continuity equations, then it generates a unique solution $\vec E(t,\vec x)$ and $\vec B(t,\vec x)$ to maxwell equations for all times. I have proven that on my electromagnetism ll undergrad course. But now I have the question of what happens when the currents are not fixed. To sleep in peace, I need a theorem that says something like:
Given $\rho(t,\vec x)$, $\vec J(t,\vec x)$ for $t<0$ (satisfying the continuity equations for those times), and $\vec E(0,\vec x)$ and $\vec B(0,\vec x)$, with $\nabla \cdot \vec E(0,\vec x) = 4\pi\rho(0,\vec x)$ and $\nabla \cdot \vec B(0,\vec x) = 0$, then it generates a unique solution $\vec E(t,\vec x)$, $\vec B(t,\vec x)$, $\rho(t,\vec x)$ and  $\vec J(t,\vec x)$ to maxwell equations for $t>0$.
Did somebody prove that? Is it true? Is it provable?
I am thinking on a simple situation in which I have two point like particles, with positive charge $q$, and a distance $L$ between them. They are fixed in space from $t = - \infty$, so for those times there are no radiation fields. At time $t = 0$, I kick both giving them instantaneously different velocities, and are no longer fixed in space. They are going to repel, move, create a magnetic field, accelerate, radiate some fields, accelerate the other with that fields, etc. The trayectories of these particles should be determined. I mean, the world is deterministic, and to be that way, there should be a theorem like the one I am asking.
Note: I am completely in the framework of classic electromagnetism, but fully relativistic (don't want any approximation), so I would like to solve the problem with no quantum mechanics appearing.
 A: There is a missing ingredient to your question which is: given the initial conditions (fields, currents, and densities at $t=0$), what determines the evolution of the current and density? The continuity equation relates the current and density, but is not itself sufficient to determine the evolution completely.
Typically we take the the Lorentz force to be this missing ingredient (though material properties like conductivity can complicate matters). With this in mind, the equations of motion aren't just Maxwell's equation, but Maxwell's equations plus the Lorentz force equation. In general this makes problems extremely difficult and so is left as a numerical problem by and large (there's also an issue with point charges behaving exceptionally poorly).
Once we have all this in mind, however, there's a simple way to see that the dynamics will be uniquely determined assuming a solution exists at all (though it may not satisfy some people's desire for rigor). Notice that in Maxwell's equations and the Lorentz force law, the time derivatives of all quantities can be isolated, meaning we can write things in a form $\frac{\partial E}{\partial t}=(\cdots),\frac{\partial B}{\partial t}=(\cdots),\ldots$. The right-hand-side of each of these equations will involve the fields and their spatial derivatives, but that won't be a problem.
Imagine now we want to solve this system numerically. Since we start knowing the values of the fields at $t=0$, we can compute all necessary spatial derivatives on the RHS of our equations and write the time derivatives as a finite difference between the field value (at each spatial point) at $t=0$ and $t=0+\Delta t$. From the form of our equations (the time derivatives being isolated) we can clearly solve for the values of the fields at the time $t=\Delta t$.
Now, this is a very loose argument based on numerical thinking, but you could imagine arguing for the existence of a temporal power series by the same means, from which it follows that a solution will exist (there are various assumptions about analyticity that go into this, but more rigor can always be added to any argument and I just want to convey the idea). The only worry we might have is that the constraints are not preserved by the time evolution. This is where my assumption that a solution exists at all comes in. If time evolution does not respect the constraints (essentially the conditions you point out your initial conditions must satisfy), then a solution will never exist. But if a solution does exist, then clearly its time evolution must satisfy the temporal-derivative relations I described above, which are all uniquely determined, and hence the fields at future times are all uniquely determined.
It is a perfectly good question to ask whether or not the constraints are preserved by time evolution, and there may even be some slick argument showing this in the special case of Maxwell's equations, but unfortunately I don't know of one. What I can say, however, is that the correct way to analyze these constraints would be through the Hamiltonian/Lagrangian formalism. There it can be shown that the constraints are indeed respected by time evolution, but that involves the formalism of constrained Hamiltonian systems which I think is beyond the scope I wanted this answer to have. A nice paper discussing this can be found here. It has some nice comments on Noether's theorem as a bonus.
Finally, since I mentioned Hamiltonian systems, let me point out that Hamiltonian systems always (at least, I can't think of a loophole off the top of my head) have the property I described above involving isolated time derivatives. This follows immediately from Hamilton's equations:
$$
\dot x=\frac{\partial H}{\partial p},\ \ \ \ \dot p=-\frac{\partial H}{\partial x}
$$
or the analogous equations for a field system (replace partial derivatives by functional derivatives). These equations always have the temporal derivatives isolated on the left while the right may, at most, depend on spatial derivatives.
