What are the existence of solutions to Maxwell's equations?

[timeant]

Are there any references?

 

[vanhees71]

A. Sommerfeld, Lectures of Theoretical Physics, vol. 3

 

[timeant]

Thank @ vanhees71 . However , I don't find the existence conditions, only found uniqueness .
Electrodynamics. Lectures on Theoretical Physics, Vol. 3 by Arnold Sommerfeld

 

[DrClaude]

https://arxiv.org/abs/0905.1798

 

[timeant]

Thank all replies :) @DrClaude @vanhees71
We all know that charge conservation is one of necessary conditions for the existence of Maxwell equations!
I want to know: is charge conservation one of sufficient conditions for the existence of Maxwell equations?

 

[vanhees71]

Together with reasonable boundary conditions of course. I guess, there is rigorous math literature on this subject. Usually the Cauchy problem of all kinds of partial-differential systems occurring in physics is an interesting topic for mathematical physicists. Particularly famous is the question about existence and uniqueness for General Relativity.

 

[timeant]

Together with reasonable boundary conditions of course. I guess, there is rigorous math literature on this subject. Usually the Cauchy problem of all kinds of partial-differential systems occurring in physics is an interesting topic for mathematical physicists. Particularly famous is the question about existence and uniqueness for General Relativity.

rigorous math literature ? Really ?
By bing.com, I cannot find the existene of field equations.
For electromagnetic field, is the charge conservation ($\partial_\alpha J^\alpha=0$) one of sufficient conditions for the existence of Maxwell equations?
For gravitational field, is the soure's conservation ($\nabla_\alpha T^{\alpha\beta}=0$) one of sufficient conditions for the existence of Einstein field equations?

As an analogy, for $\nabla\cdot \mathbf{u}=\rho, \nabla\times \mathbf{u}=\mathbf{S}$, is $\nabla\cdot \mathbf{S}=0$ one of sufficient conditions for the existence of div-curl system?

I think so! They are all sufficient conditions.

The EXISTENCE should be talked, e.g. http://www.claymath.org/millennium-problems/navier–stokes-equation

 

[vanhees71]

Charge conservation is a necessary condition for the consistency of the Maxwell equations. It's a Bianchi identity of gauge symmetry. For the gravitational field the energy-momentum tensor, which is necessarily symmetric, as a source is necessarily locally conserved. The correct equation is $\nabla_{\mu} T^{\mu \nu}=0$, i.e., you must use the covariant derivative of the tensor-field components. This is again the consequence of a Bianchi symmetry of gauge symmetry of GR (general covariance).

 

[timeant]

For single div--curl system, i.e. div u = f, curl u = B, theorem 3.3 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 : div B =0 is one of sufficient conditions of existence.

Maxwell equations are double div--curl systems. I think that the charge continuity equation is also one of sufficient conditions of existence.