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

• timeant
In summary, the conversation discusses the existence conditions for Maxwell equations and the role of charge conservation and energy-momentum tensor conservation in their consistency. It also mentions the existence of rigorous mathematical literature on this subject and the relevance of the Cauchy problem in physics. The conversation also makes an analogy to the div-curl system and discusses a theorem regarding its existence conditions.

#### timeant

Are there any references?

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

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

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?

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.

vanhees71 said:
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

Last edited:
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).

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.