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

timeant
Are there any references?

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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

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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?

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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

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