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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.

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Are there any references?

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

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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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vanhees71 said:

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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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.

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