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

  • Thread starter timeant
  • Start date
  • #1
timeant
16
2
Are there any references?
 

Answers and Replies

  • #2
vanhees71
Science Advisor
Insights Author
Gold Member
2022 Award
22,062
12,966
A. Sommerfeld, Lectures of Theoretical Physics, vol. 3
 
  • #3
timeant
16
2
Thank @ vanhees71 . However , I don't find the existence conditions, only found uniqueness .
Electrodynamics. Lectures on Theoretical Physics, Vol. 3 by Arnold Sommerfeld
 
  • #5
timeant
16
2
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?
 
  • #6
vanhees71
Science Advisor
Insights Author
Gold Member
2022 Award
22,062
12,966
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.
 
  • #7
timeant
16
2
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:
  • #8
vanhees71
Science Advisor
Insights Author
Gold Member
2022 Award
22,062
12,966
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).
 
  • #9
timeant
16
2
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.
 

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

Replies
4
Views
482
Replies
7
Views
1K
  • Last Post
Replies
19
Views
1K
Replies
3
Views
626
Replies
3
Views
503
  • Last Post
Replies
26
Views
4K
Top