Abstract.
We consider the class of nonlinear models of electromagnetism that has been described by Coleman & Dill [7]. A model is completely determined by its energy density W(B,D). Viewing the electromagnetic field (B,D) as a 3×2 matrix, we show that polyconvexity of W implies the local well-posedness of the Cauchy problem within smooth functions of class H s with s>1+d/2.
The method follows that designed by Dafermos in his book [9] in the context of nonlinear elasticity. We use the fact that B×D is a (vectorial, non-convex) entropy, and we enlarge the system from 6 to 9 equations. The resulting system admits an entropy (actually the energy) that is convex.
Since the energy conservation law does not derive from the system of conservation laws itself (Faraday’s and Ampère’s laws), but also needs the compatibility relations div B=div D=0 (the latter may be relaxed in order to take into account electric charges), the energy density is not an entropy in the classical sense. Thus the system cannot be symmetrized, strictly speaking. However, we show that the structure is close enough to symmetrizability, so that the standard estimates still hold true.
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Communicated by C.M. Dafermos
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Serre, D. Hyperbolicity of the Nonlinear Models of Maxwell’s Equations. Arch. Rational Mech. Anal. 172, 309–331 (2004). https://doi.org/10.1007/s00205-003-0303-4
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DOI: https://doi.org/10.1007/s00205-003-0303-4