Hyperbolicity of the nonlinear models of Maxwell's equations (Q1884718)

The class of nonlinear models of electromagnetism is considered [see \textit{B. D. Coleman} and \textit{E. H. Dill}, Z. Angew. Math. Phys. 22, 691--702 (1971; Zbl 0218.35072)]. To describe the electromagnetic field \((B,D)\) its energy density \(W(B,D)\) is used. The models are constructed on basis of conservation laws. It is shown that polyconvexity of \(W\) implies the local-posedness of the Cauchy problem within smooth functions of Sobolev space. The analogous technique has been used in the theory of nonlinear elasticity in the book [\textit{C. Dafermos}, Hyperbolic conservation laws in continuum physics. (Grundlehren der mathematischen Wissenschaften, 325. Springer-Verlag, Heidelberg) (2000; Zbl 0940.35002)]. Some open questions are given in a special part of the article. The particular case of planar waves is considered in detail.