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Global existence for the nonlinear equations of crystal optics. (English) Zbl 0688.35091
Journ. Équ. Dériv. Partielles, St.-Jean-De-Monts 1989, Exp. No. 5, 11 p. (1989).
Global existence of small smooth solutions to the equations of crystal optics is considered, where the dielectric tensor \(\epsilon\) depends in a nonlinear way on the electric field. Proofs of the results are sketched, but not given in detail. The main result is that if the dielectric tensor satisfies \[ (*)\quad \epsilon (E)=\epsilon (0)+O(| E|^ 3) \] and is independent of x and t, then a global in time solution exists to all small initial data. The proof follows the general line of argument of S. Klainerman and G. Ponce [Commun. Pure Appl. Math. 36, 133- 141 (1983; Zbl 0509.35009)], and is based on the estimate \[ | u(x,t)| \leq ct^{-1/2}\sum_{| \alpha | \leq k}| D^{\alpha}_ xu(x,0)|_{L^ 1} \] for solutions u of the linear equations of crystal optics. This decay estimate is weaker than the estimates obtained for the wave equation, and explains why second order terms are not allowed in (*).
Reviewer: H.-D.Alber

35Q99 Partial differential equations of mathematical physics and other areas of application
35L60 First-order nonlinear hyperbolic equations
78A10 Physical optics
35L45 Initial value problems for first-order hyperbolic systems
35B45 A priori estimates in context of PDEs
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