Abstract
This note deals with first order hyperbolic systems with constant multiplicities. We assume that the coefficients of the operator depend just on the time variable, and they verify Zygmund-type regularity conditions. For such operators, well-posedness in the framework of Sobolev spaces is known to hold, possibly with a finite loss of derivatives. The main purpose of the present paper is to establish finite propagation speed, which is the key to build up a local theory. The argument of the proof relies on showing fine estimates about the propagation of the support of the solution: this is achieved by passing in Fourier variables and applying Paley-Wiener theorem. In particular, our approach is specific of the time-dependent case, and it cannot be extended to operators whose coefficients depend also on the space variables.
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- 1.
For instance, already for m = 1, take the function S(ξ) = ξ∕ | ξ | : then S(ζ) is well-defined, but it is no more self-adjoint.
- 2.
We know that S(t, ξ) is a symmetrizer for A(t, ξ), but this does not imply that S(t, ζ) is a symmetrizer for A(t, ζ) = A(t, ξ) + iA(t, η).
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Acknowledgements
The author wishes to express all his gratitude to T. Alazard, whose relevant question motivated the study presented in this note about finite propagation speed. He whishes to thanks also G. Métivier and T. Nishitani for enlightening discussions on hyperbolic systems.
Finally, he is deeply grateful to the anonymous referee for the careful reading and the constructive remarks, which helped to improve the presentation of the paper.
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Fanelli, F. (2017). A Few Remarks on Hyperbolic Systems with Zygmund in Time Coefficients. In: Colombini, F., Del Santo, D., Lannes, D. (eds) Shocks, Singularities and Oscillations in Nonlinear Optics and Fluid Mechanics. Springer INdAM Series, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-319-52042-1_4
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