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Title: Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, I: well-posedness
Authors: Garetto, Claudia
Jah, Christian
Ruzhansky, Michael
Keywords: Hyperbolic systems
Fourier integral operators
Sobolev spaces
Issue Date: 2018
Publisher: Springer © The Author(s) 2018
Citation: GARETTO, C., JAH, C. and RUZHANSKY, M., 2018. Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, I: well-posedness. Mathematische Annalen, 372 (3-4), pp.1597–1629.
Abstract: In this paper we analyse the well-posedness of the Cauchy problem for a rather general class of hyperbolic systems with space-time dependent coefficients and with multiple characteristics of variable multiplicity. First, we establish a well-posedness result in anisotropic Sobolev spaces for systems with upper triangular principal part under interesting natural conditions on the orders of lower order terms below the diagonal. Namely, the terms below the diagonal at a distance k to it must be of order −k . This setting also allows for the Jordan block structure in the system. Second, we give conditions for the Schur type triangularisation of general systems with variable coefficients for reducing them to the form with an upper triangular principal part for which the first result can be applied. We give explicit details for the appearing conditions and constructions for 2×2 and 3×3 systems, complemented by several examples.
Description: This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Sponsor: Michael Ruzhansky was supported in parts by EPSRC Grant EP/R003025/1 and by the Leverhulme Grant RPG-2017-151.
Version: Published
DOI: 10.1007/s00208-018-1672-1
URI: https://dspace.lboro.ac.uk/2134/28240
Publisher Link: https://doi.org/10.1007/s00208-018-1672-1
Appears in Collections:Published Articles (Maths)

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