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On hyperbolicity and Gevrey well-posedness. Part two: scalar or degenerate transitions

Morisse, Baptiste 2018. On hyperbolicity and Gevrey well-posedness. Part two: scalar or degenerate transitions. Journal of Differential Equations 264 (8) , pp. 5221-5263. 10.1016/j.jde.2018.01.011

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Abstract

For first-order quasi-linear systems of partial differential equations, we formulate an assumption of a transition from initial hyperbolicity to ellipticity. This assumption bears on the principal symbol of the first-order operator. Under such an assumption, we prove a strong Hadamard instability for the associated Cauchy problem, namely an instantaneous defect of Hölder continuity of the flow from Gσ to L2 , with 0<σ<σ0 , the limiting Gevrey index σ0 depending on the nature of the transition. We restrict here to scalar transitions, and non-scalar transitions in which the boundary of the hyperbolic zone satisfies a flatness condition. As in our previous work for initially elliptic Cauchy problems [B. Morisse, On hyperbolicity and Gevrey well-posedness. Part one: the elliptic case, arXiv:1611.07225], the instability follows from a long-time Cauchy–Kovalevskaya construction for highly oscillating solutions. This extends recent work of N. Lerner, T. Nguyen, and B. Texier [The onset of instability in first-order systems, to appear in J. Eur. Math. Soc.].

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Publisher: Elsevier
ISSN: 0022-0396
Funders: EPSRC
Date of First Compliant Deposit: 22 October 2018
Date of Acceptance: 20 September 2017
Last Modified: 22 Jan 2019 16:05
URI: http://orca.cf.ac.uk/id/eprint/116059

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