Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Propagation of support and singularity formation for a class of $ 2$D quasilinear hyperbolic systems


Authors: M. C. Lopes Filho and H. J. Nussenzveig Lopes
Journal: Quart. Appl. Math. 57 (1999), 229-243
MSC: Primary 35L60; Secondary 35A20
DOI: https://doi.org/10.1090/qam/1686187
MathSciNet review: MR1686187
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Abstract: In this paper we consider a class of quasilinear, non-strictly hyperbolic $ 2 \times 2$ systems in two space dimensions. Our main result is finite speed of propagation of the support of smooth solutions for these systems. As a consequence, we establish nonexistence of global smooth solutions for a class of sufficiently large, smooth initial data. The nonexistence result applies to systems in conservation form, which satisfy a convexity condition on the fluxes. We apply the nonexistence result to a prototype example, obtaining an upper bound on the lifespan of smooth solutions with small amplitude initial data. We exhibit explicit smooth solutions for this example, obtaining the same upper bound on the lifespan and illustrating loss of smoothness through blow-up and through shock formation.


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DOI: https://doi.org/10.1090/qam/1686187
Article copyright: © Copyright 1999 American Mathematical Society


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