Propagation of support and singularity formation for a class of 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 | References | Similar Articles | Additional Information

Abstract: In this paper we consider a class of quasilinear, non-strictly hyperbolic 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.

**[1]**S. Alinhac,*Temps de vie des solutions régulières des équations d'Euler compressibles axisymmétriques en dimension deux*, Invent. Math.**11l**, 627-670 (1993) MR**1202138****[2]**P. Daripa, J. Glimm, B. Lindquist, and O. McBryan,*Polymer Floods: A case study of nonlinear wave analysis and of instability control in tertiary oil recovery*, D.O.E. Research and Development Report 03077-275, New York University, November 1986**[3]**F. John,*Nonlinear wave equations, formation of singularities*, A.M.S. University Lecture Series, vol. 2, Amer. Math. Soc., Providence, RI, 1990 MR**1066694****[4]**K. Kajitani,*The Cauchy problem for nonlinear hyperbolic systems*, Bull. Sci. Math., série,**110**, 3-48 (1986) MR**861667****[5]**B. L. Keyfitz,*A survey of nonstrictly hyperbolic conservation laws*, Lecture Notes in Mathematics, vol. 1270, Springer-Verlag, New York, 1986, pp. 152-162 MR**910112****[6]**B. L. Keyfitz and M. C. Filho Lopes,*How to use symmetry to find models for multidimensional systems of conservation laws*, Lectures in Applied Math.**29**, 273-284 (1993) MR**1247731****[7]**A. J. Krener and H. Schättler,*The structure of small-time reachable sets in low dimensions*, SIAM J. Control and Optim.**27**, No. 1, 120-147 (1989) MR**980227****[8]**M. C. Lopes Filho and H. J. Nussenzveig Lopes,*Multidimensional hyperbolic systems with degenerate characteristic structure*, Matemática Contemporanea**8**, 225-238 (1995) MR**1330039****[9]**M. C. Lopes Filho and H. J. Nussenzveig Lopes,*Singularity formation for a system of conservation laws in two space dimensions*, J. Math. Anal. Appl.**200**, 538-547 (1996) MR**1393100****[10]**M. C. Lopes Filho and H. J. Nussenzveig Lopes,*Smooth solutions of multidimensional conservation laws*, Z.A.M.M., special issue in applied analysis, Proceedings of the III ICIAM, Hamburg, 1996, pp. 141-144**[11]**M. A. Rammaha,*Formation of singularities in compressible fluids in two-space dimensions*, Proc. Amer. Math. Soc.**107**, 3, 705-714 (1989) MR**984811****[12]**T. C. Sideris,*Formation of singularities in solutions to nonlinear hyperbolic equations*, Arch. Rat. Mech. Anal.**86**, 4, 369-382 (1984) MR**759769****[13]**D. Tan and T. Zhang,*Two-dimensional Riemann problem for a hyperbolic system of conservation laws*, Acta Math. Sci. (English edition)**11**4, 369-392 (1991) MR**1174368**

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DOI:
https://doi.org/10.1090/qam/1686187

Article copyright:
© Copyright 1999
American Mathematical Society