Hyperbolic conservation laws with large initial data. Is the Cauchy problem well-posed?
Author:
Charis Tsikkou
Journal:
Quart. Appl. Math. 68 (2010), 765-781
MSC (2000):
Primary 35L65
DOI:
https://doi.org/10.1090/S0033-569X-2010-01208-9
Published electronically:
September 17, 2010
MathSciNet review:
2761243
Full-text PDF Free Access
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Abstract: We present an example in which the Glimm estimate for a strictly hyperbolic system of two conservation laws is violated.
References
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References
- P. Baiti and H.K. Jenssen, Blowup in $L^{\infty }$ for a class of genuinely nonlinear hyperbolic systems of conservation laws. Discrete Contin. Dynam. Systems 7 (2001), 837–853. MR 1849664 (2003m:35155)
- S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. of Math. 161 (2005), 1–120. MR 2150387 (2007i:35160)
- A. Bressan, Hyperbolic System of Conservation Laws. The One Dimensional Cauchy Problem. Oxford University Press, 2000. MR 1816648 (2002d:35002)
- C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Grundlehren Math. Wissenschaften Series, Second Edition, Springer Verlag, 325, 2005. MR 2169977 (2006d:35159)
- C.M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law. J. Math. Anal. Appl. 38 (1972), 33–41. MR 0303068 (46:2210)
- J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math. 18 (1965), 697–715. MR 0194770 (33:2976)
- J. Glimm and P.D. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws. Memoirs AMS, No. 101 (1970). MR 0265767 (42:676)
- H. Holden and N.H. Risebro, Front Tracking for Hyperbolic Conservation Laws. New York: Springer, 2002. MR 1912206 (2003e:35001)
- A. Jeffrey, Breakdown of the solution to a completely exceptional system of hyperbolic equations. J. Math. Anal. Appl. 45 (1974), 375–381. MR 0333472 (48:11797)
- H.K. Jenssen, Blowup for systems of conservation laws. SIAM J. Math. Anal. 31 (2000), 894–908. MR 1752421 (2001a:35114)
- J.-L. Joly, G. Metivier, and J. Rauch, A nonlinear instability for $3\times 3$ systems of conservation laws. Comm. Math. Phys. 162 (1994), 47–59. MR 1272766 (95f:35145)
- S. Kruzkov, First-order quasilinear equations with several space variables. Mat. Sbornik 123 (1970), 228-255. English translation: Math. USSR Sbornik 10 (1970), 217–273. MR 0267257 (42:2159)
- T.-P. Liu, Admissible solutions of hyperbolic conservation laws. Memoirs AMS 30 (1981), No. 240. MR 603391 (82i:35116)
- T.-P. Liu, Z. Xin and T. Yang, Vacuum states for compressible flow. Discrete Contin. Dynam. Systems 4 (1998), no. 1, 1–32. MR 1485360 (98k:76116)
- J. Smoller, Shock Waves and Reaction-diffusion Equations, Springer, New York, 1994. MR 1301779 (95g:35002)
- K. Trivisa, A priori estimates in hyperbolic systems of conservation laws via generalized characteristics. Comm. PDE 22 (1997), 235–267. MR 1434145 (98a:35086)
- K. Trivisa, BV estimates for n$\times$n systems of conservation laws. Contemp. Math. 327 (2003), 341–358. MR 1991553 (2004i:35222)
- C. Tsikkou, BV Estimates for the P-System, preprint.
- R. Young, On elementary interactions for hyperbolic conservation laws, preprint.
- R. Young, Isentropic gas dynamics with large data. In Hyperbolic Problems: Theory, Numerics, Applications, T. Hou and E. Tadmor, eds., pp. 929–939. Springer, 2003. MR 2053240
- R. Young, Blowup of solutions and boundary instabilities in nonlinear hyperbolic equations. Comm. Math. Sci. 1 (2003), 269–292. MR 1980476 (2004j:35190)
- R. Young and W. Szeliga, Blowup with small BV data in hyperbolic conservation laws. Arch. Ration. Mech. Anal. 179, No. 1, 31–54 (2006). MR 2208288 (2007h:35228)
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Additional Information
Charis Tsikkou
Affiliation:
Department of Mathematics, Brown University, Providence, RI 02912
Email:
Charis.Tsikkou@mis.mpg.de
Keywords:
Nonlinear PDE,
hyperbolic systems of conservation laws,
Glimm-Lax estimates
Received by editor(s):
July 18, 2009
Published electronically:
September 17, 2010
Additional Notes:
The author is indebted to her advisor, Professor Constantine M. Dafermos, for his constant help, advice, motivation and patience during her graduate student life at Brown University.
Article copyright:
© Copyright 2010
Brown University