Hyperbolic conservation laws with large initial data. Is the Cauchy problem well-posed?

Tsikkou, Charis

doi:10.1090/S0033-569X-2010-01208-9

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
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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

Paolo Baiti and Helge Kristian Jenssen, Blowup in $L^\infty $ for a class of genuinely nonlinear hyperbolic systems of conservation laws, Discrete Contin. Dynam. Systems 7 (2001), no. 4, 837–853. MR 1849664, DOI https://doi.org/10.3934/dcds.2001.7.837
Stefano Bianchini and Alberto Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math. (2) 161 (2005), no. 1, 223–342. MR 2150387, DOI https://doi.org/10.4007/annals.2005.161.223
Alberto Bressan, Hyperbolic systems of conservation laws, Oxford Lecture Series in Mathematics and its Applications, vol. 20, Oxford University Press, Oxford, 2000. The one-dimensional Cauchy problem. MR 1816648
Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2005. MR 2169977
Constantine M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl. 38 (1972), 33–41. MR 303068, DOI https://doi.org/10.1016/0022-247X%2872%2990114-X
James Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697–715. MR 194770, DOI https://doi.org/10.1002/cpa.3160180408
James Glimm and Peter D. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Memoirs of the American Mathematical Society, No. 101, American Mathematical Society, Providence, R.I., 1970. MR 0265767
Helge Holden and Nils Henrik Risebro, Front tracking for hyperbolic conservation laws, Applied Mathematical Sciences, vol. 152, Springer-Verlag, New York, 2002. MR 1912206
Alan Jeffrey, Breakdown of the solution to a completely exceptional system of hyperbolic equations, J. Math. Anal. Appl. 45 (1974), 375–381. MR 333472, DOI https://doi.org/10.1016/0022-247X%2874%2990079-1
Helge Kristian Jenssen, Blowup for systems of conservation laws, SIAM J. Math. Anal. 31 (2000), no. 4, 894–908. MR 1752421, DOI https://doi.org/10.1137/S0036141099352339
J.-L. Joly, G. Métivier, and J. Rauch, A nonlinear instability for $3\times 3$ systems of conservation laws, Comm. Math. Phys. 162 (1994), no. 1, 47–59. MR 1272766
S. N. Kružkov, First order quasilinear equations with several independent variables., Mat. Sb. (N.S.) 81 (123) (1970), 228–255 (Russian). MR 0267257
Tai Ping Liu, Admissible solutions of hyperbolic conservation laws, Mem. Amer. Math. Soc. 30 (1981), no. 240, iv+78. MR 603391, DOI https://doi.org/10.1090/memo/0240
Tai-Ping Liu, Zhouping Xin, and Tong Yang, Vacuum states for compressible flow, Discrete Contin. Dynam. Systems 4 (1998), no. 1, 1–32. MR 1485360, DOI https://doi.org/10.3934/dcds.1996.2.1
Joel Smoller, Shock waves and reaction-diffusion equations, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York, 1994. MR 1301779
Konstantina Trivisa, A priori estimates in hyperbolic systems of conservation laws via generalized characteristics, Comm. Partial Differential Equations 22 (1997), no. 1-2, 235–267. MR 1434145, DOI https://doi.org/10.1080/03605309708821262
Konstantina Trivisa, BV estimates for $n\times n$ systems of conservation laws, Advances in differential equations and mathematical physics (Birmingham, AL, 2002) Contemp. Math., vol. 327, Amer. Math. Soc., Providence, RI, 2003, pp. 341–358. MR 1991553, DOI https://doi.org/10.1090/conm/327/05826

C. Tsikkou, BV Estimates for the P-System, preprint.

R. Young, On elementary interactions for hyperbolic conservation laws, preprint.

Robin Young, Isentropic gas dynamics with large data, Hyperbolic problems: theory, numerics, applications, Springer, Berlin, 2003, pp. 929–939. MR 2053240

Robin Young, Blowup of solutions and boundary instabilities in nonlinear hyperbolic equations, Commun. Math. Sci. 1 (2003), no. 2, 269–292. MR 1980476

Robin Young and Walter Szeliga, Blowup with small BV data in hyperbolic conservation laws, Arch. Ration. Mech. Anal. 179 (2006), no. 1, 31–54. MR 2208288, DOI https://doi.org/10.1007/s00205-005-0370-9