de Lellis, Camillo Blowup of the BV norm in the multidimensional Keyfitz and Kranzer system. (English) Zbl 1074.35073 Duke Math. J. 127, No. 2, 313-339 (2005). This paper deals with the Cauchy problem for a system of conservation laws with a particular class of flux functions in \(m\) space dimensions. It is shown that initial data, with arbitrarily small total variations and uniformly close to a non-zero constant, such that the BV norm of admissible solution blows up instantaneously, no matter what criterion of admissibility is chosen among the ones proposed in the literature. In certain cases this occurs unless the system reduces to decoupled transport equations with constant coefficients. This work should be of interest to some one working on conservation laws. Reviewer: V. D. Sharma (Mumbai) Cited in 4 Documents MSC: 35L65 Hyperbolic conservation laws 35L45 Initial value problems for first-order hyperbolic systems Keywords:Cauchy problem; bounded variation (BV) norm; blow up of solutions; transport equations; conservation laws × Cite Format Result Cite Review PDF Full Text: DOI References: [1] L. Ambrosio, Transport equation and Cauchy problem for \(BV\) vector fields , Invent. Math. 158 (2004), 227–260. · Zbl 1075.35087 · doi:10.1007/s00222-004-0367-2 [2] L. Ambrosio, F. Bouchut, and C. De Lellis, Well-posedness for a class of hyperbolic systems of conservation laws in several space dimensions , Comm. Partial Differential Equations 29 (2004), 1635–1651. · Zbl 1072.35116 · doi:10.1081/PDE-200040210 [3] L. Ambrosio and G. Dal Maso, A general chain rule for distributional derivatives , Proc. Amer. Math. Soc. 108 (1990), 691–702. JSTOR: · Zbl 0685.49027 · doi:10.2307/2047789 [4] L. Ambrosio and C. De Lellis, Existence of solutions for a class of hyperbolic systems of conservation laws in several space dimensions , Int. Math. Res. Not. 2003 , no. 41, 2205–2220. · Zbl 1061.35048 · doi:10.1155/S1073792803131327 [5] L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems , Oxford Math. Monogr., Oxford Univ. Press, New York, 2000. · Zbl 0957.49001 [6] P. Brenner, The Cauchy problem for the symmetric hyperbolic systems in \(L^p\) , Math. Scand. 19 (1966), 27–37. · Zbl 0154.11304 [7] A. Bressan, An ill posed Cauchy problem for a hyperbolic system in two space dimensions , Rend. Sem. Mat. Univ. Padova 110 (2003), 103–117. · Zbl 1114.35123 [8] C. M. Dafermos, Stability for systems of conservation laws in several space dimensions , SIAM J. Math. Anal. 26 (1995), 1403–1414. · Zbl 0844.35065 · doi:10.1137/S0036141093258471 [9] ——–, Hyperbolic Conservation Laws in Continuum Physics , Grundlehren Math. Wiss. 325 , Springer, Berlin, 2000. · Zbl 0940.35002 [10] C. De Lellis and M. Westdickenberg, On the optimality of velocity averaging lemmas , Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), 1075–1085. · Zbl 1041.35019 · doi:10.1016/S0294-1449(03)00024-6 [11] R. J. Di Perna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989), 511–547. · Zbl 0696.34049 · doi:10.1007/BF01393835 [12] H. Frid, Asymptotic stability of non-planar Riemann solutions for multi-D systems of conservation laws with symmetric nonlinearities , J. Hyperbolic Differ. Equ. 1 (2004), 567–579. · Zbl 1066.35056 · doi:10.1142/S0219891604000238 [13] B. L. Keyfitz and H. C. Kranzer, A system of nonstrictly hyperbolic conservation laws arising in elasticity theory , Arch. Rational Mech. Anal. 72 (1980), 219–241. · Zbl 0434.73019 · doi:10.1007/BF00281590 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.