Duke Mathematical Journal

Blowup of the BV norm in the multidimensional Keyfitz and Kranzer system

Camillo De Lellis

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We consider the Cauchy problem for the system ∂tui + divz(g(|u|)ui) = 0, i ∈ {1,…, k}, in m space dimensions and with gC3. When k ≥ 2 and m = 2, we show a wide choice of g's for which the bounded variation (BV) norm of admissible solutions can blow up, even when the initial data have arbitrarily small oscillation and arbitrarily small total variation, and are bounded away from the origin. When m ≥ 3, we show that this occurs whenever g is not constant, that is, unless the system reduces to k decoupled transport equations with constant coefficients.

Article information

Duke Math. J., Volume 127, Number 2 (2005), 313-339.

First available in Project Euclid: 23 March 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35L65: Conservation laws
Secondary: 35L45: Initial value problems for first-order hyperbolic systems 35L40: First-order hyperbolic systems


De Lellis, Camillo. Blowup of the BV norm in the multidimensional Keyfitz and Kranzer system. Duke Math. J. 127 (2005), no. 2, 313--339. doi:10.1215/S0012-7094-04-12724-1. https://projecteuclid.org/euclid.dmj/1111609854

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