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

De Lellis, C (2005). Blowup of the BV norm in the multidimensional Keyfitz and Kranzer system. Duke Mathematical Journal, 127(2):313-339.

Abstract

We consider the Cauchy problem for the system ∂tui + divz(g(|u|)ui) = 0, i ∈ {1,…, k}, in m space dimensions and with g ∈ C3. 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.

Abstract

We consider the Cauchy problem for the system ∂tui + divz(g(|u|)ui) = 0, i ∈ {1,…, k}, in m space dimensions and with g ∈ C3. 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.

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