Preprint 2004-056

Blow-Up of the BV Norm in the Multidimensional Keyfitz and Kranzer System

Abstract: We consider the Cauchy problem for the system ∂_t u_i + ∇_z (g(|u|) u_i) = 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 BV norm of admissible solutions can blow up, even when the initial data have arbitrarily small oscillation, arbitrarily small total variation, and are bounded away from the origin. When m≥3 we show that this occurs whenever g is not constant, i.e. unless the system reduces to k decoupled transport equations with constant coefficients.

Paper:

Available as gzipped PostScript (120 Kbytes; uncompress using gunzip).

Author(s):

Camillo De Lellis, <delellis@math.unizh.ch>

Publishing information:

To appear in Duke Mathematical Journal

Comments:

Submitted by:

<delellis@math.unizh.ch> October 12 2004.