Existence and uniqueness of BV solutions for a conservation law with \(\sigma\)-finite Borel measures as initial conditions (Q1265119)

Existence and uniqueness of the bounded-variation solution to the Cauchy problem for the nonlinear conservation law \(\partial u/\partial t+ \partial u^m/ \partial x=0\) with \(m>1\) and with a Borel measure \(\mu\geq 0\) as initial condition is proved provided \(\sup_{R>r} R^{m/(1-m)} \int^0_{-R} d\mu\) is finite for some \(r\) positive. On the other hand, for \(\mu=\sum^\infty_{i=1} 2^{i\lambda} \delta (x+2^i)\) with \(\lambda> m/(m-1)\) and \(\delta (\cdot)\) denoting the Dirac mass, the nonexistence of any solution is shown.