Gamma-entropy cost for scalar conservation laws

Giovanni Bellettini, Lorenzo Bertini, Mauro Mariani, Matteo Novaga

Published 2007-12-07, updated 2008-09-08Version 2

We are concerned with a control problem related to the vanishing viscosity approximation to scalar conservation laws. We investigate the $\Gamma$-convergence of the control cost functional, as the viscosity coefficient tends to zero. A first order $\Gamma$-limit is established, which characterizes the measure-valued solutions to the conservation laws as the zeros of the $\Gamma$-limit. A second order $\Gamma$-limit is then investigated, providing a characterization of entropic solutions to conservation laws as the zeros of the $\Gamma$-limit.