Vanishing viscosity solutions of scalar conservation laws at a junction

31/05/2017, ore 12, aula 2

We consider the Cauchy problem for scalar conservation laws on a junction where m incoming and n outgoing edges meet. In the first part of this talk we present a  well-posedness result for solutions obtained as limits of the vanishing viscosity approximations considered by Coclite and Garavello (SIAM, 2010).

The proof of our main result relies on the introduction of a family of Kruzhkov-type adapted entropies at the junction and a suitable definition of admissible solution. The key step in our construction consists in the description and analysis of the set of stationary solutions for the inviscid problem from the point of view developed  by Andreianov, Karlsen, Risebro and Cancès to deal with scalar conservation laws with discontinuous flux.

Numerical tests, obtained by a finite volumes scheme, are presented to show the typical behavior of solutions.

In the second part we explain how we can obtain a different class of limit solutions by changing the transmission condition at the junction for the parabolic (approximate) problems.

This research project is developed in collaboration with Boris P. Andreianov (Univ. Tours), Giuseppe M. Coclite (Politec. Bari) and Sabrina F. Pellegrino (Univ. Bari