Inviscid Burgers equation on a metric tree

Consider the classical inviscid Burgers equation u_t+uu_x=0 in a monodimensional case. The basic interpretation of the system explains the motion of one wave, creation of shocks and rarefaction waves, but does not capture the interaction of waves, since big waves always overtake smaller ones. Hoping to receive passing through phenomenon between waves, think now about the extension of the monodimensional structure of the domain to a graph, giving the possibility for the solution to take different paths.

The above reasoning serves us as a motivation to develop the theory of the Burgers equation on the metric graph. Alike many network problems, the mian difficulty here is to determine the behaviour in vertices to make the problem well-posed. In the talk we will show some interesting examples as well as key points of mathematical theory.

The talk is based on joint work with Piotr B. Mucha.