Burgers’ Equation Revisited: Extension of Mono-Dimensional Case on a Network

Abstract

The paper deals with the analysis of Burgers’ equation on acyclic metric graphs. The main goal is to establish the existence of weak solutions in the TV—class of regularity. A key point is transmission conditions in vertices obeying the Kirchhoff law. First, we consider positive solutions at arbitrary acyclic networks and highlight two kinds of vertices, describing two mechanisms of flow splitting at the vertex. Next we design rules at vertices for solutions of arbitrary sign for any subgraph of hexagonal grid, which leads to a construction of general solutions with TV—regularity for this class of networks. Introduced transmission conditions are motivated by the change of the energy estimation.