Burgers' equation revisited: extension of mono-dimensional case on a network

TL;DR

This paper extends the analysis of Burgers' equation to acyclic networks, establishing weak solutions with total variation regularity and defining transmission conditions at vertices based on Kirchhoff law.

Contribution

It introduces new transmission conditions for Burgers' equation on networks and constructs solutions with TV-regularity for arbitrary sign solutions on hexagonal grid subgraphs.

Findings

Existence of weak solutions with TV-regularity on acyclic networks.

Transmission conditions at vertices based on Kirchhoff law.

Construction of solutions for arbitrary sign on hexagonal grid subgraphs.

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.