Abstract

Relative entropy is a rather coarse but flexible tool for studying stability properties of systems of hyperbolic conservation laws. In particular, it provides weak strong uniqueness. It is commonly applied to problems on the whole of Euclidean space or on the flat torus, since its interaction with boundary conditions is not straightforward. We will discuss to which extent it can be extended to hyperbolic balance laws on networks.

Systems of hyperbolic balance laws on networks play a significant role in applications such as traffic flow or gas networks. In problems of this type, the networks are described by graphs with (spatially one dimensional) hyperbolic systems on each edge and (algebraic) coupling conditions at the nodes. A frequently studied example are isothermal Euler equations on the edges. In this case, one obvious coupling condition is conservation of mass at the nodes but further coupling conditions are not obvious - indeed, conservation of momentum cannot be expected due to interaction of the flow with the pipe walls and a lack of information on junction geometry.

A few years ago, energy consistent coupling conditions have been proposed by Reigstad. We will show that energy consistent coupling conditions do not ensure that a relative entropy stability framework is available for such systems. This is due to a lack of control on sums of relative entropy fluxes at the nodes.

We will discuss how (certain aspects of) a relative entropy framework can be recovered, when attention is restricted to subsonic solutions of the isothermal Euler equations. This is, indeed, a relevant setup for the operation of natural gas networks. In this setting, the relative entropy guarantees that Lipschitz solutions depend continuously on their initial data and it can be used for studying low Mach/large friction limits. At the same time, this variant of relative entropy no longer provides weak entropic-strong uniqueness.