Conservation law and Hamilton-Jacobi equations on a junction: the convex case

TL;DR

This paper explores the relationship between Hamilton-Jacobi equations and scalar conservation laws on network junctions, establishing convergence results and analyzing the impact of junction complexity on solution properties.

Contribution

It extends the known link between HJ equations and SCL to convex cases on 1:1 junctions, introduces new convergence results for numerical schemes, and characterizes effective flux conditions.

Findings

The correspondence between HJ and SCL solutions holds on 1:1 junctions.

Numerical schemes converge to solutions with effective flux conditions.

For more than two edges, the HJ germ is not L1-dissipative.

Abstract

The goal of this paper is to study the link between the solution to an Hamilton-Jacobi (HJ) equation and the solution to a Scalar Conservation Law (SCL) on a special network. When the equations are posed on the real axis, it is well known that the space derivative of the solution to the Hamilton-Jacobi equation is the solution to the corresponding scalar conservation law. On networks, the situation is more complicated and we show that this result still holds true in the convex case on a 1:1 junction. The correspondence between solutions to HJ equations and SCL on a 1:1 junction is done showing the convergence of associated numerical schemes. A second direct proof using semi-algebraic functions is also given. Here a 1:1 junction is a simple network composed of two edges and one vertex. In the case of three edges or more, we show that the associated HJ germ is not a L 1-dissipative germ, while it is the case for only two edges. As an important byproduct of our numerical approach, we get a new result on the convergence of numerical schemes for scalar conservation laws on a junction. For a general desired flux condition which is discretized, we show that the numerical solution with the general flux condition converges to the solution of a SCL problem with an effective flux condition at the junction. Up to our knowledge, in previous works the effective condition was directly implemented in the numerical scheme. In general the effective flux condition differs from the desired one, and is its relaxation, which is very natural from the point of view of Hamilton-Jacobi equations. Here for SCL, this effective flux condition is encoded in a germ that we characterize at the junction.