Well-posedness of general 1D initial boundary value problems for scalar balance laws

Abstract

We focus on the initial boundary value problem for a general scalar balance law in one space dimension. Under rather general assumptions on the flux and source functions, we prove the well-posedness of this problem and the stability of its solutions with respect to variations in the flux and in the source terms. For both results, the initial and boundary data are required to be bounded functions with bounded total variation. The existence of solutions is obtained from the convergence of a Lax-Friedrichs type algorithm with operator splitting. The stability result follows from an application of Kružkov's doubling of variables technique, together with a careful treatment of the boundary terms.

A study on minimum time regulation of a bounded congested road with upstream flow control

“…A well-posedness theorem is presented in [17] for general IBVP posed on a finite segment. Below we adapt the result to the IBVP under Definition II.4.…”

Well‐posedness of IBVP for 1D scalar non‐local conservation laws

“…The analysis of the non‐local problem is carried out exploiting the same strategy used in both [] and []. As already mentioned [], studies the non‐local IBVP where the non‐local operator is the standard convolution product, while [] considers the local problem for a balance law, i.e. a one dimensional IBVP where the flux function has the form and there is also a source term.…”

“…However, in this way the a priori estimates on the solution would be less precise than those presented in this work. Namely, a positivity result and an ‐bound on the solution are missing in []. Moreover, the ‐estimate recovered here depends on the first derivatives of the flux function, see Theorem , while using the results of [] yields an estimate depending on the mixed second derivatives of f .…”

“…Nevertheless, the result concerning the stability with respect to the flux function proved in [], recalled below in Theorem , is of crucial importance in this work, since it contributes significantly in the proof of the Lipschitz continuous dependence of solutions to on initial and boundary data, see Proposition , and thus in the proof of the uniqueness of solutions to . At this regard, we remark that the stability proof provided in [] is wrong, but could be fixed following the same strategy proposed here.…”

“…Section presents a sample application of the model and some numerical integrations. The final appendix A recalls some results from [] on the local IBVP, necessary throughout the paper.…”