Estimates of the Domain of Dependence for Scalar Conservation Laws

TL;DR

This paper develops outer estimates for the domain of dependence in multidimensional scalar conservation laws using controllability sets, enhancing understanding of wave support and control applications.

Contribution

It introduces a novel outer estimate for the domain of dependence based on a differential inclusion, extending classical Kruzkov theory.

Findings

Outer estimates for the domain of dependence are derived.

Reachable sets provide outer bounds for wave support.

Applications to control problems are demonstrated.

Abstract

We consider the Cauchy problem for a multidimensional scalar conservation law and construct an outer estimate for the domain of dependence of its Kruzkov solution. The estimate can be represented as the controllability set of a specific differential inclusion. In addition, reachable sets of this inclusion provide outer estimates for the support of the wave profiles. Both results follow from a modified version of the classical Kruzkov uniqueness theorem, which we also present in the paper. Finally, the results are applied to a control problem consisting in steering a distributed quantity to a given set.