The obstacle-mass constraint problem for hyperbolic conservation laws. Solvability

TL;DR

This paper introduces and analyzes the obstacle-mass constraint problem for multidimensional scalar hyperbolic conservation laws, proving the existence of solutions using a penalization/viscosity approach.

Contribution

It establishes the existence of entropy solutions for the obstacle-mass constrained problem, incorporating a nonlocal Lagrange multiplier and identifying conditions for global existence.

Findings

Existence of entropy solutions under certain conditions.

Introduction of a nonlocal parabolic problem due to the mass constraint.

Conditions on initial data and obstacle function for global existence.

Abstract

In this work we introduce the obstacle-mass constraint problem for a multidimensional scalar hyperbolic conservation law. We prove existence of an entropy solution to this problem by a penalization/viscosity method. The mass constraint introduces a nonlocal Lagrange multiplier in the penalized equation, giving rise to a nonlocal parabolic problem. We determine conditions on the initial data and on the obstacle function which ensure global in time existence of solution. These are not smoothness conditions, but relate to the propagation of the support of the initial data.