Initial-boundary value problems for conservation laws with source terms and the Degasperis-Procesi equation

TL;DR

This paper establishes the existence and uniqueness of entropy solutions for conservation laws with source terms and Dirichlet boundary conditions, including applications to the Degasperis-Procesi shallow water equation, using kinetic formulation and compactness methods.

Contribution

It introduces a novel approach to boundary conditions for conservation laws with source terms and proves well-posedness for the Degasperis-Procesi equation with nonlocal sources.

Findings

Proved existence of strong boundary traces for solutions.

Established well-posedness of entropy solutions with boundary conditions.

Applied results to the Degasperis-Procesi equation with nonlocal source terms.

Abstract

We consider conservation laws with source terms in a bounded domain with Dirichlet boundary conditions. We first prove the existence of a strong trace at the boundary in order to provide a simple formulation of the entropy boundary condition. Equipped with this formulation, we go on to establish the well-posedness of entropy solutions to the initial-boundary value problem. The proof utilizes the kinetic formulation and the compensated compactness method. Finally, we make use of these results to demonstrate the well-posedness in a class of discontinuous solutions to the initial-boundary value problem for the Degasperis-Procesi shallow water equation, which is a third order nonlinear dispersive equation that can be rewritten in the form of a nonlinear conservation law with a nonlocal source term.