2008/11/04 by G. C. Coclite, Kenneth H. Karlsen, Coclite, G. C. +3
Mathematics · Physics and Astronomy · #35G25 #35L65 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.0811.0549
openalex publication_date 2008/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.