Journal
JOURNAL OF FUNCTIONAL ANALYSIS
Volume 257, Issue 12, Pages 3823-3857Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jfa.2009.09.022
Keywords
Conservation laws with source terms; Trace theorem; Kinetic formulation; Boundary value problems; Averaging lemma; Degasperis-Procesi equation
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Funding
- Research Council of Norway
- CSCAMM
- Dong-A University
- Necas Center
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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 averaging lemma. 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. (C) 2009 Elsevier Inc. All rights reserved.
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