Initial–boundary value problems for conservation laws with source terms and the Degasperis–Procesi equation |
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Authors: | G.M. Coclite K.H. Karlsen Y.-S. Kwon |
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Affiliation: | aDipartimento di Matematica, Università degli Studi di Bari, Via E. Orabona 4, 70125 Bari, Italy;bCentre of Mathematics for Applications, University of Oslo, PO Box 1053, Blindern, N–0316 Oslo, Norway;cInstitute of Mathematics of the Academy of Sciences of the Czech Republic, Zitna' 25, 115 67 Praha 1, Czech Republic;dDepartment of Mathematics, Dong-A University, Busan 604-714, Republic of Korea |
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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 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. |
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Keywords: | Conservation laws with source terms Trace theorem Kinetic formulation Boundary value problems Averaging lemma Degasperis– Procesi equation |
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