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

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

2008-11-04

arxiv.org/abs/0811.0549v1

Mathematics

Analysis of PDEs

24 pages

Scientific paper

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.

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