×

Boundary layers in weak solutions of hyperbolic conservation laws. (English) Zbl 0959.35119

This paper is concerned with the initial-boundary value problem for a nonlinear hyperbolic system of conservation laws. The authors study the boundary layers that may arise in approximations of entropy discontinuous solutions. They consider both the vanishing viscosity method and finite difference schemes. They demonstrate that different regularization methods generate different boundary layers. Assuming solely uniform \(L^\infty\) bounds on the approximate solutions and so dealing with \(L^\infty\) solutions, they derive several entropy inequalities satisfied by the boundary layer in each case under consideration. A Young measure is introduced to describe the boundary trace. When a uniform bound on the total variation is available, the boundary Young measure reduces to a Dirac mass.
From the above analysis, they deduce several formulations for the boundary condition which apply whether the boundary is characteristic or not. Each formulation is based on a set of admissible boundary values, following the terminology of F. Dubois and P. Le Flock [J. Differ. Equations 71, No. 1, 93-122 (1988; Zbl 0649.35057)]. The local structure of these sets and the well-posedness of the corresponding initial-boundary value problem are investigated. The results are illustrated with convex and non convex conservative laws and examples from continuum mechanics.

MSC:

35L65 Hyperbolic conservation laws
35L50 Initial-boundary value problems for first-order hyperbolic systems

Citations:

Zbl 0649.35057
PDFBibTeX XMLCite
Full Text: DOI arXiv