Boundary Layers in Weak Solutions of Hyperbolic Conservation Laws

Abstract

Abstract. This paper is concerned with the initial-boundary value problem for a nonlinear hyperbolic system of conservation laws. We study the boundary layers that may arise in approximations of entropy discontinuous solutions. We consider both the vanishing viscosity method and finite difference schemes (Lax-Friedrichs type schemes, Godunov scheme). We demonstrate that different regularization methods generate different boundary layers. Hence, the boundary condition can be formulated only if an approximation scheme is selected first. Assuming solely uniform L ∞ bounds on the approximate solutions and so dealing with L ∞ solutions, we 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. Form the above analysis, we deduce several formulations for the boundary condition which apply whether the boundary is characteristic or not. Each formulation is based a set of admissible boundary values, following Dubois and LeFloch's terminology in "Boundary conditions for nonlinear hyperbolic systems of conservation laws", J. Diff. Equa. 71 (1988), 93-122. The local structure of those sets and the well-posedness of the corresponding initial-boundary value problem are investigated. The results are illustrated with convex and nonconvex conservation laws and examples from continuum mechanics.

Numerical boundary layers for hyperbolic systems in 1-D

“…These layers turn out to be stable under a smallness condition, except in the scalar case where no condition is required. Hence this paper is a rigorous justification of the formal analysis developed in [9] and is a numerical counterpart to [8]. It also completes the analysis of [6].…”

“…Near 0, C vis is a manifold, whose tangent space is spanned by the eigenvectors of f (0) with negative corresponding eigenvalues. We refer to [4], [8] and [9] for more details.…”

Remarks on the Contributions of Constantine M. Dafermos to the Subject of Conservation Laws

“…[28] and [8]). We next discuss the implications of applying the entropy admissibility criterion (3) with a designated entropy pair (η(U ), q(U )) on the selection of admissible wave fans.…”

Scalar Conservation Laws with Boundary Conditions and Rough Data Measure Solutions

“…The corresponding conditions of the trace for the initial data was derived in [11] and further studied in [8], [22]. Young measures have been used also to study the behavior of solutions to initial boundary value problems of some hyperbolic systems of conservation laws, see [7], [13].…”

“…Note that the new ν ε and σ ε satisfy (13) in Ω δ + , for δ ≥ ε ⊥ , since with a two dimensional mollification parameter ε = (ε ⊥ ,ε, ) and a planar boundary only the x ⊥ -convolution part, ω ε ⊥ , of the mollification interfere with the boundary. Take the limit ε ⊥ → 0+ in this version of (13) to obtain Claim 1b:…”