The Mathematical Theory of Self-similar Boundary Layers for Nonlinear Hyperbolic Systems with Viscosity and Capillarity
by Anupam Pal Choudhury   K.T. Joseph   Philippe G. LeFloch  

Vol. 10 No. 4 (2015) P.639~P.693

ABSTRACT

We study the vanishing viscosity-capillarity limit under the assumption of self-similarity when the underlying nonlinear hyperbolic system of conservation laws is formulated as a boundary value problem on the half-line. We establish a uniform bound on the total variation of solutions for the corresponding viscous-capillary boundary Riemann problem, provided the capillarity coefficient does not exceed a critical threshold. This leads us to a convergence theorem, as well as an existence result for the boundary Riemann problem for systems with sufficiently small Riemann data and sufficiently small capillarity. Furthermore, allowing for a possibly large capillarity coefficient, we then derive an equation governing the boundary layer and we introduce the notion of "viscous-capillary set of admissible boundary states", which, following Dubois and LeFloch, represents all possible boundary states arising in the vanishing viscosity-capillarity limit. This set may involve, both, classical (compressive) and nonclassical (undercompressive) shock layers, the latter being typically determined by a kineric relation associated with the problem.

KEYWORDS
nonlinear hyperbolic system, Riemann problem, boundary layer, viscosity, capilarity, total variation bound.

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 35L65, 76N10.

MILESTONES

Received: 2015-03-31
Revised : 2015-06-29
Accepted: 2015-06-23

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