2015 / December Volume 10 No.4
The Mathematical Theory of Self-similar Boundary Layers for Nonlinear Hyperbolic Systems with Viscosity and Capillarity
| Published Date |
2015 / December
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| Title | The Mathematical Theory of Self-similar Boundary Layers for Nonlinear Hyperbolic Systems with Viscosity and Capillarity |
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| Pagination | 639-693 |
| 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.
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| AMS Subject Classification |
35L65, 76N10.
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| Received |
2015-06-29
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| Accepted |
2015-06-23
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