Singular limits in phase dynamics with physical viscosity and capillarity

Joseph, K.T. ; LeFloch, P. G. (2007) Singular limits in phase dynamics with physical viscosity and capillarity Proceedings of the Royal Society of Edinburgh - A - Mathematics, 137 (6). pp. 1287-1312. ISSN 0308-2105

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Following pioneering work by Fan and Slemrod who studied the effect of artificial viscosity terms, we consider the system of conservation laws arising in liquid-vapor phase dynamics with physical viscosity and capillarity effects taken into account. Following Dafermos we consider self-similar solutions to the Riemann problem and establish uniform total variation bounds, allowing us to deduce new existence results. Our analysis cover both the hyperbolic and the hyperbolic-elliptic regimes and apply to arbitrarily large Riemann data. The proofs rely on a new technique of reduction to two coupled scalar equations associated with the two wave fans of the system. Strong L1 convergence to a weak solution of bounded variation is established in the hyperbolic regime, while in the hyperbolicelliptic regime a stationary singularity near the axis separating the two wave fans, or more generally an almost-stationary oscillating wave pattern (of thickness depending upon the capillarityviscosity ratio) are observed which prevent the solution to have globally bounded variation.

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Deposited On:12 Nov 2010 13:28
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