2006/12/30 by Kuruvilla Joseph, Philippe G. LeFloch, Joseph, K. T. +1
Earth and Planetary Sciences · Engineering · Mathematics · #35L65 #76L05 #76N #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Thin Films #Navier-Stokes equation solutions #Numerical Analysis (math.NA) #Ocean Waves and Remote Sensing
paper · pdf · doi:10.48550/arxiv.math/0701007
openalex publication_date 2006/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \sl 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 hyperbolic-elliptic 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 capillarity-viscosity ratio) are observed which prevent the solution to have globally bounded variation.