2004/08/11 by Peter Howard, Howard, Peter, Kevin Zumbrun +1 · 1 citation
Mathematics · Physics and Astronomy · #35L65 (35B25 35B30 35K65) #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AP #math.MP #msc:35B30 #msc:35L65
paper · pdf · doi:10.48550/arxiv.math/0408150
Corrected typos, added brief remarks on physical models and applications
arxiv created 2004/08/14 · arxiv updated 2009/12/01
Using a simplified pointwise iteration scheme, we establish nonlinear phase-asymptotic orbital stability of large-amplitude Lax, undercompressive, overcompressive, and mixed under--overcompressive type shock profiles of strictly parabolic systems of conservation laws with respect to initial perturbations |u0(x)|≤ E0 (1+|x|)-3/2 in C0+α, E0 sufficiently small, under the necessary conditions of spectral and hyperbolic stability together with transversality of the connecting profile. This completes the program initiated by Zumbrun and Howard in \citeZH, extending to the general undercompressive case results obtained for Lax and overcompressive shock profiles in \citeSzX, \citeL, \citeZH, \citeZ.2, \citeRa, \citeMZ.1--\citeMZ.5, and for special undercompressive profiles in \citeLZ.1--\citeLZ.2, \citeHZ. In particular, together with spectral results of \citeZ.6, our results yield nonlinear stability of large-amplitude undercompressive phase-transitional profiles near equilibrium of Slemrod's model \citeSl.5 for van der Waal gas dynamics or elasticity with viscosity--capillarity.