Stability of undercompressive shock profiles
Peter Howard, Kevin Zumbrun
Abstract
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 ZH, extending to the general undercompressive case results obtained for Lax and overcompressive shock profiles in SzX, L, ZH, Z.2, Ra, MZ.1--MZ.5, and for special undercompressive profiles in LZ.1--LZ.2, HZ. In particular, together with spectral results of Z.6, our results yield nonlinear stability of large-amplitude undercompressive phase-transitional profiles near equilibrium of Slemrod's model Sl.5 for van der Waal gas dynamics or elasticity with viscosity--capillarity.
Create a lesson
Related papers
Global existence of smooth solutions to the Navier-Stokes equations for initial data with infinite energy
Alexey Cheskidov, Hedong Hou
Sharp Global Well-Posedness for the Logarithmically Regularized 2D Euler Equations
Zachary Radke
Optimal transport to non-convex polyhedral domains
Shibing Chen, Qi-Rui Li, Yuanyuan Li et al.
Generalized minimizers for the capillarity energy outside convex obstacles
Gabriele Nani, Simone Vincini
Generic boundary behavior of Dirichlet eigenfunctions
Alec Payne, Zihui Zhao
Near-boundary rigidity and strict maximum modulus for Neumann eigenfunctions on convex planar domains
Haiyun Deng, Xuyong Jiang, Xiaoping Yang