On the problem of stability of viscous shocks

2025/11/24 by Bolotin, Sergey, Treschev, Dmitry
Engineering · Mathematics · #Advanced Differential Equations and Dynamical Systems #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations

paper · doi:10.48550/arxiv.2511.19167

openalex publication_date 2025/11/24 · openalex created_date 2025/11/27 · openalex updated_date 2026/07/28

Abstract

We consider the problem of spectral stability of traveling wave solutions u=γ(x-Wt) for a system of viscous conservation laws ∂_t u + ∂_x F(u) = ∂²_x u. Such solutions correspond to heteroclinic trajectories γ of a system of ODE. In general conditions of stability can be obtained only numerically. We propose a model class of piece-wise linear (discontinuous) vector fields F for which the stability problem is reduced to a linear algebra problem. We show that the stability problem makes sense in such low regularity and construct several examples of stability loss. Every such example can be smoothed to provide a smooth example of the same phenomenon.

Citations

• Waves, structures, and the Riemann problem for a system of hyperbolic conservation laws
• Stability of undercompressive shock profiles
• Nonlinear asymptotic stability of viscous shock profiles for conservation laws

Related

• Galloping instability of viscous shock waves
• Nonlinear stability of periodic traveling wave solutions of systems of viscous conservation laws in the generic case
• Conditional stability of unstable viscous shocks
• Nonlinear stability of periodic traveling wave solutions of viscous conservation laws in dimensions one and two
• Planar stability critera for viscous shock waves of systems with real viscosity