Nonlinear Asymptotic Stability of General Small-Amplitude Viscous Laxian Shock Waves

Nonlinear Stability of Viscous Shock Waves

Spectral stability of small-amplitude viscous shock waves in several space dimensions

A planar viscous shock profile of a hyperbolic-parabolic system of conservation laws is a steady solution in a moving coordinate frame. The asymptotic stability of viscous profiles and the related vanishing-viscosity limit are delicate questions already in the well understood case of one space dimension and even more so in the case of several space dimensions. It is a natural idea to study the ...

Nonlinear Stability of Shock Waves for Viscous Conservation Laws

where u = u{x,t) E R , the flux f(u) is a smooth n-vector-valued function, and the viscosity B(u) is a smooth n x n matrix. We are interested in the stability of traveling waves, the "viscous shock waves", for (1). It is shown that when the initial data are a perturbation of viscous shock waves, then the solution converges to these viscous shock waves, properly translated in space, in the unifo...

Stability and Dynamics of Viscous Shock Waves

We examine from a classical dynamical systems point of view stability, dynamics, and bifurcation of viscous shock waves and related solutions of nonlinear pde. The central object of our investigations is the Evans function: its meaning, numerical approximation, and behavior in various asymptotic limits.

Nonlinear Stability of Viscous Roll Waves

Extending results of Oh–Zumbrun and Johnson–Zumbrun for parabolic conservation laws, we show that spectral stability implies nonlinear stability for spatially periodic viscous roll wave solutions of the one-dimensional St. Venant equations for shallow water flow down an inclined ramp. The main new issues to be overcome are incomplete parabolicity and the nonconservative form of the equations, w...