Nonlinear Stability of Viscous Roll Waves

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...

Nonlinear Stability of Viscous Shock Waves

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...

Persistence of Generalized Roll-Waves under Viscous Perturbation

The purpose of this article is to study the persistence of solution of a hyperbolic system under small viscous perturbation. Here, the solution of the hyperbolic system is supposed to be periodic: it is a periodic perturbation of a roll-wave. So, it has an infinity of shocks. The proof of the persistence is based on an expansion of the viscous solution and estimates on Green’s functions. Keywor...

Stability of Rarefaction Waves in Viscous Media

We study the time-asymptotic behavior of weak rarefaction waves of systems with strictly hyperbolic ux functions in one dimensional viscous uids. Our main result is to show that solutions of perturbed rarefaction data converge to an approximate, \Burgers"-rarefaction wave, for initial perturbations w 0 with small mass and localized as w 0 (x) = O(jxj ?1). The proof proceeds by iteration of a po...