Advances in Differential Equations
- Adv. Differential Equations
- Volume 9, Number 9-10 (2004), 1167-1184.
Asymptotic stability of traveling waves for viscous conservation laws with dispersion
This work is concerned with the asymptotic stability of traveling waves for scalar viscous conservation laws with a convex flux function and a dispersion term. First we prove the existence of solutions locally in time of the initial-value problem for initial data near a constant solution by Fourier analysis. Using the semigroup method the local existence for initial data that are an $L^2$ perturbation of a traveling-wave profile is proved. We also obtain a regularity property of these solutions. The solution operator generated by the linearized equation plays a crucial role. Using the energy method we establish a priori estimates. These estimates, when combined with the local existence, lead to the desired global-in-time existence as well as the time-asymptotic decay of solutions with initial data close to a monotone traveling wave.
Adv. Differential Equations Volume 9, Number 9-10 (2004), 1167-1184.
First available in Project Euclid: 18 December 2012
Permanent link to this document
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 35B35: Stability 47N20: Applications to differential and integral equations 76E99: None of the above, but in this section
Pan, Jun; Warnecke, Gerald. Asymptotic stability of traveling waves for viscous conservation laws with dispersion. Adv. Differential Equations 9 (2004), no. 9-10, 1167--1184. https://projecteuclid.org/euclid.ade/1355867917.