Viscous profiles for traveling waves of scalar balance laws: The uniformly hyperbolic case

• Main Author: Joerg Haerterich
• Format: Article
• Language: English
• Published: Texas State University 2000-04-01
• Series: Electronic Journal of Differential Equations
• Subjects: Hyperbolic conservation laws; source terms; traveling waves; viscous profiles; singular perturbations.
• Online Access: http://ejde.math.txstate.edu/Volumes/2000/30/abstr.html

We consider a scalar hyperbolic conservation law with a nonlinear source term and viscosity $varepsilon$. For $varepsilon=0$, there exist in general different types of heteroclinic entropy traveling waves. It is shown that for $varepsilon>0$ sufficiently small the viscous equation possesses similar traveling wave solutions and that the profiles converge in exponentially weighted $L^1$-norms as $varepsilon$ decreases to zero. The proof is based on a careful study of the singularly perturbed second-order equation that arises from the traveling wave ansatz.