Margaret Beck, C. Eugene Wayne
Invariant Manifolds and the Stability of Traveling Waves in 
Scalar Viscous Conservation Laws
(327K, PDF)

ABSTRACT.  The stability of traveling wave solutions of scalar, viscous conservation laws is investigated 
by decomposing perturbations into three components: two far-field components 
and one near-field component. The linear operators associated to the far-field components 
are the constant coeficient operators determined by the asymptotic spatial limits 
of the original operator. Scaling variables can be applied to study the evolution of these 
components, allowing for the construction of invariant manifolds and the determination 
of their temporal decay rate. The large time evolution of the near-field component is 
shown to be governed by that of the far-field components, thus giving it the same temporal 
decay rate. We also give a discussion of the relationship between this geometric 
approach and previous results, which demonstrate that the decay rate of perturbations 
can be increased by requiring that initial data lie in appropriate algebraically weighted 
spaces.