Invariant manifolds and the stability of traveling waves in scalar viscous conservation laws (Q2466489)

In the present study the stability of travelling 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 determined by the asymptotic spatial limits of the original operator. By applying scaling variables to these operators, a spectral gap is created, thus allowing for the use of invariant manifold theory to determine the temporal decay rate of the far-field components. The linear operator associated to the near-field component has the spectrum contained entirely within the left-half plane, and so the associated semigroup decays exponentially in time. Moreover, the full perturbation is shown to decay at the same rate as the far-field components.