Persistence of generalized roll-waves under viscous perturbation
The author studies spatially periodic solutions of a hyperbolic conservation law \(u_t+f(u)_x=g(u)\) with a linear source under small viscous perturbations \(\varepsilon u_{xx}\). She is interested in the persistence of so-called roll waves, which are solutions with several shock waves of arbitrary strength. Under the assumption that the shocks remain non-interacting and possess a viscous profile which is linearly stable, she constructs an approximate solution up to second order and derives error estimates for the Green's function of the linearized operator around this approximate solution. To this end a careful splitting into nine different Green's kernels is necessary which describe the different wave productions and interactions near the shocks. Using these estimates existence of a solution to the original problem can be shown.
scientific article; zbMATH DE number 6189155