On the closeness of trajectories for model quasi-gasdynamic equations

Abstract

On a model example of a linear hyperbolic equation with small parameter multiplying the highest time derivative it is shown that the closeness of individual trajectories to the dynamics of the limiting parabolic equation essentially depends on the Fourier spectra of the initial data. The trajectories stay close if the higher modes decay sufficiently fast. If the initial data are irregular and there are relatively high modes, then the convergence of the trajectories becomes non-uniform. Namely, the boundary layer is formed and there exist small moments of time such that the difference of the solutions reaches in the mean a finite value as the coefficient of the highest time derivative tends to zero. These results reflect the difficulties that may arise in the analysis of the systems of non-linear quasi-gasdynamic equations.