On Second-Order Parabolic and Hyperbolic Perturbations of a First-Order Hyperbolic System

Abstract

We study the Cauchy problems for a first-order symmetric hyperbolic system of equations with variable coefficients and its singular perturbations that are second-order strongly parabolic and hyperbolic systems of equations with a small parameter $\tau > 0$ in front of the second derivatives with respect to $x$ and t. The properties of solutions of all three systems are formulated, and estimates of order $O({{\tau }^{{\alpha /2}}})$ are given for the difference between the solutions of the original system and systems with perturbations for an initial function w₀ of smoothness α in the sense of ${{L}^{2}}({{\mathbb{R}}^{n}})$, $0 < \alpha \leqslant 2$. For $\alpha = 1{\text{/}}2$, a broad class of discontinuous functions w₀ is covered. Applications to the linearized system of gas dynamics equations and to the linearized parabolic and hyperbolic second-order quasi-gasdynamic systems of equations are given.