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

Doklady Mathematics. 2022. Vol. 106. No. 2. P. 308–314.

A. A. Zlotnik, Chetverushkin B. N.

We study the Cauchy problems аor 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 the solutions of all three systems are formulated, and estimates of order $O(\tau^{\alpha/2})$ are given for the difference between solutions of the original system and systems with perturbations for an initial function $\*w_0$ of smoothness $\alpha$ in the sense of $L^2(\mathbb{R}^n)$, $0<\alpha\leq 2$. For $\alpha=1/2$, a broad class of discontinuous $\*w_0$ 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.

Keywords: small parameter, linear systems of partial differential equations, estimates for the difference of solutions, quasi-gasdynamic systems of equations