Solution to a linearized system of two-dimensional dynamics of viscous gas

A linear system of partial differential equations approximately describing the dynamics of small perturbations of a nonstationary viscous barotropic gas in a neighborhood of the steady state is considered in the paper. Analytic formulas for the solution are obtained for initial conditions of special type, the asymptotics of the rate of convergence to the stationary solution is studied. Similar assertions are proved for a finite-difference approximation of the original problem constructed on grids of V. I. Lebedev. In addition, the presence of analytical formulas for the solution allows us to explain why a perturbation of velocity jump type decreased significantly better than that for a pressure jump. The obtained results form a basis for studying the problem of asymptotic stabilization of solutions to two-dimensional equations of gas dynamics with dissipation terms.