Stabilization of the Solution to the Initial-Boundary Value Problem for One-Dimensional Isothermal Equations of Viscous Compressible Multicomponent Media

Abstract

The paper considers the initial-boundary value problem for one-dimensional isothermal equations of viscous compressible multicomponent media, commonly known as a generalization of the Navier — Stokes equations. The studied equations include higher derivatives of the velocities of all components, unlike the Navier — Stokes equations, where viscosity is a scalar variable. Viscosity forms a matrix of elements responsible for viscous friction due to the composite structure of viscous stress tensors for the multicomponent case. Diagonal elements of the matrix stand for viscous friction within each component, and off-diagonal elements stand for friction between components. Such complication does not allow the automatic extension of the known results for the Navier — Stokes equations to the multicomponent case. Thus, for the diagonal matrix, the equations would be linked only through the lower terms. The paper considers a more complex case of an off-diagonal viscosity matrix. We prove the stabilization of the solution of the initial-boundary value problem with an unlimited increase in time without simplifying assumptions about the structure of the viscosity matrix, except for the standard physical requirements of symmetry and positive definiteness.