Asymptotic Behavior of the Solution to the Initial-boundary Value Problem for One-dimensional Motions of a Barotropic Compressible Viscous Multifluid

Abstract

An initial-boundary value problem is considered for one-dimensional barotropic equations of compressible viscous multicomponent media, which are a generalization of the Navier–Stokes equations of the dynamics of a single-component compressible viscous fluid. In the equations under consideration, higher order derivatives of the velocities of all components are present due to the composite structure of the viscous stress tensors. Unlike the single-component case in which the viscosities are scalars, in the multicomponent case they form a matrix whose entries describe viscous friction. Diagonal entries describe viscous friction within each component, and non-diagonal entries describe friction between the components. This fact does not allow to automatically transfer the known results for the Navier–Stokes equations to the multicomponent case. In the case of a diagonal viscosity matrix, the momentum equations are possibly connected via the lower order terms only. In the paper the more complicated case of an off-diagonal (filled) viscosity matrix is under consideration. The stabilization of the solution to the initial-boundary value problem with unbounded time increase is proved without simplifying assumptions on the structure of the viscosity matrix, except for the standard physical requirements of symmetry and positive definiteness.