Uniform estimates and stabilization of solutions to equations of one-dimensional motion of a multicomponent barotropic mixture

Abstract

Numerous papers deal with equations of motion of a barotropic gas (see [1]-[3] and the references therein). The asymptotic behavior as t --* +oo of the solutions as well as their existence and uniqueness, is of great interest, especially if (all or some of) the data are not assumed to be small. These topics were studied for the one-dimensional motion in [4-7] and in some other papers. Equations of motion of gas mixtures are more complicated and also play an important role. The unique solvability "in the large" was proved in [8] for the initial-boundary value problem for model equations describing a multicomponent mixture of viscous barotropic gases. The present paper contains some results concerning the asymptotic behavior of the solutions to a similar (but more general) system. These results have been obtained by combining the methods of [7] with the use of Lagrangian mass coordinates [8] for each component of the mixture. Note that attempts to solve this problem by using the methods in [4] have failed (even in the absence of mass forces). w In the half-strip H = (0, V) x R + , we study the system of quasilinear differential equations