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We establish that the quasihydrodynamic system of equations of motion of a perfect polytropic gas is parabolic (in the sense of Petrovskii). We study the stability of small perturbations on a constant background and, for the Cauchy problem and the initial boundary-value problems for the corresponding linearized system, we obtain uniform (on the infinite time interval) estimates of relative perturbations. The corresponding results are also derived in the barotropic case for a general equation of state.

1. An extensive literature is devoted to systems of quasilinear equations of motion of viscous compressible media (see [1, 2]). One of the problems of current interest is the development of the theory of global (with respect to time and data) weak solutions, including questions of their existence, uniqueness, and continuous dependence on data (such as initial data, the coefficients of the equations, the constant terms, etc.). For the system of equations of one-dimensional motion of a viscous heat-conducting gas, the last two questions were studied in [3–6] (see also [7, 8]). For more complicated systems of equations for reacting gases (in other words, in combustion problems) weak solutions were studied in [9, 10]. Quite recently, in [11], the existence of weak solutions was obtained for an even more complicated system of equations, namely, for a reacting binary gas mixture. Note that regular solutions of problems concerning binary mixtures were studied earlier in [12, 13]. The present paper is devoted to the questions of uniqueness of these weak solutions and of their continuous dependence on data. The system of quasilinear equations of one-dimensional motion of a viscous compressible reacting binary gas mixture is of the form

η(x, t) → η± ,v (x, t) → v± ,θ (x, t) → θ± as x →± ∞, 0

1. A. V. Babin and M. I. Vishik, Proc. Roy. Soc. Edinburgh. Sect. A, 116, No. 3-4, 221-243 (1990). 2. M. A. Efendiev and A. Miranville, Discrete Contin. Dynam. Systems, 5, No. 2, 399-424 (1999). 3. S. Merino, J. DifferentiM Equations, 132, 87-106 (1996). 4. H. Triebel, Interpolatlon Theory, _b'~nction Spaces, Differential Operators, Birkhs Berlin (1977). 5. A. N. Kolmogorov and V. M. Tikhomirov, Uspek.hi Mat. Nauk [Russian Math. Surveys], 14, No. 2, 3-86 (1959). 6. M. I. Vishik and V. V. Chepyzhov, Mat. Sb. [Russian Acad. ScJ. Sb. Math.], 189, No. 2, 81-110 (1998). 7. R. Temam, Intinite-dimensional Dynamical Systems in Mechanics and Physics, Springer, New York (1988).

Nonhomogeneous initial boundary value problems for a specific quasilinear system of equations of composite type are studied. The system describes the one-dimensional motion of a viscous perfect polytropic gas. We assume that the initial data belong to the spacesL ∞(Ω) orL 2(Ω) and the problems under consideration have generalized solutions only. For such solutions, a theorem on strong stability is proved, i.e., estimates for the norm of the difference of two solutions are expressed in terms of the sums of the norms of the differences of the corresponding data. Uniqueness of generalized solutions is a simple consequence of this theorem.

For the multidimensional heat equation in a parallelepiped, optimal error estimates inL2(Q) are derived. The error is of the order of τ+¦h¦2 for any right-hand sidef ∈L2(Q) and any initial function\(u_0 \in \mathop {W_2^1 }\limits^ \circ \left( \Omega \right)\); for appropriate classes of less regularf andu0, the error is of the order of ((τ+¦h¦2γ), 1/2≤γ

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