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ABSTRACT. In the present paper we study the qualitative behavior as t ---+ c~ of the solution of the Cauchy problem for a system of equations describing a dynamics of a two-component viscous fluid. The model under consideration takes into account the mutual diffusion of the fluid components as well as their capillary interaction. We describe the w-limit set of trajectories of the dynamical system generated by the problem. It is proved that the stationary solution of the problem, is a homogeneous stationary distribution of one of the components, is asymptotically stable. Any other stationary solution is not asymptotically stable and is even unstable if there are no close stationary solutions corresponding to a smaller energy level.KEY WORDS: fluid dynamics, capillary forces, stabilization problems, asymptotic stability Statement of the problemIt is known that the solvability of the problem of the dynamics of two immiscible fluids separated by a boundary possessing surface tension is proved in the classical statement only in the small. In this connection, the following model based on ideas due to Van ~, + ~ v~ = ~(-~ + w'(~))The model takes into account the capillary interaction of fluids as well as their mutual diffusion. The functions occurring in the equations have the following physical meaning: ~ is the velocity, p is the pressure, ~o is the concentration of one of the components, and W'(~) = dW/d~v, where the function W(qo) is the so-called double-well potential. In general, according to its physical meaning, the concentration cannot take values outside the interval [0, 1]; this can be achieved by a suitable choice of W. In this paper we neglect this condition and assume that qa may take any real value. We assume the following conditions on W: The function W(~v) = ~v2(~o -1) 2 satisfies all these conditions. To be definite, we assume that Eq. (3) contains precisely this function.In the paper, the Cauchy problem for system (1)- (3) is studied in the case of two space variables: --(xl, x2) 9 R 2 , ~ --(vl, v2) 9 R 2 9 Let us pose the initial conditions: ~(~, 0) = ~0(~), ~(~, 0) = ~0(~).

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