Abstract-We study a partially invariant solution of rank 2 and defect 3 of the equations of a viscid heat-conducting liquid. It is interpreted as a two-dimensional motion of three immiscible liquids in a flat channel bounded by fixed solid walls, the temperature distribution on which is known. From a mathematical point of view, the resulting initial-boundary value problem is a nonlinear inverse problem. Under some assumptions (often valid in practical applications), the problem can be replaced by a linear problem. For the latter we obtain some a priori estimates, find an exact steady solution, and prove that the solution approaches the steady regime as time increases, provided that the temperature on the walls stabilizes. DOI: 10.1134/S1990478916010026Keywords: thermocapillarity, a priori estimates, conjugate initial-boundary value problem, asymptotic behaviorAs is known, motion begins in a nonuniformly heated liquid. In applications, we often encounter the situations when motion is originated in the two or more liquid media that contact one another along certain interfaces. If the liquids do not mix during their interaction then they form some more or less visible interfaces. The oil-water system is a typical example of this situation. The need for modeling multiphase flows, taking into account the differences in physical and chemical factors, arises in the design of cooling systems and power plants, in studying the growth of crystals and films, or in the aerospace industry [1][2][3][4].Some exact solutions of the equations of Marangoni convection are known [5][6][7]. One of the first was obtained in [8] which is a steady Poiseuille flow of two immiscible liquids in an oblique channel. As a rule, almost all flows were steady and unidirectional. The stability of these flows was investigated in [9,10]. As far as the nonsteady thermocapillary flows are concerned, their study have begun rather recently [11,12].The problem of thermocapillary convection of two incompressible liquids in a container separated by a closed separation surface was studied in [13]. The time-local unique solvability of the problem was obtained in the H¨older classes of functions. The problem of thermocapillary motion of a drop in the entire three-dimensional space was studied in [14]. Moreover, its unique solvability was established in the H¨older classes with a power-like weight. It turned out that the velocity vector field decreases at infinity in same manner as the initial data and mass forces, whereas the temperature tends to a constant equal to the limit of the initial temperature at infinity.This article is devoted to the study of solutions of the two-dimensional conjugate boundary value problem which results from the linearization of the Navier-Stokes system of equations supplemented with the heat transfer equation. Motion is initiated by thermocapillary forces applied along two interfaces which cause nonsteady Marangoni convection. Such a convection can prevail under conditions of microgravity or in the movement of thin liquid films....