The paper is dedicated to the one-dimensional problem of the nonisothermal flow of a two-phase mixture of viscous incompressible fluids with inhomogeneous boundary conditions. The mathematical model describing the two viscous fluids mixture flow is based on the equations of mass conservation, momentum conservation for each phase, and on the energy conservation equation, in the large. Local in time solvability of the initial boundary value problem in S.L.Sobolev and Helder spaces is proved.

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... on 1 sets the problem set up and provides the short literature review on the topic close papers and the main result formulation. Section 2 explains the transformation of the original system of equations. Sections 3, 4 prove the existence of the strong and classic solutions on a small time interval with constant true density using the Bubnov-Galerkin method. Notionally, the proof of the theorem is based on the similar result proof for viscous heatconducting gas (Antonsev S.N., Kazhihov A.V., Monahov V.N. Boundary value problems of heterogeneous fluid mechanic). The particularity of the considered problem is the presence of inhomogeneous boundary conditions.