This paper presents a mathematical model development for description of the two-layer stationary liquid and gas flows with evaporation. The system is restricted by rigid impenetrable borders and is subjected to longitudinal gradients of temperature. The modeling is performed on the basis of the Navier-Stokes equations in the Oberbeck-Boussinesq approximation with Soret and Dufour effects in the upper gas-vapor layer. These effects are also taken into account when determining heat and mass transfer conditions at the thermocapillary interface. Kinematic and dynamic conditions are also assumed to be fulfilled on the interface. Concentrations of the saturated vapor are defined from the Clapeyron-Clausius equation. The gas flow rate in the upper layer is considered a given value. Exact solutions of equations that take into account the influence of longitudinal temperature gradients, gravity, concentration and temperature effects on the structure of the flows, and evaporation in the system are elaborated.