A priori estimates of the adjoint problem describing the slow flow of a binary mixture and a fluid in a channel

We obtain a priori estimates of the solution in the uniform metric for a linear conjugate initial-boundary inverse problem describing the joint motion of a binary mixture and a viscous heat-conducting liquid in a plane channel. With their help, it is established that the solution of the non-stationary problem with time growth tends to a stationary solution according to the exponential law when the temperature on the channel walls stabilizes with time.