On an adjoint initial-boundary value problem

We study an adjoint initial-boundary value problem for linear parabolic equations; which arises when modeling the unidirectional motion of two viscous fluids with a common interface under the action of a pressure gradient. Under some conditions on the pressure gradient, we obtain a priori estimates and show that the solution enters a stationary mode. For semibounded layers, we find the solution in closed form and indicate the case of a self-similar solution. We determine the volume flow rates in the layers.