Properties of the solution of the inverse adjoint boundary-value problem of thermal convection in a tube

A problem is posed on a joint unsteady unidirectional motion of two immiscible fluids in a cylindrical tube with a constant temperature difference on the solid surface of the tube. From the mathematical viewpoint, this is an adjoint and inverse problem with respect to the pressure gradient of one of the fluids along the tube. The condition of problem overdetermination is a specified unsteady total flow rate of both fluids. A steady solution is found. A priori estimates of the solution of the unsteady problem in a uniform metric are obtained. Based on these estimates, sufficient conditions for input data are formulated at which the steady solution is exponentially stable.