Abstract: In this paper we consider an evolutionary continuous casting problem with convection
∂^tx+b∂_yx-△k(u)+v·▽u=0 coupled with Stokes equation in the liquid phase
∂_tv-ν△v+▽p=f(u) The mixed boundary condition is put on temprature. The existence of a weak solution is obtained by using methods of regularization, temperature dependent penalty form in the Stokes equation and compact arguments.

Key words: Boussinesq-Stefan problem, continuous extraction, nonstationary