On the strict modelling for Stefan problems with random convection and its temperature control

摘要

The paper considers the strict modelling of Stefan problems proposed by Rubinstein (1971) with random convection, and the temperature control problems for the proposed model. It is well known that the parabolic equation has an infinite thermal propagation speed. In order to avoid this physically unacceptable aspect, the heat conduction model of the hyperbolic type is derived from the physical point of view. First, taking the randomness in the velocity of the fluid by convection into consideration, the hyperbolic heat conduction model with random convection is proposed. Next, the free boundary problem for the proposed model is studied. It is shown that the considered free boundary problem is formulated by the stochastic variational inequality of a new type. The existence and uniqueness theorem of the solution to the stochastic variational inequality is given. Finally, the temperature control problem for the hyperbolic Stefan system with random convection is considered and a simple but very useful temperature control method is proposed.