Karolina Pawlak
supervisor: Ewa Zadrzyńska-Piętka, Adam Kubica
Stefan problem is one of the most famous free boundary problem of evolution type. It describes the process of melting of solid. In general, at the beginning the domain [0,R] x [0,T], where R,T are positive, is divided into the solid and liquid part. By the left boundary we provide the heat, which makes the ice melt. As the result of melting of solid, the interphase between solid and liquid part is moving. In the classical formulation, there are two unknowns: the temperature and the function describing the free boundary.
In my work, instead of considering the equation separately in the solid and liquid part and looking for the free boundary, we consider the problem in the whole domain. We have only one unknown: the enthalpy function, which encode the information in which state the material is. Enthalpy function is not continuous, because it has the jump at the critical temperature. As the result, we need consider weak solutions, which are more general than classical one.
We prove the existence and uniqueness of weak solutions of the fractional Stefan problem with the distributed order Caputo derivative. We consider the one phase problem, so we assume that temperature = max{0, enthalpy – 1} and our problem is nonlinear. Our result holds only on a sufficiently small interval of time. In order to get the global existence, we will need to show that the solutions of evolution equations with distributed order fractional derivative are Hölder continuous.