Two Problems in non-linear PDE’s with Phase Transitions

This thesis is in the field of non-linear partial differential equations (PDE), and specifically focusing on problems which show some type of phase-transition. One of the problems deals with a multiple phase Hele-Shaw flow. A single phase Hele-Shaw flow models a Newtoninan fluid which is being injected in the space between two narrowly separated parallel planes. The time evolution of the space that the fluid occupies can be modelled by the PDE −∆u+χ{u>0} = tμ, where t is the time parameter and μ describes the injection of fluid. The nonregular term χ{u>0} in this problem is related to problems in the field of free boundary problems. In the multi-phase problem we generalize this situation and consider the mathematical modelling of the time-evolution of a system of multiple phases which interact according to the principle that the joint boundary which emerges when two phases meet is fixed for all future times. The problem is handled by introducing a parameterized equation where the parameter controls regularization and penalization. The penalization term is non-local in time and tracks the history of the system, penalizing the joint support of two different phases in space-time. In the first paper we give the detailed description of the mathematical model of the mulitple phase Hele-Shaw flow. The main result is the existence theory of a weak solution to the parameterized equations in a Bochner space using functional analytic tools such as the implicit function theorem in infinite dimensions. It is shown that the family of solutions to the parameterized problem is uniformly bounded in the considered Bochner space, allowing us to extract a weakly convergent subsequence for the case when the penalization tends to infinity. The limiting flow is defined as the multiple phase Hele-Shaw flow. The second problem deals with a parameterized highly oscillatory quasilinear elliptic equation in divergence form −div(Qη(x, x/ , u)∇u) = f together with Dirichlet data on a bounded domain Ω in R. As the regularization parameter η → 0, the equation gets a jump in the conductivity Qη induced by a term of the type χ{u>ψ }, where ψ is a locally -periodic obstacle. Hence the solution has two different phases. As the oscillation parameter → 0 the solution to the equation experiences high frequency jumps in the conductivity, resulting in the corresponding solutions showing an effective global behaviour. The global behavior is related to the so called homogenized solution. In the second paper we study the quasi-linear equation both qualitatively and quantitatively. We show that the parameterized equation has a weak solution in