Highly nonlinear evolutionary problems

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In the project “Highly nonlinear evolutionary problems”, we are mainly concerned with time dependent partial differential equations and minimizing properties. We will treat three different kinds of problems. First, we consider doubly nonlinear parabolic equations, which appear in the modelling of several physical phenomena as, for instance, in plasma physics or the analysis of turbulent filtration of a gas or a liquid through porous media. Those equations also find application in the characterization of ground water problems, heat radiation in plasmas, or the motion of viscous fluids. One main goal of this project is to investigate a self-improving property of solutions of these equations by using the method of Expansion of Positivity.In the last years, the interest in problems that are related with partial differential equations formulated in domains that change in time grew. This is partly due to the fact that a number of problems in mathematical biology are naturally posed on growing domains (e.g. developing organisms or proliferating cells) or domains that evolve in some particular way. There are also some classical engineering applications like fluids or gases in settings as channels or pipes with confining walls that may be displaced, removed or brought in at will. The aim is to show existence results for variational solutions to evolutionary problems that describe these phenomena. The approach will be based on DeGeorgi’s method of minimizing movements.Finally, we will consider functionals with an exponential growth rate. The purpose is to show that parabolic minimizers that are associated to such solutions are in some way smooth, assuming that the growth rate is not “too large”. To prove this, we will make use of a parabolic version of DeGeorgi classes and a variant of the Moser iteration.