Physical laws are mathematically encoded into Partial Differential Equations (PDEs). They tell us how certain quantities – like heat, water, or cars – depend on position and time. Precise information on the fundamental processes of the natural world is based to a large extent on PDEs; in turn, these processes will hint at solutions to mathematical problems. The project will study fine properties of irregular solutions of certain PDEs. Its research will seek to answer if initially irregular solutions become regular after some time, and if the PDEs are well-posed for growing (large) initial data. It will also investigate how solutions behave in the most quantitative way, by using explicit barriers or by understanding the long-time behaviour of the PDEs.