Time-periodic solutions of advection-diffusion equations on moving hypersurfaces

In this paper we study time-periodic solutions to advection-diffusion equations of a scalar quantity $u$ on a periodically moving $n$-dimensional hypersurface $\Gamma(t) \subset \mathbb{R}^{n+1}$. We prove existence and uniqueness of solutions in suitable Hölder spaces

Parabolic PDEs on evolving spaces

This thesis is concerned with the well-posedness of solutions to certain linear and nonlinear parabolic PDEs on evolving spaces. We first present an abstract framework for the formulation and well-posedness of linear parabolic PDEs on abstract evolving Hilbert spaces. We introduce new function spaces and a notion of a weak time derivative called the weak material derivative for this purpose. We apply this general theory to moving hypersurfaces and Sobolev spaces and study four different linear problems including a coupled bulk-surface system and a dynamical boundary problem. Then we formulate a Stefan problem itself on an evolving surface and consider weak solutions given integrable data through the enthalpy approach, using a generalisation to the Banach space setting of the function spaces introduced in the abstract framework. We finish by studying a nonlocal problem: a porous medium equation with a fractional diffusion posed on an evolving surface and we prove well-posedness for bounded initial data