Abstract

Many physical models give rise to the need to solve partial differential equations in time dependent regions. The complex morphology of biological membranes and cells coupled with biophysical mathematical models present significant computational challenges as evidenced within the Newton Institute programme “Coupling Geometric PDEs with Physics for Cell Morphology, Motility and Pattern Formation”. In this talk we discuss the mathematical issues associated with the formulation of PDEs in time dependent domains in both flat and curved space. Here we are thinking of problems posed on time dependent d-dimensional hypersurfaces Gamma(t) in R^{. The surface Gamma(t) may be the boundary of the bounded open bulk region Omega(t). In this setting we may also view Omega(t) as (d+1)dimensional sub-manifold in R}{d+2}. Using this observation we may develop a discretisation theory applicable to both surface and bulk equations. We will present an abstract framework for treating the theory of well posedness of solutions to abstract parabolic partial differential equations on evolving Hilbert spaces using generalised Bochner spaces. This theory is applicable to variational formulations of PDEs on evolving spatial domains including moving hyper-surfaces. We formulate an appropriate time derivative on evolving spaces called the material derivative and define a weak material derivative in analogy with the usual time derivative in fixed domain problems; our setting is abstract and not restricted to evolving domains or surfaces. Then we show well-posedness to a certain class of parabolic PDEs under some assumptions on the parabolic operator and the data. Specifically, we study in turn a surface heat equation, an equation posed on a bulk domain, a novel coupled bulk-surface system and an equation with a dynamic boundary condition. We give some background to applications in cell biology. We describe how the theory may be used in the development and numerical analysis of evolving surface finite element spaces which unifies the discrtetisation methodology for evolving surface and bulk equations. In order to have good discretisation one needs good meshes. We will indicate how geometric PDEs may be used to compute high quality meshes. We give some computational examples from cell biology involving the coupling of surface evolution to processes on the surface.