Error estimates for generalised non-linear Cahn-Hilliard equations on evolving surfaces

In this paper, we consider the Cahn--Hilliard equation on evolving surfaces with prescribed velocity and a general non-linear potential which has locally Lipschitz derivatives. High-order evolving surface finite elements are used to discretise the weak equation system in space, and a modified matrix--vector formulation for the semi-discrete problem is derived. The anti-symmetric structure of the equation system is preserved by the spatial discretisation. A new stability proof, based on this structure, combined with consistency bounds proves optimal-order and uniform-in-time error estimates. To demonstrate the advantages of the new stability analysis, an extension of the stability and convergence results is given for generalised non-linear Cahn--Hilliard-type equations on evolving surfaces. The paper is concluded by a variety of numerical experiments

Short Time Existence for Coupling of Scaled Mean Curvature Flow and Diffusion

We prove a short time existence result for a system consisting of a geometric evolution equation for a hypersurface and a parabolic equation on this evolving hypersurface. More precisely, we discuss a mean curvature flow scaled with a term that depends on a quantity defined on the surface coupled to a diffusion equation for that quantity. The proof is based on a splitting ansatz, solving both equations separately using linearization and a contraction argument. Our result is formulated for the case of immersed hypersurfaces and yields a uniform lower bound on the existence time that allows for small changes in the initial value of the height function.Comment: 40 page

Surface, Bulk, and Geometric Partial Differential Equations: Interfacial, stochastic, non-local and discrete structures

Partial differential equations in complex domains with free\linebreak boundaries and interfaces continue to be flourishing research areas at the interfaces between PDE theory, differential geometry, numerical analysis and applications. Main themes of the workshop have been PDEs on evolving domains, phase field approaches, interactions of bulk and surface PDEs, curvature driven evolution equations. Applications particular from biology, such as cell and cancer modelling and fluid as well solid mechanics have been subjects of the conference

Short time existence for coupling of scaled mean curvature flow and diffusion

We prove a short time existence result for a system consisting of a geometric evolution equation for a hypersurface and a parabolic equation on this evolving hypersurface. More precisely, we discuss a mean curvature flow scaled with a term that depends on a quantity defined on the surface coupled to a diffusion equation for that quantity. The proof is based on a splitting ansatz, solving both equations separately using linearization and a contraction argument. Our result is formulated for the case of immersed hypersurfaces and yields a uniform lower bound on the existence time that allows for small changes in the initial value of the height function