On Evolution Equations for Moving Domains

We introduce a general framework for the description of the autonomous motion of closed surfaces which are diffeomorphic images of spheres. The governing surface motion laws are in general nonlocal and lead therefore to nonlocal evolution equations for a perturbation function on a fixed reference domain. Although this evolution equation is fully nonlinear, the invariance of the problem with respect to a certain class of reparametrizations and the corresponding chain rule allow a quasilincarization of the evolution equation. Hence, as far as short-time existence and uniqueness of the solution and stability of equilibria are concerned, the analysis of the problems is reduced to the study of their linearizations and the application of known techniques for quasilinear Cauchy problems. Using a priori estimates and Ga.lerkin approximations in Sobolev spaces, both parabolic and first-order hyperbolic equations can be treated. In the case of parabolic problems, the smoothing property of the evolution can be proved. This general approach can be applied to a broad class of moving boundary problems. We will briefly discuss lIele-Shaw flow and Stokes flow driven by surface tension as well as classical Hele-Shaw flow with advancing liquid boundary as examples for parabolic evolutions: