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It is shown that surface tension effects on the free boundary have a regularizing effect for Hele-Shaw models, which implies existence and uniqueness of classical solutions for general initial domains.

Let X be a Banach space and let A be a closed linear operator on X. It is shown that the abstract Cauchy probleṁif and only if it has the property of maximal L p -regularity. Moreover, it is also shown that the derivation operator D = d/dt admits an H ∞ -calculus in weighted L p -spaces.Introduction. Let X be a Banach space and let A be a closed linear operator on X with domain D(A). We consider the abstract Cauchy probleṁ

Abstract. We show existence and uniqueness of classical solutions for the motion of immersed hypersurfaces driven by surface diffusion. If the initial surface is embedded and close to a sphere, we prove that the solution exists globally and converges exponentially fast to a sphere. Furthermore, we provide numerical simulations showing the creation of singularities for immersed curves.

The Mullins Sekerka model is a nonlocal evolution model for hypersurfaces, which arises as a singular limit for the Cahn Hilliard equation. We show that classical solutions exist globally and tend to spheres exponentially fast, provided that they are close to a sphere initially. Our analysis is based on center manifold theory and on maximal regularity.1998 Academic Press

We show convergence of solutions to equilibria for quasilinear parabolic evolution equations in situations where the set of equilibria is non-discrete, but forms a finite-dimensional C 1 -manifold which is normally hyperbolic. Our results do not depend on the presence of an appropriate Lyapunov functional as in the Łojasiewicz-Simon approach, but are of local nature.

Abstract. By means of a center manifold analysis we investigate the averaged mean curvature flow near spheres. In particular, we show that there exist global solutions to this flow starting from non-convex initial hypersurfaces.

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