Gieri Simonett

Classical solutions of multidimensional Hele-Shaw models

Existence and uniqueness of classical solutions for the multidimensional expanding Hele{Shaw problem are proved.

The surface diffusion flow for immersed hypersurfaces

We show existence and uniqueness of classical solutions for the motion of immersed hypersurfaces driven by surface diusion. 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.

Analytic solutions for a Stefan problem with Gibbs-Thomson correction

We provide existence of a unique smooth solution for a class of oneand two-phase Stefan problems with Gibbs-Thomson correction in arbitrary space dimensions. In addition, it is shown that the moving interface depends analytically on the temporal and spatial variables. Of crucial importance for the analysis is the property of maximal Lpregularity for the linearized problem, which is fully developed in this paper as well.

The volume preserving mean curvature flow near spheres

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.

ON THE TWO-PHASE NAVIER-STOKES EQUATIONS WITH SURFACE TENSION

The two-phase free boundary problem for the Navier-Stokes system is considered in a situation where the initial interface is close to a halfplane. By means of Lp-maximal regularity of the underlying linear problem we show local well-posedness of the problem, and prove that the solution, in particular the interface, becomes instantaneously real analytic. In this paper we consider a free boundary problem that describes the motion of two viscous incompressible capillary Newtonian fluids. The fluids are separated by an interface that is unknown and has to be determined as part of the problem. Let 1(0) ⊂ R n+1 (n ≥ 1) be a region occupied by a viscous incompressible fluid, fluid1, and let 2(0) be the complement of the closure of 1(0) in R n+1 , corre- sponding to the region occupied by a second incompressible viscous fluid, fluid2. We assume that the two fluids are immiscible. Let 0 be the hypersurface that bounds 1(0) (and hence also 2(0)) and let ( t) denote the position of 0 at time t. Thus, ( t) is a sharp interface which separates the fluids occupying the regions 1(t) and 2(t), respectively, where 2(t) := R n+1 \ 1(t). We denote the normal field on ( t), pointing from 1(t) into 2(t), by ν(t, � ). Moreover, we de- note by V (t, � ) and κ(t, � ) the normal velocity and the mean curvature of ( t) with respect to ν(t, � ), respectively. Here the curvature κ(x, t) is assumed to be negative when 1(t) is convex in a neighborhood of x ∈ ( t). The motion of the fluids is governed by the following system of equations for i = 1,2 :    

On convergence of solutions to equilibria for quasilinear parabolic problems

Abstract 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.

Analyticity of the interface in a free boundary problem

Of concern is a class of free boundary problems which arise, for instance, in connection with the flow of an incompressible fluid in porous media. More precisely, we consider the following situation: Let F0 denote a fixed, impermeable layer in a homogeneous and isotropic porous medium. We assume that some part of the region above F0 is occupied with an incompressible Newtonian fluid. In addition, we suppose that there is a sharp interface, Ff, separating the wet region I2f enclosed by F0 and Ff, respectively, from the dry part, i.e., we consider a saturated fluid-air flow. The fluid moves under the influence of gravity and we assume that the motion is governed according to Darcys law. The standard model encompassing this situation consists of an elliptic equation for a velocity potential, to be solved in a domain with a free boundary, and of an evolution equation for the free boundary. In order to give a concise mathematical description let us introduce the following class of admissible interfaces:

A numerical scheme for axisymmetric solutions of curvature-driven free boundary problems, with applications to the Willmore flow

We present a numerical scheme for axisymmetric solutions to curvature-driven moving boundary problems governed by a local law of motion, e.g. the mean curvature flow, the surface diffusion flow, and the Willmore flow. We then present several numerical experiments for the Willmore flow. In particular, we provide numerical evidence that the Willmore flow can develop singularities in finite time.

Maximal regularity for a free boundary problem

This paper is concerned with the motion of an incompressible fluid in a rigid porous medium of infinite extent. The fluid is bounded below by a fixed, impermeable layer and above by a free surface moving under the influence of gravity. The laminar flow is governed by Darcys law.We prove existence of a unique maximal classical solution, using methods from the theory of maximal regularity, analytic semigroups, and Fourier multipliers. Moreover, we describe a state space which can be considered as domain of parabolicity for the problem under consideration.

Qualitative Behavior of Solutions for Thermodynamically Consistent Stefan Problems with Surface Tension

We study the qualitative behavior of a thermodynamically consistent two-phase Stefan problem with surface tension and with or without kinetic undercooling. It is shown that these problems generate local semiflows in well-defined state manifolds. If a solution does not exhibit singularities in a sense made precise herein, it is proved that it exists globally in time and its orbit is relatively compact. In addition, stability and instability of equilibria are studied. In particular, it is shown that multiple spheres of the same radius are unstable, reminiscent of the onset of Ostwald ripening.