Joachim Escher

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.

Symmetry of steady deep-water waves with vorticity

For a large class of vorticities we prove that a steady periodic deep-water wave must be symmetric if its profile is monotone between crests and troughs.

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.

Well-posedness, blow-up phenomena, and global solutions for the b-equation

Abstract In the paper we first establish the local well-posedness for a family of nonlinear dispersive equations, the so called b-equation. Then we describe the precise blow-up scenario. Moreover, we prove that for the b-equation we do have the coexistence of global in time solutions and blow-up phenomena: Depending on the initial data solutions may exist for ever, while other data force the solution to produce a singularity in finite time. Finally, we prove the uniqueness and existence of global weak solution to the equation provided the initial data satisfy certain sign conditions.

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:

Steady water waves with multiple critical layers

We construct small-amplitude periodic water waves with multiple critical layers. In addition to waves with arbitrarily many critical layers and a single crest in each period, multimodal waves with several crests and troughs in each period are found. The setting is that of steady two-dimensional finite-depth gravity water waves with vorticity.

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.

ON THE PARABOLICITY OF THE MUSKAT PROBLEM: WELL-POSEDNESS, FINGERING, AND STABILITY RESULTS

We consider in this paper the Muskat problem in a periodic geometry and incorporate capillary as well as gravity effects in the modelling. The problem re-writes as an abstract evolution equation and we use this property to prove well-posedness of the problem and to establish exponential stability of some flat equilibrium. Using bifurcation theory we also find finger shaped steady-states which are all unstable.