G Georg Prokert

Existence results for Hele–Shaw flow driven by surface tension

This paper addresses short-time existence and uniqueness of a solution to the N-dimensional Hele–Shaw flow problem with surface tension as driving mechanism. Global existence in time and exponential decay of the solution near equilibrium are also proved. The results are obtained in Sobolev spaces Hs with sufficiently large s. The main tools are perturbations of a fixed reference domain, linearization with respect to these perturbations, a quasilinearization argument based on a geometric invariance property, and a priori energy estimates.

Existence results for the quasistationary motion of a free capillary liquid drop

We consider instationary creeping flow of a viscous liquid drop with free boundary driven by surface tension. This yields a nonlocal surface motion law involving the solution of the Stokes equations with Neumann boundary conditions given by the curvature of the boundary. The surface motion law is locally reformulated as a fully nonlinear parabolic (pseudodifferential) equation on a smooth manifold. Using analytic expansions, invariance properties, and a priori estimates we give, under suitable presumptions, a short-time existence and uniqueness proof for the solution of this equation in Sobolev spaces of sufficiently high order. Moreover, it is shown that if the initial shape of the drop is near the ball then the evolution problem has a solution for all positive times which exponentially decays to the ball.

ON A HELE-SHAW-TYPE DOMAIN EVOLUTION WITH CONVECTED SURFACE ENERGY DENSITY ∗

Interest is directed to a moving boundary problem with a gradient flow structure which generalizes surface tension-driven Hele-Shaw flow to the case of nonconstant surface tension coefficient taken along with the liquid particles at the boundary. In the case with kinetic undercooling regularization, well-posedness of the resulting evolution problem in Sobolev scales is proved, including cases in which the surface tension coefficient degenerates. The problem is reformulated as a vector- valued, degenerate parabolic Cauchy problem. To solve this, we prove and apply an abstract result on Galerkin approximations with variable bilinear forms.

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:

A moving boundary problem for the Stokes equations involving osmosis:Variational modelling and short-time well-posedness

Within the framework of variational modelling we derive a one-phase moving boundary problem describing the motion of a semipermeable membrane enclosing a viscous liquid, driven by osmotic pressure and surface tension of the membrane. For this problem we prove the existence of classical solutions for a short-time.

On a Hele-Shaw type domain evolution with convected surface energy density : the third-order problem

We investigate a moving boundary problem with a gradient flow structure which generalizes Hele-Shaw flow driven solely by surface tension to the case of nonconstant surface ten- sion coefficient taken along with the liquid particles at boundary. The resulting evolution problem is first order in time, contains a third-order nonlinear pseudodifferential operator and is degenerate parabolic. Well-posedness of this problem in Sobolev scales is proved. The main tool is the con- struction of a variable symmetric bilinear form so that the third-order operator is semibounded with respect to it. Moreover, we show global existence and convergence to an equilibrium for solutions near trivial equilibria (balls with constant surface tension coefficient). Finally, numerical examples in 2D and 3D are given.

Hyperbolic evolution equations for moving boundary problems

Short-time existence and uniqueness results in Sobolev spaces are proved for Hele-Shaw flow with kinetic undercooling and for Stokes flow without surface tension. In both cases, the flow is driven by arbitrarily distributed sources and sinks in the interior of the liquid domain. The proofs are based on a general approach consisting of the reformulation of the problem as a Cauchy problem for a nonlinear, nonlocal evolution equation on the unit sphere, quasilinearization by equivariance, investigation of the linearization, and Galerkin approximations. In the situation discussed here, the linearized evolution operator is a first-order differential operator, and thus the evolution equation is of hyperbolic type. Finally, a brief survey of the properties of the evolution equations that arise from Hele-Shaw flow and Stokes flow with and without regularization is given.

On Stokes flow driven by surface tension in the presence of a surfactant

We consider short-time existence, uniqueness, and regularity for a moving boundary problem describing Stokes flow of a free liquid drop driven by surface tension. The surface tension coefficient is assumed to be a nonincreasing function of the surfactant concentration, and the surfactant is insoluble and moves by convection along the boundary. The problem is reformulated as a fully nonlinear, nonlocal Cauchy problem for a vector-valued function on a fixed reference manifold. This problem is, in general, degenerate parabolic. Existence and uniqueness results are obtained via energy estimates in Sobolev spaces of sufficiently high order. In the two-dimensional case, the problem is strictly parabolic, and we prove instantaneous smoothing of the free boundary, using maximal regularity results in little Holder spaces.

A New Model for Fungal Hyphae Growth Using the Thin Viscous Sheet Equations

In this paper, we model the growth of single nonbranching fungal hypha cell. The growth proceeds as an elongating expansion in a single direction. Modelling of hyphae growth consists out of two parts: transport of cell wall building material to the cell wall and growth of the cell wall as new cell wall building material arrives. In this paper we present a new model for hyphae growth using the work of Barnicki-Garcia et al. (1989), which assumes that cell wall building material is transported in straight lines by an isotropic point source, and the work of Campas and Mahadevan (2009), which assumes that the cell wall is a thin viscous sheet. Furthermore, we include a novel equation which models the hardening of the cell wall with age. We show numerically that these governing equations have solutions corresponding to hyphae growth. We also compute asymptotic expansions near the apex and the base of the cell.

Stability of equilibria of a two-phase Stokes-osmosis problem

Within the framework of variational modelling we derive a two-phase moving boundary problem that describes the motion of a semipermeable membrane separating two viscous liquids in a fixed container. The model includes the effects of osmotic pressure and surface tension of the membrane. For this problem we prove that the manifold of steady states is locally exponentially attractive.