Two-dimensional Free-boundary Problem Research Articles

Bounds for steady water waves with vorticity

The two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. Bounds on the free-surface profiles and on the total head are obtained under minimal assumptions about properties of solutions to the problem and the vorticity distribution.

Nonlinear stability analysis of a two-dimensional diffusive free boundary problem

We explore global existence and stability of planar solutions to a multi-dimensional Case II polymer diffusion model which takes the form of a one-phase free boundary problem with phase onset. Due to a particular boundary condition, convergence cannot be expected on the whole domain. A boundary integral formulation derived in [13] is shown to remain valid in the present context and allows us to circumvent this difficulty by restricting the analysis to the free boundary. The integral operators arising in the boundary integral formulation are analyzed by methods of pseudodifferential calculus. This is possible as explicit symbols are available for the relevant kernels. Spectral analysis of the linearization can then be combined with a known principle of linearized stability [12] to obtain local exponential stability of planar solutions with respect to two-dimensional perturbations.

Mean Curvature Motion of Triple Junctions of Graphs in Two Dimensions

We consider a system of three surfaces, graphs over a bounded domain in ℝ2, intersecting along a time-dependent curve and moving by mean curvature while preserving the pairwise angles at the curve of intersection (equal to 2π/3.) For the corresponding two-dimensional parabolic free boundary problem we prove short-time existence of classical solutions (in parabolic Hölder spaces), for sufficiently regular initial data satisfying a compatibility condition.

Boundary dynamics of a two-dimensional diffusive free boundary problem

Numerous models of industrial processes such as diffusion in glassy polymers or solidification phenomena, lead to general one-phase free boundary value problems with phase onset. In this paper we develop a framework viable to prove global existence and stability of planar solutions to one such multi-dimensional model whose application is in controlled-release pharmaceuticals. We utilize a boundary integral reformulation to allow for the use of maximal regularity. To this effect, we view the operators as pseudo-differential and exploit knowledge of the relevant symbols. Within this framework, we give a local existence and continuous dependence result necessary to prove planar solutions are locally exponentially stable with respect to two-dimensional perturbations.

Fundamental bounds for steady water waves

The two-dimensional free-boundary problem of steady gravity waves on water of finite depth is considered. Bounds on the free-surface profiles and on the values of Bernoulli’s constant are obtained under minimal assumptions about properties of solutions to the problem.

On the continuity of the free boundary in the problem Δ p u = − ( h ( x , y ) χ ( u )) x

We consider a two-dimensional free boundary problem involving the p-Laplacian. We show that the free boundary is the graph x = φ(y) of a continuous function.

On a class of free boundary problems of type $div(a(X) \nabla u) = -div(\chi (u)H(X))$

We consider a class of two-dimensional free boundary problems of type $div(a(X) \nabla u) = -div(\chi (u)H(X))$, where $H$ is a Lipschitz vector function satisfying $div(H(X))\geq 0$. We prove that the free boundary $\partial [u>0] \cap\Omega$ is represented locally by a family of continuous functions.

On the continuity of the free boundary in problems of type [formula omitted

We consider a class of two-dimensional free boundary problems including the heterogeneous dam, lubrication and aluminium electrolysis problems. We prove the Lipschitz continuity of the solution and the continuity of the free boundary.

Application of Green's differential equation to the analysis of ion-matrix sheaths around wedge-shaped cathodes

A relation between the gradient of the electric field and mean curvature of equipotential surfaces (Green's differential equation) is applied to a two-dimensional free-boundary problem arising in the study of ion sheaths around wedge-shaped cathodes. With the assumption that the equipotential lines are hyperbolae, this relation leads to a nonlinear ordinary differential equation for the potential along the bisector line of the wedge. An approximate solution is found, which yields, in particular, the sheath width along this line as a function of the wedge angle. The resulting values are in good agreement with published results obtained by numerically solving Poisson's equation.

Investigation Of The Motion Of A Fixed RotatingSolid Body Over Ice With A Thin Fluid Interface

This article contains a theoretical investigation of the frictional forces on a solid body, such as a curling stone, rotating about a fixed axis, where the retarding force is due to the viscous drag of a thin water layer overlying a thick ice layer. Most introductory physics texts state that this deceleration is due to frictional surface forces, with a constant coefficient of sliding friction. Here, we proceed under the assumption that a fluid boundary layer is formed between the surface of the solid body and the ice as a result of pressure melting. In addition, the interface between the water and ice phases is considered to be a Stefan problem. The resulting fluid layer is responsible for the viscous drag forces that slow and eventually stop the rotating solid body. As a special case, a similar problem obtained by neglecting curvature is examined where the problem may be expressed in Cartesian coordinates rather than radial coordinates. Thus instead of a rotating curling rock, the physical problem is represented by a finite skate blade moving in a straight line over an ice surface with a fluidized interface. To calculate the drag forces experienced by the skate blade we first determine the temperature distribution of the water/ice interface, which requires the solution of a two-dimensional free boundary problem. A simplified problem is considered where the depth of the boundary layer is assumed to be independent of the horizontal position, and thereby we find expressions for the temperature of the water and ice in addition to an expression for the position of the ice water interface. Upon obtaining these results we can calculate the approximate drag experienced by a finite skate blade. At this point we can perform a similar analysis in radial coordinates providing a first approximation of the viscous drag forces of a rotating curling stone.

Univalent Functions in Two-Dimensional Free Boundary Problems

The main goal of the paper is to bring together methods of the classical theory of univalent functions and some problems of fluid mechanics. Our interest centers on free boundary problems. We study the time evolution of the free boundary of a viscous fluid in the zero- and nonzero-surface-tension models for planar flows in Hele-Shaw cells either with an extending to infinity free boundary or with a bounded free boundary. We consider special classes of univalent functions that admit an explicit geometric interpretation to characterize the shape of the free interface. Another model is two-dimensional solidification/melting of a nucleus in a forced flow.

On the Existence of a Weak Solution to a Two-Dimensional Free-Boundary Problem with a Nonlinear Flux Condition

The main result of this paper is a global existence theorem for a two-dimensional free-boundary problem with a nonlinear boundary condition in suitable Sobolev spaces. The existence result is proved by using some a priori estimates and the Schauder fixed point theorem.

On a Class of Geometry-Driven Free Boundary Problems

An abstract class of two-dimensional geometry-driven free boundary problems is considered in which the evolution of the Cauchy transform of the time-evolving domain satisfies a partial differential equation of conservation type. This idea generalizes an association between free boundary problems and the inverse gravitational problem. It is shown that this class of free boundary problems admits exact solutions expressible in terms of a finite set of time-evolving parameters. In addition, even when exact solutions are not available, a theorem is established which shows that solutions corresponding to certain initial conditions possess nontrivial conserved quantities. Two distinct problems of physical interest are shown to fall within this abstract class: the evolution of a fluid blob in a rotating Hele--Shaw cell and the evolution of highly viscous fluid under the effects of surface tension. Various aspects of these two problems are discussed.

FROM THE IMPLIED VOLATILITY SKEW TO A ROBUST CORRECTION TO BLACK-SCHOLES AMERICAN OPTION PRICES

We describe a robust correction to Black-Scholes American derivatives prices that accounts for uncertain and changing market volatility. It exploits the tendency of volatility to cluster, or fast mean-reversion, and is simply calibrated from the observed implied volatility skew. The two-dimensional free-boundary problem for the derivative pricing function under a stochastic volatility model is reduced to a one-dimensional free-boundary problem (the Black-Scholes price) plus the solution of a fixed boundary-value problem. The formal asymptotic calculation that achieves this is presented here. We discuss numerical implementation and analyze the effect of the volatility skew.

Instabilities in a Two-Dimensional Combustion Model with Free Boundary

We prove instability of the planar travelling wave solution in a two-dimensional free boundary problem modelling the propagation of near- equidiffusional premixed flames in the whole plane. We reduce the problem to a fixed boundary fully nonlinear parabolic system. The spectrum of the linearized operator contains an interval [0,ω c ], ω c > 0, so we cannot construct backward solutions. We use an argument about stability of dynamical systems in Banach spaces in order to prove pointwise instability of the moving front.

Numerical solution of a nonlocal problem arising in plasma physics

In this paper, we present a numerical method to approximate the equilibrium of a plasma magnetically confined in a stellarator device. We consider a two-dimensional nonlocal free-boundary problem which involves both relative and decreasing rearrangements. We use a finite-element discretization to the differential operator and an iterative algorithm in the nonlinear terms. An approximation scheme for the computation of the rearrangement has been implemented and tested. Finally we give numerical results for an helical system.

On a nonlocal elliptic problem arising in the magnetic confinement of a plasma in a stellarator

This paper is a review of recent results (including the author's work) on a two-dimensional free-boundary problem, modeling magnetohydrodynamic equilibrium in a stellarator nuclear fusion device. The main tools used in the paper under review are decreasing rearrangement (applied to the magnetic flux) and relative rearrangement (with respect to the magnetic flux)

Solvability of some two-dimensional quasistationary free boundary problems for the Navier-Stokes equations with moving contact points

The paper is concerned with some quasistationary two-dimensional free boundary problems of viscous flow with moving contact points and with contact angle equal to π. A typical example of such a flow is filling a capillary tube in the presence of surface tension. The proof of the solvability of these problems is based on the analysis (made by the author and V. V. Pukhnachëv about 10 years ago) of the asymptotic formulas for the solutions of the Navier-Stokes equations in a neigborhood of contact points. Bibliography: 10 titles.

A note on the classical solution of a two-dimensional superconductor free boundary problem

In this paper we consider a superconductor free boundary problem. Under isothermal conditions, a superconductor material (of ‘type I’) will develop two phases separated by a sharp interface Γ(t). In the ‘normal’ conducting phase the magnetic field is divergence free and satisfies the heat equation, whereas on the interface Γ(t), curl , where n is the normal and Vn is the velocity of Γ(t) in the direction of n; further, (constant) on Γ(t). Existence and uniqueness of a classical solution locally in time are established by Newton's iteration method under assumptions which enable us to reduce the 3-dimensional problem to a problem depending on essentially two space variables.

Approximate method for a two-dimensional parabolic free boundary problem

An approximate method is used to obtain the numerical solution of a two-dimensional parabolic free boundary problem arising from the filtration of a compressible fluid through a porous medium. A numerical scheme is presented and numerical results are given.