Showing papers on "Free boundary problem published in 1999"
TL;DR: In this article, the rate and profile of blow-up of diffusion equations with nonlocal nonlinear reaction terms were investigated and sharp estimates on the size of boundary layer and on the asymptotic behavior of the solution in the boundary layer were derived.
215 citations
TL;DR: In this paper, the authors studied the initial boundary value problem for Einstein's vacuum field equation and showed that there exists a smooth solution to Ric[g] = 0 on a manifold which has (after an identification) a neighbourhood of S in T∪S as a boundary.
Abstract: We study the initial boundary value problem for Einstein's vacuum field equation. We prescribe initial data on an orientable, compact, 3-dimensional manifold S with boundary Σ≠? and boundary conditions on the manifold T= Re+
0×Σ. We assume the boundaries Σ and { 0 }×, Σ of S and T to be identified in the natural way. Furthermore, we prescribe certain gauge source functions which determine the evolution of the fields. Provided that all data are smooth and certain consistency conditions are met on Σ, we show that there exists a smooth solution to Einstein's equation Ric[g] = 0 on a manifold which has (after an identification) a neighbourhood of S in T∪S as a boundary. The solution is such that S is space-like, the initial data are induced by the solution on S, and, in the region of T where the solution is defined, T is time-like and the boundary conditions are satisfied.
206 citations
TL;DR: In this article, a wide class of boundary-value problems for which the application of elliptic theory can be reduced to elementary algebraic operations is described and characterized by the following polynomial property: the sesquilinear form corresponding to the problem degenerates only on some finite-dimensional linear space of vector polynomials.
Abstract: We describe a wide class of boundary-value problems for which the application of elliptic theory can be reduced to elementary algebraic operations and which is characterized by the following polynomial property: the sesquilinear form corresponding to the problem degenerates only on some finite-dimensional linear space of vector polynomials. Under this condition the boundary-value problem is elliptic, and its kernel and cokernel can be expressed in terms of . For domains with piecewise-smooth boundary or infinite ends (conic, cylindrical, or periodic), we also present fragments of asymptotic formulae for the solutions, give specific versions of general conditional theorems on the Fredholm property (in particular, by modifying the ordinary weighted norms), and compute the index of the operator corresponding to the boundary-value problem. The polynomial property is also helpful for asymptotic analysis of boundary-value problems in thin domains and junctions of such domains. Namely, simple manipulations with permit one to find the size of the system obtained by dimension reduction as well as the orders of the differential operators occurring in that system and provide complete information on the boundary layer structure. The results are illustrated by examples from elasticity and hydromechanics.
160 citations
TL;DR: In this article, the authors studied the asymptotic behavior of the solution to boundary-value problem for the second order elliptic equation in the bounded domain with a very rapidly oscillating locally periodic boundary and showed that the limiting problem can involve Dirichlet, Fourier or Neumann boundary conditions depending on the structure of the coefficient of the original problem.
120 citations
TL;DR: In this paper, the authors define the following albedo operator for a given pair (σα,A;) is admissible, i.e. (χ,υ') G X x V and σp(x, υ') := / v jfeίa:,!;',^)^.
Abstract: niΛ \\-υ'Vχf{x,v)-σa(x,v)f{x,v)+ ί k(x,υ f ,v)f(x,v')dv' =0 i n l x F , U -U \\ Jv { /|r_=/Here /_ is a given function on Γ_. We assume that the pair (σα,A;) is admissible, i.e. (i) 0<σa G L ° ° ( I x F ) , (ii) 0 < k(x,υ',-) G L(V) for a.e. (χ,υ') G X x V and σp(x,υ') := / v jfeίa:,!;',^)^ belongs to L°°(X x F). If the direct problem (1.1) is solvable, one can define the following albedo operator
119 citations
TL;DR: In this paper, the authors introduce a weak solution for a boundary value problem, where the motion of a rigid body immersed in a viscous fluid is not known a priori.
Abstract: We introduce a concept of weak solution for a boundary value problem modelling the motion of a rigid body immersed in a viscous fluid. The time variation of the fluid's domain (due to the motion of the rigid body) is not known a priori, so we deal with a free boundary value problem. Our main theorem asserts the existence of at least one weak solution for this problem. The result is global in time provided that the rigid body does not touch the boundary.
118 citations
111 citations
TL;DR: General absorbing boundary conditions will be developed for the Schrodinger equation with one spatial dimension, using group velocity considerations, and previously published absorbing boundary Conditions will be shown to reduce to special cases of this absorbing boundary condition.
Abstract: A large number of differential equation problems which admit traveling waves are usually defined on very large or infinite domains. To numerically solve these problems on smaller subdomains of the original domain, artificial boundary conditions must be defined for these subdomains. One type of artificial boundary condition which can minimize the size of such subdomains is the absorbing boundary condition. The imposition of absorbing boundary conditions is a technique used to reduce the necessary spatial domain when numerically solving partial differential equations that admit traveling waves. Such absorbing boundary conditions have been extensively studied in the context of hyperbolic wave equations. In this paper, general absorbing boundary conditions will be developed for the Schrodinger equation with one spatial dimension, using group velocity considerations. Previously published absorbing boundary conditions will be shown to reduce to special cases of this absorbing boundary condition. The well-posedness of the initial boundary value problem of the absorbing boundary condition, coupled to the interior Schrodinger equation, will also be discussed. Extension of the general absorbing boundary condition to higher spatial dimensions will be demonstrated. Numerical simulations using initial single Gaussian, double Gaussian, and a narrow Gaussian pulse distributions will be given, with comparision to exact solutions, to demonstrate the reflectivity properties of various orders of the absorbing boundary condition.
97 citations
TL;DR: In this article, a relaxation of this nonconvex constraint of a phase distribution with a sharp interface on a macroscopic scale is proposed and shown to predict shape and evolution of the mixing profile as a function of the density difference and mobility quotient.
Abstract: We study the flow of two immiscible fluids of different density and mobility in a porous medium. If the heavier phase lies above the lighter one, the interface is observed to be unstable. The two phases start to mix on a mesoscopic scale and the mixing zone grows in time—an example of evolution of microstructure. A simple set of assumptions on the physics of this two-phase flow in a porous medium leads to a mathematically ill-posed problem—when used to establish a continuum free boundary problem. We propose and motivate a relaxation of this “nonconvex” constraint of a phase distribution with a sharp interface on a macroscopic scale. We prove that this approach leads to a mathematically well-posed problem that predicts shape and evolution of the mixing profile as a function of the density difference and mobility quotient. © 1999 John Wiley & Sons, Inc.
87 citations
TL;DR: It is shown that the clean and modular separation of geometric information from the numerical procedures results in a solution technique that is essentially meshfree and allows an almost effortless modification of geometrical shapes, boundary conditions, and the governing equations.
Abstract: Spatial discretization of the domain and/or boundary conditions prevents application of many numerical techniques to physical problems with time-varying geometry and boundary conditions. By contrast, the R-functions method (RFM) for solving boundary and initial value problems discretizes not the domain but the underlying functional space, while the prescribed boundary conditions are satisfied exactly. The clean and modular separation of geometric information from the numerical procedures results in a solution technique that is essentially meshfree and allows an almost effortless modification of geometrical shapes, boundary conditions, and the governing equations. We show that these properties of the RFM make it highly suitable for automated modeling and simulation of non-stationary physical problems with time-varying geometries and boundary conditions.
84 citations
TL;DR: In this paper, a new class of generalized Atiyah-Patodi-Singer boundary value problems is introduced, motivated by the work of Vishik on the analytic torsion.
Abstract: Motivated by the work of Vishik on the analytic torsion we introduce a new class of generalized Atiyah-Patodi-Singer boundary value problems. We are able to derive a full heat expansion for this class of operators generalizing earlier work of Grubb and Seeley. As an application we give another proof of the gluing formula for the eta invariant. Our class of boundary conditions contains as special cases the usual (nonlocal) Atiyah-Patodi-Singer boundary value problems as well as the (local) relative and absolute boundary conditions for the Gauss-Bonnet operator.
TL;DR: In this article, the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane was studied and the uniqueness of solution was established.
Abstract: We consider the Dirichlet boundary–value problem for the Helmholtz equation in a non–locally perturbed half–plane. This problem models time–harmonic electromagnetic scattering by a one–dimensional infinite rough perfectly conducting surface; the same problem arises in acoustic scattering by a sound–soft surface. Chandler–Wilde and Zhang have suggested a radiation condition for this problem, a generalization of the Rayleigh expansion condition for diffraction gratings, and uniqueness of solution has been established. Recently, an integral equation formulation of the problem has also been proposed and, in the special case when the whole boundary is both Lyapunov and a small perturbation of a flat boundary, the unique solvability of this integral equation has been shown by Chandler–Wilde and Ross by operator perturbation arguments. In this paper we study the general case, with no limit on surface amplitudes or slopes, and show that the same integral equation has exactly one solution in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident fields including the incident plane wave, the Dirichlet boundary–value problem for the scattered field has a unique solution.
TL;DR: In this paper, the authors consider the free boundary problem for the evolution of a nearly straight slender fiber of viscous fluid and show that the shape of the cross-section can be described by means of a conformal map which depends on time and distance along the fibre axis.
Abstract: We consider the free boundary problem for the evolution of a nearly straight slender fibre of viscous fluid. The motion is driven by prescribing the velocity of the ends of the fibre, and the free surface evolves under the action of surface tension, inertia and gravity. The three-dimensional Navier-Stokes equations and free-surface boundary conditions are analysed asymptotically, using the fact that the inverse aspect ratio, defined to be the ratio between a typical fibre radius and the initial fibre length, is small. This first part of the paper follows earlier work on the stretching of a slender viscous fibre with negligible surface tension effects. The inclusion of surface tension seriously complicates the problem for the evolution of the shape of the cross-section. We adapt ideas applied previously to two-dimensional Stokes flow to show that the shape of the cross-section can be described by means of a conformal map which depends on time and distance along the fibre axis. We give some examples of suitable relevant maps and present numerical solutions of the resulting equations. We also use analytic methods to examine the coupling between stretching and the evolution of the cross-section shape.
TL;DR: In this article, the authors study the initial value problem for the system of compressible adiabatic flow through porous media in the one space dimension with fixed boundary condition and establish the global existence and large time behavior for the classical solutions via the combination of characteristic analysis and energy estimate methods.
Journal Article•
TL;DR: In this article, the authors considered a one-phase quasi-steady Stefan free boundary problem with surface tension, when the initial position of the free boundary is close to the unit sphere in R~ (v > 2), and expressed in the form r = It is proved that the problem has a unique global solution with free boundary which is analytic and converges exponentially fast.
Abstract: We consider a one-phase quasi-steady Stefan free boundary problem
with surface tension, when the initial position of the free boundary is close to the
unit sphere in R~ (v > 2), and expressed in the form r = It is proved
that the problem has a unique global solution with free boundary which is analytic
in E and which converges exponentially fast, as t -~ oo, to a sphere whose center
and radius can both be expressed as power series in c. The methods developed
here clearly extend to a general class of free boundary problems.
TL;DR: In this article, the full von Karman system accounting for in plane acceleration and thermal effects is considered and the main results are: (i) the wellposedness of regular and weak (finite energy) solutions, (ii) the uniform decay rates obtained for the energy function in the presence of boundary damping affecting only the velocity field representing in plane displacements of the plate.
Abstract: The full von Karman system accounting for in plane acceleration and thermal effects is considered. The main results of the paper are: (i) the wellposedness of regular and weak (finite energy) solutions, (ii) the uniform decay rates obtained for the energy function in the presence of boundary damping affecting only the velocity field representing in plane displacements of the plate. The key role in these results is played by: (i) new sharp regularity estimates for the boundary traces of elastic systems and (ii) newly established properties of analyticity of semigroups arising in thermoelastic systems with free boundary conditions.
TL;DR: In this paper, a regularized Newton iteration scheme based on a boundary integral equation approach for the initial Neumann boundary value problem for the heat equation is proposed for the inverse problem to determine the shape of an insulated inclusion within a heat conducting medium.
Abstract: We consider the inverse problem to determine the shape of an insulated inclusion within a heat conducting medium from overdetermined Cauchy data of solutions for the heat equation on the accessible exterior boundary of the medium. For the approximate solution of this ill-posed and nonlinear problem we propose a regularized Newton iteration scheme based on a boundary integral equation approach for the initial Neumann boundary value problem for the heat equation. For a foundation of the Newton method we establish the differentiability of the solution to the initial Neumann boundary value problem with respect to the interior boundary curve in the sense of a domain derivative and investigate the injectivity of the linearized mapping. Some numerical examples for the feasibility of the method are presented.
TL;DR: In this paper, the effect of relevant boundary terms on the nature of branes is discussed for toroidal and orbifold compactifications of the bosonic string and the results are subjected to T-duality tests and the special role of the points in moduli space fixed under T -duality is explained from least-action considerations.
TL;DR: In this paper, an inverse source problem for the Helmholtz equation in a bounded domain is considered, where the problem is to reconstruct the shape of the support of a source term from the Cauchy data on the boundary of the solution of the governing equation.
Abstract: We consider an inverse source problem for the Helmholtz equation in a bounded domain. The problem is to reconstruct the shape of the support of a source term from the Cauchy data on the boundary of the solution of the governing equation. We prove that if the shape is a polygon, one can calculate its support function from such data. An application to the inverse boundary value problem is also included.
TL;DR: In this paper, the fundamental solutions of the governing equations and their normal derivatives, which are required in the formulation of the MFS, can be expressed in terms of complete elliptic integrals.
Abstract: In this paper, we investigate the application of the Method of Fundamental Solutions (MFS) to two classes of axisymmetric potential problems. In the first, the boundary conditions as well as the domain of the problem, are axisymmetric, and in the second, the boundary conditions are arbitrary. In both cases, the fundamental solutions of the governing equations and their normal derivatives, which are required in the formulation of the MFS, can be expressed in terms of complete elliptic integrals. The method is tested on several axisymmetric problems from the literature and is also applied to an axisymmetric free boundary problem. Copyright © 1999 John Wiley & Sons, Ltd.
TL;DR: The dynamical behavior of the laser beam fusion cutting process of metals is investigated in this paper, where integral methods such as the variational formulation are applied to the partial differential equations for the free boundary problem and a finite dimensional approximation of the dynamical system is obtained.
Abstract: The dynamical behaviour of the laser beam fusion cutting process of metals is investigated Integral methods such as the variational formulation are applied to the partial differential equations for the free boundary problem and a finite dimensional approximation of the dynamical system is obtained The model describes the shape of the evolving cutting kerf and the melt flow The analysis is aimed at revealing the characteristic features of the resultant cut, for example, ripple formation and adherent dross The formation of the ripples in the upper part of the cut, where no resolidified material is detectable, is discussed in detail A comparison with numerical simulations and experiments is made
TL;DR: In this paper, the authors discuss the application of complex variable methods to Hele-Shaw flows and two-dimensional Stokes flows, both with and without surface tension, and explore the parallels between the two problems.
Abstract: We discuss the application of complex variable methods to Hele-Shaw flows and twodimensional Stokes flows, both with free boundaries. We outline the theory for the former, in the case where surface tension effects at the moving boundary are ignored. We review the application of complex variable methods to Stokes flows both with and without surface tension, and we explore the parallels between the two problems. We give a detailed discussion of conserved quantities for Stokes flows, and relate them to the Schwarz function of the moving boundary and to the Baiocchi transform of the Airy stress function. We compare the results with the corresponding results for Hele-Shaw flows, the principal consequence being that for Hele-Shaw flows the singularities of the Schwarz function are controlled in the physical plane, while for Stokes flow they are controlled in an auxiliary mapping plane. We illustrate the results with the explicit solutions to specific initial value problems. The results shed light on the construction of solutions to Stokes flows with more than one driving singularity, and on the closely related issue of momentum conservation, which is important in Stokes flows, although it does not arise in Hele-Shaw flows. We also discuss blow-up of zero-surface-tension Stokes flows, and consider a class of weak solutions, valid beyond blow-up, which are obtained as the zero-surface-tension limit of flows with positive surface tension.
03 Aug 1999
TL;DR: In this article, the authors considered the fourth order ordinary differential equation together with the boundary conditions and obtained existence results for the problem (E −(B) for more than just sublinear and superlinear cases discussed by most other authors.
Abstract: In this paper, we consider the fourth order ordinary differential equation together with the boundary conditions Some existence results for the problem (E)—(B) are obtained. The results here apply to more than just the sublinear and superlinear cases discussed by most other authors.
TL;DR: A bootstrapping algorithm for the extraction of potentials and arbitrary derivatives of the Cauchy data of regular three-dimensional second order elliptic boundary value problems in connection with corresponding boundary integral equations.
Abstract: In this paper we develop and analyze a bootstrapping algorithm for the extraction of potentials and arbitrary derivatives of the Cauchy data of regular three-dimensional second order elliptic boundary value problems in connection with corresponding boundary integral equations. The method rests on the derivatives of the generalized Green's representation formula, which are expressed in terms of singular boundary integrals as Hadamard's finite parts. Their regularization, together with asymptotic pseudohomogeneous kernel expansions, yields a constructive method for obtaining generalized jump relations. These expansions are obtained via composition of Taylor expansions of the local surface representation, the density functions, differential operators and the fundamental solution of the original problem, together with the use of local polar coordinates in the parameter domain. For boundary integral equations obtained by the direct method, this method allows the recursive numerical extraction of potentials and their derivatives near and up to the boundary surface.
TL;DR: In this paper, the authors obtained uniform decay rates for the energy function associated with a three-dimensional structural acoustic model described by coupled system consisting of the wave equation and plate equation with the coupling on the interface between the acoustic chamber and the wall.
TL;DR: In this article, it was shown that certain non-equilibrium expectation values in the boundary sine-Gordon model coincide with associated equilibrium-state expectation values for systems which differ from the BGS in that certain extra boundary degrees of freedom (q-oscillators) are added.
TL;DR: In this article, a general framework for boundary conditions is described, in which all methods can be written either in terms of temporal derivatives (temporal form}) or in termsof spatial derivatives ({\em spatial form}) and a direct formal comparison of boundary treatments which have been originaly proposed in completely different forms.