Showing papers on "Free boundary problem published in 2004"

Large Scale Dynamics of Precipitation Fronts in the Tropical Atmosphere: A Novel Relaxation Limit

Abstract: A simplifled set of equations is derived systematically below for the interaction of large scale ∞ow flelds and precipitation in the tropical atmosphere. These equations, the Tropical Climate Model, have the form of a shallow water equation and an equation for moisture coupled through a strongly nonlinear source term. This source term, the precipitation, is of relaxation type in one region of state space for the temperature and moisture, and vanishes identically elsewhere in the state space of these variables. In addition, the equations are coupled nonlinearly to the equations for barotropic incompressible ∞ow. Several mathematical features of this system are developed below including energy principles for solutions and their flrst derivatives independent of relaxation time. With these estimates, the formal inflnitely fast relaxation limit converges to a novel hyperbolic free boundary problem for the motion of precipitation fronts from a large scale dynamical perspective. Elementary exact solutions of the limiting dynamics involving precipitation fronts are developed below and include three families of waves: fast drying fronts as well as slow and fast moistening fronts. The last two families of waves violate Lax's Shock Inequalities; nevertheless, numerical experiments presented below conflrm their robust realizability with realistic flnite relaxation times. From the viewpoint of tropical atmospheric dynamics, the theory developed here provides a new perspective on the fashion in which the prominent large scale regions of moisture in the tropics associated with deep convection can move and interact with large scale dynamics in the quasi-equilibrium approximation.

Dynamics and breakup of a contracting liquid filament

Abstract: Contraction of a filament of an incompressible Newtonian liquid in a passive ambient fluid is studied computationally to provide insights into the dynamics of satellite drops created during drop formation. This free boundary problem, which is composed of the Navier–Stokes system and the associated initial and boundary conditions that govern the evolution in time of the filament shape and the velocity and pressure fields within it, is solved by the method of lines incorporating the finite element method for spatial discretization. The finite element algorithm developed here utilizes an adaptive elliptic mesh generation technique that is capable of tracking the dynamics of the filament up to the incipience of pinch-off without the use of remeshing. The correctness of the algorithm is verified by demonstrating that its predictions accord with (a) previously published results of Basaran (1992) on the analysis of finite-amplitude oscillations of viscous drops, (b) simulations of the dynamics of contracting filaments carried out with the well-benchmarked algorithm of Wilkes et al. (1999), and (c) scaling laws governing interface rupture and transitions that can occur from one scaling law to another as pinch-off is approached. In dimensionless form, just two parameters govern the problem: the dimensionless half-length which measures the relative importance of viscous force to capillary force. Regions of the parameter space are identified where filaments (a) contract to a sphere without breaking into multiple droplets, (b) break via the so-called endpinching mechanism where daughter drops pinch-off from the ends of the main filament, and (c) break after undergoing a series of complex oscillations. Predictions made with the new algorithm are also compared to those made with a model based on the slender-jet approximation. A region of the parameter space is found where the slender-jet approximation fares poorly, and its cause is elucidated by examination of the vorticity dynamics and flow fields within contracting filaments.

Time‐Dependent Problems with the Boundary Integral Equation Method

Abstract: Time-dependent problems that are modeled by initial-boundary value problems for parabolic or hyperbolic partial differential equations can be treated with the boundary int egral equation method. The ideal situation is when the right-hand side in the partial differential equation and th e initial conditions vanish, the data are given only on the boundary of the domain, the equation has constant coefficien ts, and the domain does not depend on time. In this situation, the transformation of the problem to a boundary integral equation follows the same well-known lines as for the case of stationary or time-harmonic problems modeled by elliptic boundary value problems. The same main advantages of the reduction to the boundary prevail: Reduction of the dimension by one, and reduction of an unbounded exterior domain to a bounded boundary. There are, however, specific difficulties due to the addition al time dimension: Apart from the practical problems of increased complexity related to the higher dimension, there can appear new stability problems. In the stationary case, one often has unconditional stability for reasonable approximation methods, and this stability is closely related to variational formulations based on the elliptici ty of the underlying boundary value problem. In the timedependent case, instabilities have been observed in practi ce, but due to the absence of ellipticity, the stability analysis is more difficult and fewer theoretical results are available. In this article, the mathematical principles governing the construction of boundary integral equation methods for time-dependent problems are presented. We describe some of the main algorithms that are used in practice and have been analyzed in the mathematical literature.

Boundary regularity for the Ricci equation, geometric convergence, and Gel’fand’s inverse boundary problem

Abstract: This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The second is to establish geometric convergence of a (sub)sequence of manifolds with boundary with such geometrical bounds and also an upper bound on the diameter and a lower bound on injectivity and boundary injectivity radius, making use of the first part. The third theme involves the uniqueness and conditional stability of an inverse problem proposed by Gel'fand, making essential use of the results of the first two parts.

Multidimensional boundary layers for a singularly perturbed Neumann problem

Abstract: We continue the study of [34], proving concentration phenomena for the equation − e2 Δu + u = up in a smooth bounded domain Ω ⊆ $\mathbb{R}^n$ and with Neumann boundary conditions. The exponent p is greater than or equal to 1, and the parameter e is converging to zero. For a suitable sequence ej → 0, we prove the existence of positive solutions uj concentrating at the whole boundary of Ω or at some of its components.

THE mKdV EQUATION ON THE HALF-LINE

Abstract: An initial boundary-value problem for the modified Korteweg–de Vries equation on the half-line, . It is shown that for a particular class of boundary conditions, the linearizable boundary conditions, all the spectral functions can be computed from the given initial data by using algebraic manipulations of the global relation; thus, in this case, the problem on the half-line can be solved as efficiently as the problem on the whole line.AMS 2000 Mathematics subject classification: Primary 37K15; 35Q53. Secondary 35Q15; 34A55

Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions

Abstract: This paper adresses the construction and study of a Crank-Nicolson-type discretization of the two-dimensional linear Schrödinger equation in a bounded domain Ω with artificial boundary conditions set on the arbitrarily shaped boundary of Ω. These conditions present the features of being differential in space and nonlocal in time since their definition involves some time fractional operators. After having proved the well-posedness of the continuous truncated initial boundary value problem, a semi-discrete Crank-Nicolson-type scheme for the bounded problem is introduced and its stability is provided. Next, the full discretization is realized by way of a standard finite-element method to preserve the stability of the scheme. Some numerical simulations are given to illustrate the effectiveness and flexibility of the method.

Diffusion of a fluid through an elastic solid undergoing large deformation

Abstract: This paper is concerned with the modeling of slow diffusion of a fluid into a swelling solid undergoing large deformation. Both the stress in the solid as well as the diffusion rates are predicted. The approach presented here, based on the balance laws of a single continuum with mass diffusion, overcomes the difficulties inherent in the theory of mixtures in specifying boundary conditions. A “natural” boundary condition based upon the continuity of the chemical potential is derived by the use of a variational approach, based on maximizing the rate of dissipation. It is shown that, in the absence of inertial effects, the differential equations resulting from the use of mixture theory can be recast into a form that is identical to the equations obtained in our approach. The boundary value problem of the steady flow of a solvent through a gum rubber membrane is solved and the results show excellent agreement with the experimental data of Paul and Ebra-Lima (J. Appl. Polym. Sci. 14 (1970) 2201) for a variety of solvents.

Regularity of a free boundary in parabolic potential theory

Abstract: We study the regularity of the free boundary in a Stefan-type problem 1u − ∂t u = χ in D ⊂ Rn × R, u = |∇u| = 0 on D \ with no sign assumptions on u and the time derivative ∂t u.

Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations

Abstract: We establish the existence and stability of multidimensional transonic shocks (hyperbolic-elliptic shocks) for the Euler equations for steady compressible potential fluids in infinite cylinders. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for velocity, can be written as a second order nonlinear equation of mixed elliptic-hyperbolic type for the velocity potential. The transonic shock problem in an infinite cylinder can be formulated into the following free boundary problem: The free boundary is the location of the multidimensional transonic shock which divides two regions of C1,α flow in the infinite cylinder, and the equation is hyperbolic in the upstream region where the C1,α perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem in unbounded domains. Our results indicate that there exists a solution of the free boundary problem such that the equation is always elliptic in the unbounded downstream region, the uniform velocity state at infinity in the downstream direction is uniquely determined by the given hyperbolic phase, and the free boundary is C1,α, provided that the hyperbolic phase is close in C1,α to a uniform flow. We further prove that, if the steady perturbation of the hyperbolic phase is C2,α, the free boundary is C2,α and stable under the steady perturbation. © 2003 Wiley Periodicals Inc.

Spatial segregation limit of a competition-diffusion system with Dirichlet boundary conditions

Abstract: We consider a competition-diffusion system with inhomogeneous Dirichlet boundary conditions for two competitive species and show that they spatially segregate as the interspecific competition rates become large. The limit problem turns out to be a free boundary problem.

Stability of transonic shock fronts in two-dimensional Euler systems

Abstract: We study the stability of stationary transonic shock fronts under two-dimensional perturbation in gas dynamics. The motion of the gas is described by the full Euler system. The system is hyperbolic ahead of the shock front, and is a hyperbolic-elliptic composed system behind the shock front. The stability of the shock front and the downstream flow under two-dimensional perturbation of the upstream flow can be reduced to a free boundary value problem of the hyperbolic-elliptic composed system. We develop a method to deal with boundary value problems for such systems. The crucial point is to decompose the system to a canonical form, in which the hyperbolic part and the elliptic part are only weakly coupled in their coefficients. By several sophisticated iterative processes we establish the existence and uniqueness of the solution to the described free boundary value problem. Our result indicates the stability of the transonic shock front and the flow field behind the shock.

On the stability of contact discontinuity for compressible Navier-Stokes equations with free boundary

Abstract: where ( ) is the velocity, ρ( ) > 0 the density, θ( ) the absolute temperature, μ > 0 the viscosity constant and κ > 0 the coefficient of heat conduction. The pressure = (ρ θ) and the internal energy = (ρ θ) are related by the second law of thermodynamics. There have been a lot of works on the asymptotic behaviors of the solutions for the system (1.1). Most of these results are concerned with the rarefaction wave and viscous shock wave. We refer to [10–15] for 2 × 2 case and [4–5, 7–8] for 3 × 3 case and references therein. However there is no result on the contact discontinuity for the system (1.1) until now due to various difficulties. Although some progress on the contact discontinuity were obtained by Liu and Xin [9] and Xin [17] in which the asymptotic toward the contact discontinuity was investigated for the initial value problem (IVP) of viscous conservation laws with uniformly artificial viscosity, no result is known for the physical system, especially for the compressible N-S equations (1.1). Therefore we really want to give a positive result on the contact discontinuity for the physical system (1.1). To simplify our problem, we focus our attention on the perfect gas. In this situation,

Symmetric hyperbolicity and consistent boundary conditions for second-order Einstein equations

Abstract: We present two families of first-order in time and second-order in space formulations of the Einstein equations (variants of the Arnowitt-Deser-Misner formulation) that admit a complete set of characteristic variables and a conserved energy that can be expressed in terms of the characteristic variables. The associated constraint system is also symmetric hyperbolic in this sense, and all characteristic speeds are physical. We propose a family of constraint-preserving boundary conditions that is applicable if the boundary is smooth with tangential shift. We conjecture that the resulting initial-boundary value problem is well-posed.

Uniqueness and boundary behavior of large solutions to elliptic problems with singular weights

Abstract: We consider the elliptic problems $\Delta u=a(x)u^m$, $m>1$, and $\Delta u=a(x)e^u$ in a smooth bounded domain $\Omega$, with the boundary condition $u=+\infty$ on $\partial\Omega$. The weight function $a(x)$ is assumed to be Holder continuous, growing like a negative power of $d(x)=$ dist $(x,\partial\Omega)$ near $\partial\Omega$. We show existence and nonexistence results, uniqueness and asymptotic estimates near the boundary for both the solutions and their normal derivatives.

Doubly Nonlinear Thin-Film Equations in One Space Dimension

Abstract: We consider a free-boundary problem for a class of fourth-order nonlinear parabolic equations which are degenerate both with respect to the unknown and to its third derivative The problem is relevant in the description of the surface-tension driven spreading of a non-Newtonian liquid over a solid surface in the “complete wetting” regime Relying solely on global and local energy estimates and on Bernis’ inequalities, we prove existence of solutions to this problem, and obtain sharp upper bounds for the propagation of their support A necessary condition for the occurrence of waiting-time phenomena is also derived

Design of Absorbing Boundary Conditions for Schrödinger Equations in $\mathbbR$ d

Abstract: We construct a family of absorbing boundary conditions for the linear Schrodinger equation on curved boundaries in any dimension which are local both in space and time. We give a convergence result and show that the corresponding initial boundary value problems are well posed. We also prove the nonlinear Schrodinger equation with the absorbing boundary conditions of the linearized problem to be well posed. We finally present numerical results both for the linear and the nonlinear case.

On Invertibility of Elastic Single-Layer Potential Operator

Abstract: The question of unique solvability of the boundary integral equation of the first kind given by the single-layer potential operator is studied in the case of plane isotropic elasticity. First, a sufficient condition of the positivity, and hence invertibility, of this operator is presented. Then, considering a scale transformation of the domain boundary, the well known formula for scaling the Robin constant in potential theory is generalized to elasticity. Subsequently, an explicit equation for evaluation of critical scales for a given boundary, when the single-layer operator fails to be invertible, is deduced. It is proved that there are either two simple critical scales or one double critical scale for any domain boundary. Numerical results, obtained applying a symmetric Galerkin boundary element code, confirm the propositions of the theory developed for both single and multi-contour boundaries.

Boundary Element Methods: Foundation and Error Analysis

Abstract: This chapter gives an expository introduction to the Galerkin-BEM for the elliptic boundary value problems from the mathematical point of view. Emphasis will be placed on the variational formulations of the boundary integral equations and the general error estimates for the approximate solution in appropriate Sobolev spaces. A classification of boundary integral equations will be given on the basis of the Sobolev index. The simple relations between the variational formulations of the boundary integral equations and the corresponding partial differential equations under consideration will be indicated. Basic concepts such as stability, consistency, and convergence, as well as the condition numbers and ill-posedness, will be discussed by using elementary examples. Keywords: boundary integral equations; fundamental solutions; variational formulations; Garding's inequality; Galerkin's method; boundary elements; stability; ill-posedness; asymptotic error estimates

The Free Boundary Problem in the Optimization of Composite Membranes

Abstract: In this paper, continuing our earlier article [CGIKO], we study qualitative properties of solutions of a certain eigenvalue optimization problem. Especially we focus on the study of the free boundary of our optimal solutions on general domains. (Contemporary Math. of the American Math. Soc., v. 268(2000))