Pointwise regularity for a parabolic equation with log-term singularity

We consider a free boundary problem for the heat equation with a given non-negative external heat source, which is often called a parabolic Bernoulli free boundary problem. On the free boundary, we impose the zero Dirichlet condition and the fixed normal derivative so that heat escapes from the boundary. In various settings, we show that there exist no solutions when the initial temperature equals the fixed temperature no matter where the initial location of the free boundary is given provided that the external heat source is bounded from above. We also note that there is a chance to have a solution when the external temperature is unbounded as time tends to zero by giving a self-similar solution.

A parabolic free boundary problem which arises in a statistical decision setting is reduced to a free boundary problem for the heat equation which is amenable to numerical solution by the method of lines. An algorithm is given and the apparently globally converging results are compared with the asymptotic expansions of H. Chernoff and J. Breakwell.

: Research on the numerical solution of free boundary problems for partial differential equations with locally or sequentially one-dimensional methods has been supported by the U.S. Army Research Office through two consecutive 3-year research contracts. The time and resources provided have made it possible to develop a reasonably comprehensive mathematical theory and flexible numerical algorithms on which to base current computational methods and future research. During the first 3-year period, the method of fractional steps and the method of lines were applied to elliptic and parabolic free boundary problems. During the past three years work was directed toward demonstrating the flexibility of the method of lines for increasingly complex problems, on examining the behavior of certain ill-posed elliptic free boundary problems, and on establishing a mathematical theory for sequentially one-dimensional algorithms. (Author)

A free boundary problem, which comes from the model of the perpetual American call options with utility functions in financial market, is investigated. It is a degenerative parabolic free boundary problem and is studied by the line method. The existence, regularity and uniqueness of the solution as well as some properties of the free boundary are established.

The field of the mathematical and numerical analysis of systems of nonlinear pdes involving interfaces and free boundaries is a burgeoning area of research. Many such systems arise from mathematical models in ma- terial science and fluid dynamics such as phase separation in alloys, crystal growth, dynamics of multiphase fluids and epitaxial growth. In applications of these mathematical models, suitable performance indices and appropriate control actions have to be specified. Mathematically this leads to optimiza- tion problems with pde constraints including free boundaries. It is now timely to consider such control problems because of the maturity of the field of com- putational free boundary problems. The aim of the mini-workshop was to bring together leading experts and young researchers from the separate fields of numerical free boundary problems and optimal control in order to estab- lish links and to identify suitable model problems to serve as paradigms for progressing knowledge of optimal control of free boundaries.

We consider the inverse Stefan type free boundary problem, where information on the boundary heat flux and the density of the sources are missing and must be found along with the temperature and the free boundary. We pursue the optimal control framework analyzed in [1, 2], where the boundary heat flux, the density of the sources, and the free boundary are components of the control vector. We prove the Frechet differentiability in Besov spaces, and derive the formula for the Frechet differential under minimal regularity assumptions on the data. The result implies a necessary condition for optimal control and opens the way to the application of projective gradient methods in Besov spaces for the numerical solution of the inverse Stefan problem.

We consider a free boundary problem of a quasi-linear strongly degenerate parabolic equation arising from a model of pressure filtration of flocculated suspensions. We provide definitions of generalized solutions of the free boundary problem in the framework of L2 divergence-measure fields. The formulation of boundary conditions is based on a Gauss--Green theorem for divergence-measure fields on bounded domains with Lipschitz deformable boundaries and avoids referring to traces of the solution. This allows one to consider generalized solutions from a larger class than BV. Thus it is not necessary to derive the usual uniform estimates of spatial and time derivatives of the solutions of the correspondingregularized problem, as required by the BV approach. We first prove the existence and uniqueness of the solution of the regularized parabolic free boundary problem and then apply the vanishing viscosity method to prove the existence of a generalized solution to the degenerate free boundary problem.

Introduction Using finite-difference methods for European options is relatively straightforward, as there is no possibility of early exercise. As we have seen, the possibility of early exercise may lead to free boundaries. The chief problem with free boundaries, from the point of view of numerical analysis, is that we do not know where they are. This makes it difficult to impose the free boundary conditions, since we have to determine where to impose them as part of the solution procedure. (Recall that in Chapter 8 we simply imposed the boundary conditions at fixed grid points.) There are two distinct strategies for the numerical solution of free boundary problems. One is to attempt to track the free boundary as part of the time-stepping process. In the context of valuation of American options this is not a particularly attractive method, as the free boundary conditions are both implicit – that is, they do not give a direct expression for the free boundary or its time derivatives. We simply note the existence of such methods here, and refer the reader to the literature for a discussion of various boundary tracking strategies for implicit free boundary problems. The other strategy is to attempt to find a transformation that reduces the problem to a fixed boundary problem from which the free boundary can be inferred afterwards . There are many transformations that do this, but we consider only the particularly elegant method involving the use of the linear complementarity formulation.

This works studies a quasilinear, reaction-diffusion type parabolic free boundary problem modelling population dynamics. Holder norm a priori estimates are established for the free boundary and the solution. Uniqueness of the solution is shown and qualitative properties of the solution are investigated.

The paper is concerned with a one-dimensional parabolic problem in a domain bounded by two lines x = 0 and x = kt, k > 0, (x, t) ∈ ℝ2, with the Neumann boundary condition on the line x = 0 and with dynamic boundary condition on the line x = kt. For the solution of this problem, a coercive estimate in a weighted Hölder norm is obtained. It is shown that this estimate can be useful for the analysis of parabolic free boundary problems. Bibliography: 7 titles.

A very simple model for cell swelling by osmosis is introduced, resulting in a parabolic free boundary problem. In case of radially symmetric initial conditions, it is shown that the model can be viewed as a gradient flow involving entropy, surface area and the Wasserstein metric. This observation is used to construct solutions and explain the presence and nature of osmosis.

One-dimensional Shape Memory Alloy Problem with Duhem Type of Hysteresis Operator.- Existence and Uniqueness Results for Quasi-linear Elliptic and Parabolic Equations with Nonlinear Boundary Conditions.- Finite Time Localized Solutions of Fluid Problems with Anisotropic Dissipation.- Parabolic Equations with Anisotropic Nonstandard Growth Conditions.- Parabolic Systems with the Unknown Dependent Constraints Arising in Phase Transitions.- The N-membranes Problem with Neumann Type Boundary Condition.- Modelling, Analysis and Simulation of Bioreactive Multicomponent Transport.- Asymptotic Properties of the Nitzberg-Mumford Variational Model for Segmentation with Depth.- The ?-Laplacian First Eigenvalue Problem.- Comparison of Two Algorithms to Solve the Fixed-strike Amerasian Options Pricing Problem.- Nonlinear Diffusion Models for Self-gravitating Particles.- Existence, Uniqueness and an Explicit Solution for a One-Phase Stefan Problem for a Non-classical Heat Equation.- Dislocation Dynamics: a Non-local Moving Boundary.- Bermudean Approximation of the Free Boundary Associated with an American Option.- Steady-state Bingham Flow with Temperature Dependent Nonlocal Parameters and Friction.- Some P.D.E.s with Hysteresis.- Embedding Theorem for Phase Field Equation with Convection.- A Dynamic Boundary Value Problem Arising in the Ecology of Mangroves.- Wave Breaking over Sloping Beaches Using a Coupled Boundary Integral-Level Set Method.- Finite Difference Schemes for Incompressible Flows on Fully Adaptive Grids.- Global Solvability of Constrained Singular Diffusion Equation Associated with Essential Variation.- Capillary Mediated Melting of Ellipsoidal Needle Crystals.- Boundary Regularity at {t = 0} for a Singular Free Boundary Problem.- Fast Reaction Limits and Liesegang Bands.- Numerical Modeling of Surfactant Effects in Interfacial Fluid Dynamics.- The Value of an American Basket Call with Dividends Increases with the Basket Volatility.- Mathematical Modelling of Nutrient-limited Tissue Growth.- Asymptotic Hysteresis Patterns in a Phase Separation Problem.- Obstacle Problems for Monotone Operators with Measure Data.- Piecewise Constant Level Set Method for Interface Problems.- Dynamics of a Moving Reaction Interface in a Concrete Wall.- Adaptive Finite Elements with High Aspect Ratio for Dendritic Growth of a Binary Alloy Including Fluid Flow Induced by Shrinkage.- A Free Boundary Problem for Nonlocal Damage Propagation in Diatomites.- Concentrating Solutions for a Two-dimensional Elliptic Problem with Large Exponent in Nonlinearity.- Existence of Weak Solutions for the Mullins-Sekerka Flow.- Existence and Approximation Results for General Rate-independent Problems via a Variable Time-step Discretization Scheme.- Global Attractors for the Quasistationary Phase Field Model: a Gradient Flow Approach.- Aleksandrov and Kelvin Reflection and the Regularity of Free Boundaries.- Solvability for a PDE Model of Regional Economic Trend.- Surface Energies in Multi-phase Systems with Diffuse Phase Boundaries.- High-order Techniques for Calculating Surface Tension Forces.- Simulation of a Model of Tumors with Virus-therapy.- Asymptotic Behavior of a Hyperbolic-parabolic Coupled System Arising in Fluid-structure Interaction.

In this paper we consider stationary ice sheet modelled as a Stokes flow in a bounded two-dimensional domain. In particular, we study the behavior of the grounding line, where different boundary conditions meet: no-slip conditions for the grounded part and force balance conditions for the floating part whose shape is a priori undetermined. This yields a free boundary problem with mixed boundary conditions and a contact line, called “grounding line” in the glaciological context, that might move along the solid substrate. We show that solutions with moving grounding lines and zero contact angle do exist and determine the shape and asymptotic properties of the free boundary.

Free boundaries are a characteristic feature in many problems in ideal hydrodynamics, e.g., in jet problems. The term "free boundary" refers to parts of the boundary of the flow region which are not known in advance and on which two boundary conditions are specified: kinematic and dynamic. Free-boundary problems are usually nonlinear, difficult to analyze, and so far have been little studied, with few exceptions, despite the voluminous literature on this subject. A survey of the classical results on these problems can be found, e.g., in the monographs [1-3]. In this paper, we consider two types of ideal-fluid plane flow with internal free boundaries in a channel of a complex shape. These flows are described by elliptical equations for the stream functions. We study flow in Lavrent'ev's modified scheme [4], which joins potential and turbulent flows with an unknown separation boundary. Vorticity is defined up to a constant, which is determined from the total circulation on the boundary of the turbulent region. A numerical algorithm is constructed, based on the fictitious region method. The flow of the two-layer fluid is modeled by the Laplace equation for the stream functions. The specific feature of the problem is the presence of nonhomogeneous matching conditions on the unknown boundary. The generalized formulation of the problem and the fictitious region method produce an efficient algorithm and numerical through schemes. The nonlinear problems are solved using an iterative process of successive approximations by the nonlinearity (the free boundary). 1. Joining Problem. Consider the problem of joining potential and turbulent flows with an unknown separation boundary (Lavrent'ev's scheme [4]). The flow region is D = D o O D1, with potential flow in D O and turbulent flow in D 1. The stream function ~p is such that ~(x) = 0 for x ~ S = OD o tA OD1,

This paper studies conditions for local (in time) solvability of a qualitatively new singularllimit problem, the free (unknown) boundary problem appearing recently. In fact, there are not so many different free boundary problems, which corresponds to not so large a variety of principally different phase transitions of the first and second kinds. Therefore, the appearance of principally new problems elicits interest. This paper studies structural features of a certain problem on the basis of a certain method developed previously, precisely, the localization method [1, 3, 9].