One-dimensional Free Boundary Problem Research Articles

Global existence and uniqueness of solutions for one-dimensional reaction-interface systems

In this paper, we provide a mathematical framework in studying the wave propagation with the annihilation phenomenon in excitable media. We deal with the existence and uniqueness of solutions to a one-dimensional free boundary problem (called a reaction–interface system) arising from the singular limit of a FitzHugh–Nagumo type reaction–diffusion system. Because of the presence of the annihilation, interfaces may intersect each other. We introduce the notion of weak solutions to study the continuation of solutions beyond the annihilation time. Under suitable conditions, we show that the free boundary problem is well-posed.

Local existence of a solution to a free boundary problem describing migration into rubber with a breaking effect

<abstract><p>We consider a one-dimensional free boundary problem describing the migration of diffusants into rubber. In our setting, the free boundary represents the position of the front delimitating the diffusant region. The growth rate of this region is described by an ordinary differential equation that includes the effect of breaking the growth of the diffusant region. In this specific context, the breaking mechanism is should be perceived as a non-dissipative way of describing eventual hyperelastic response to a too fast diffusion penetration. In recent works, we considered a similar class of free boundary problems modeling diffusants penetration in rubbers, but without attempting to deal with the possibility of breaking or accelerating the occurring free boundaries. For simplified settings, we were able to show the global existence and uniqueness as well as the large time behavior of the corresponding solutions to our formulations. Since here the breaking effect is contained in the free boundary condition, our previous results are not anymore applicable. The main mathematical obstacle in ensuring the existence of a solution is the non-monotonic structure of the free boundary. In this paper, we establish the existence and uniqueness of a weak solution to the free boundary problem with breaking effect and give explicitly the maximum value that the free boundary can reach.</p></abstract>

A free boundary problem for a diffusion–convection equation

One-dimensional free boundary problem for a nonlinear diffusion–convection equation with a Dirichlet condition at fixed face x=0, variable in time, is considered. Through several transformations the problem is reduced to a free boundary problem for a diffusion equation and the integral formulation is obtained. By using fixed point theorems, the existence of at least a solution, for small time, to a system of coupled nonlinear integral equations is obtained.

Boundary Control of the Motion of a Heavy Piston in Viscous Gas

A one-dimensional free boundary problem of the motion of a heavy piston in a tube filled with viscous gas is considered. The system of governing equations and boundary conditions is derived. The ob...

On the Existence of Weak Solutions for a 1-D Free-Boundary Concrete Carbonation Problem

In this paper we deal with a one-dimensional free-boundary problem arising in the modeling of concrete carbonation process. More precisely, we investigate global existence, uniqueness and large-time behavior of weak solutions for the problem under consideration. We also obtain the existence of a weak solution when the measure of the initial domain vanishes.

Heat Potentials Method in the Treatment of One-dimensional Free Boundry Problems Applied in Cryomedicine

Free boundary problems are considered to be the most difcult and the least researched in the eld of mathematical physics. The present article is concerned with the research of the following issue: treatment of one-dimensional free boundary problems. The treated problem contains a nonlinear evolutionary equation, which occurs within the context of mathematical modeling of cryosurgery problems. In the course of the research, an integral expression has been obtained. The obtained integral expression presents a general solution to the non-homogeneous evolutionary equation which contains the functions that represent simple-layer and double-layer heat potential density. In order to determine the free boundary and the density of potential a system of nonlinear, the second kind of Fredholm integral equations was obtained within the framework of the given work. The treated problem has been reduced to the system of integral equations. In order to reduce the problem to the integral equation system, a method of heat potentials has been used. In the obtained system of integral equations instead of K(ξ; x; τ - t) in case of Dirichlet or Neumann conditions the corresponding Greens functions G(ξ; x; τ - t) or N(ξ; x; τ - t) have been applied. Herewith the integral expression contains fewer densities, but the selection of arbitrary functions is reserved. The article contains a number of results in terms of building a mathematical model of cooling and freezing processes of biological tissue, as well as their effective solution development.

A Differential Game for a Multiclass Queueing Model in the Moderate-Deviation Heavy-Traffic Regime

We study a differential game that governs the moderate-deviation heavy-traffic asymptotics of a multiclass single-server queueing control problem with a risk-sensitive cost. We consider a cost set on a finite but sufficiently large time horizon, and show that this formulation leads to stationary feedback policies for the game. Several aspects of the game are explored, including its characterization via a (one-dimensional) free boundary problem, the semi-explicit solution of an optimal strategy, and the specification of a saddle point. We emphasize the analogy to the well-known Harrison-Taksar free boundary problem which plays a similar role in the diffusion-scale heavy-traffic literature.

On a large time behavior of a solution to a one-dimensional free boundary problem for adsorption phenomena

In this paper we consider a one-dimensional free boundary problem describing adsorption phenomena in a porous media. This problem was already proposed and we established a local existence in time and a uniqueness result in [1,4]. Here, through the discussion for the dynamics of the free boundary in wetting and drying processes we suppose a specific form of the growth rate condition for the free boundary. By using this form in our problem we can obtain the global existence of a solution in time and convergence of the free boundary as t→∞.

Control of Stirling engine. Simplified, compressible model

A one-dimensional free boundary problem on a motion of a heavy piston in a tube filled with viscous gas is considered. The system of governing equations and boundary conditions is derived. The obtained system of differential equations can be regarded as a mathematical model of an exterior combustion engine. The existence of a weak solution to this model is proved. The problem of maximization of the total work of the engine is considered.

Substrate degradation in high-Rayleigh-number reactive convection

We study buoyancy-induced convection of a solute in an ideal two-dimensional fluid-saturated porous medium, where the solute undergoes a second-order reaction with a chemical substrate that is fixed in the underlying matrix. Numerical simulations at high Rayleigh number show how a flow is established in which a thin dynamic boundary layer beneath the solute source feeds slender vertical plumes beneath. We examine how the substrate is reactively degraded, at a rate enhanced by convective mixing. For the case when the substrate is abundant, we derive a reduced-order model describing the slow degradation of the substrate, which is formulated as a novel one-dimensional free-boundary problem. Numerical simulations and the reduced model reveal how, when the reaction is rapid compared to the convective time scale, the plumes propagate deep into the flow domain with reaction confined to a narrow region at their base. In contrast, slow reaction allows plumes to fill the domain before degradation of the substrate proceeds homogeneously. An alternative model with a thin reaction front captures the rapid degradation of the substrate when the solute concentration is relatively high.

Global-in-time smoothness of solutions to the vacuum free boundary problem for compressible isentropic Navier–Stokes equations

In this paper we establish the global existence of smooth solutions to vacuum free boundary problems of the one-dimensional compressible isentropic Navier–Stokes equations for which the smoothness extends all the way to the boundaries. The results obtained in this work include the physical vacuum for which the sound speed is C1/2-Hölder continuous near the vacuum boundaries when 1 < γ < 3. The novelty of this result is its global-in-time regularity which is in contrast to the previous main results of global weak solutions in the literature. Moreover, in previous studies of the one-dimensional free boundary problems of compressible Navier–Stokes equations, the Lagrangian mass coordinates method has often been used, but in the present work the particle path (flow trajectory) method is adopted, which has the advantage that the particle paths and, in particular, the free boundaries can be traced.

A free boundary problem arising in the ecological models with N-species

This paper is concerned with the one-dimensional free boundary problem for quasilinear reaction-diffusion systems arising in the ecological models with N-species, where some of the species are made up of two separated groups and the mankind’s influence is taken into account. In the problem under consideration, there are n free boundaries, the coefficients of the equations are allowed to be discontinuous on the free boundaries and the reaction functions are mixed quasimonotone. The aim is to show the local existence of the solutions for the free boundary problem by the fixed point method, and the global existence and uniqueness of the solutions for the corresponding diffraction problem by the approximation and estimate methods.

Steady fall of isothermal, resistive-viscous, compressible fluid across magnetic field

This is a basic MHD study of the steady fall of an infinite, vertical slab of isothermal, resistive-viscous, compressible fluid across a dipped magnetic field in uniform gravity. This double-diffusion steady flow in unbounded space poses a nonlinear but numerically tractable, one-dimensional (1D) free-boundary problem, assuming constant coefficients of resistivity and viscosity. The steady flow is determined by a dimensionless number μ1 proportional to the triple product of the two diffusion coefficients and the square of the linear total mass. For a sufficiently large μ1, the Lorentz, viscous, fluid-pressure, and gravitational forces pack and collimate the fluid into a steady flow of a finite width defined by the two zero-pressure free-boundaries of the slab with vacuum. The viscous force is essential in this collimation effect. The study conjectures that in the regime μ1→0, the 1D steady state exists only for μ1∈Ω, a spectrum of an infinite number of discrete values, including μ1 = 0 that corresponds to two steady states, the classical zero-resistivity static slab of Kippenhahn and Schlüter [R. Kippenhahn and A. Schlüter, Z. Astrophys. 43, 36 (1957)] and its recent generalization [B. C. Low et al., Astrophys. J. 755, 34 (2012)] to admit an inviscid resistive flow. The pair of zero-pressure boundaries of each of the μ1→0 steady-state slabs are located at infinity. Computational evidence suggests that the Ω steady-states are densely distributed around μ1 = 0, as an accumulation point, but are sparsely separated by open intervals of μ1-values for which the slab must be either time-dependent or spatially multi-dimensional. The widths of these intervals are vanishingly small as μ1→0. This topological structure of physical states is similar to that described by Landau and Liftshitz [L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Addison-Wesley, Reading, MA, 1959)] to explain the onset of hydrodynamic turbulence. The implications of this MHD study are discussed, with an interest in the prominences in the solar atmosphere and the interstellar clouds in the Galaxy.

A one dimensional free boundary problem for adsorption phenomena

In this paper we deal with a one-dimensional free boundary problem, which is a mathematical model for an adsorption phenomena appearing in concrete carbonation process. This model was proposed in line of previous studies of three dimensional concrete carbonation process. The main result in this paper is concerned with the existence and uniqueness of a time-local solution to the free boundary problem. This result will be obtained by means of the abstract theory of nonlinear evolution equations and Banach's fixed point theorem, and especially, the maximum principle applied to our problem will play a very important role to obtain the uniform estimate to approximate solutions.

A free-boundary problem for concrete carbonation: Front nucleation and rigorous justification of the $\sqrt{{t}}$-law of propagation

We study a one-dimensional free-boundary problem describing the penetration of carbonation fronts (free reaction-triggered interfaces) in concrete. Using suitable integral estimates for the free boundary and involved concentrations, we reach a twofold aim: (1) We fill a fundamental gap by justifying rigorously the experimentally guessed \sqrt{t} asymptotic behavior. Previously we obtained the upper bound s(t)\leq C'\sqrt{t} for some constant C' ; now we show the optimality of the rate by proving the right nontrivial lower estimate, i.e. there exists C''&gt;0 such that s(t)\geq C''\sqrt{t} . (2) We obtain weak solutions to the free-boundary problem for the case when the measure of the initial domain vanishes. In this way, we allow for the nucleation of the moving carbonation front – a scenario that until now was open from the mathematical analysis point of view.

On the stability loss for an Euler beam resting on a tensionless Pasternak foundation

In the present work, the tensionless contact problem of an Euler–Bernoulli beam of finite length resting on a two-parameter Pasternak-type foundation is investigated. Owing to the tensionless character of the contact, the beam may lift-off the foundation and the point where contact ceases and detachment begins, named contact locus, needs be assessed. In this situation, a one-dimensional free boundary problem is dealt with. An extra condition, in the form of a homogeneous second-order equation in the displacement and its derivatives, is demanded to set the contact locus and it gives the problem its nonlinear feature. Conversely, the loading and the beam length may be such that the beam rests entirely supported on the foundation, which situation is governed by a classical linear boundary value problem. In this work, contact evolution is discussed for a continuously varying loading condition, starting from a symmetric layout and at a given beam length, until overturning is eventually reached. In particular, stability is numerically assessed through the energy criterion, which is shown to stand for the free boundary situation as well. At overturning, a descending pathway in the system energy appears and stability loss is confirmed.

On a non-linear moving boundary problem for a diffusion–convection equation

We study a one-dimensional free boundary problem for a non-linear diffusion–convection equation whose diffusivity is heterogeneous in space as well as being non-linear. Under the Bäcklund transformation the problem is reduced to an associated free boundary problem. We prove the existence and uniqueness, local in time, of the solution by using the Friedman Rubinstein integral representation method and the Banach contraction theorem.

A Four-Step Fixed-Grid Method for 1D Stefan Problems

This note is concerned with a fixed-grid finite difference method for the solution of one-dimensional free boundary problems. The method solves for the field variables and the location of the boundary in separate steps. As a result of this decoupling, the nonlinear part of the algorithm involves only a scalar unknown, which is the location of the moving boundary. A number of examples are used to study the applicability of the method. The method is particularly useful for moving boundary problems with various conditions at the front.

Numerical Modeling of Microsegregation for Fe-base Multicomponent Alloys with Peritectic Transformation Coupled with Thermodynamic Calculations

Numerical model was developed to simulate the microsegregation and phase transformation for an Fe-based multicomponent alloy with peritectic transformation. The present model was based on the simplest one-dimensional free boundary problem and assumed the local equilibrium condition at the interface. In addition, the model was coupled with thermodynamic calculation software (ChemAPP) in order to calculate equilibrium concentration at an interface. As the calculation region, the transverse and longitudinal cross section of columnar dendrite was approximated by a star shape, assuming dendrite envelope. The validity of the present model was evaluated by comparing with the analytical models and experimental data. In the comparison with the analytical models, which are lever rule, Gulliver–Scheil model and Clyne–Kurz model, the calculated results were close to the curve of lever rule for Fe–C binary alloy, and were close to the curve of Gulliver–Scheil model for Fe–Mn binary alloy. For both alloys, the calculated results were in good agreement with the curves of Clyne–Kurz model. The relationship between peritectic temperature range and carbon content was calculated for Fe–C binary alloy and compared with the result of a model by Fredriksson et al. [H. Fredriksson et al.: Metal Science, 16 (1982), 575]. The calculated result was in great agreement with their one. Also, the peritectic temperature range for peritectic content was not always the maximum, and the carbon content slightly shifted to hyper-peritectic side as the cooling rate became higher. For Fe–C–Mn–Si–P–Mo alloy, we calculated microsegregation and peritectic transformation, and compared with the experimental data reported by Ueshima et al. [Y. Ueshima et al.: Tetsu-to-Hagané, 73 (1987), 1551]. Temperatures of δ/γ transformation and γ-solidification decreased by adding to molybdenum, and these results were close to those of Ueshima et al. Also, the distributions of manganese, phosphorus and molybdenum calculated by the present model were in essential agreement with those of their experimental data.

Explicit solutions for one-dimensional two-phase free boundary problems with either shrinkage or expansion

Abstract We consider a one-dimensional solidification of a pure substance which is initially in liquid state in a bounded interval [ 0 , l ] . Initially, the liquid is above the freezing temperature, and cooling is applied at x = 0 while the other end x = l is kept adiabatic. At the time t = 0 , the temperature of the liquid at x = 0 comes down to the freezing point and solidification begins, where x = s ( t ) is the position of the solid–liquid interface. As the liquid solidifies, it shrinks ( 0 r 1 ) or expands ( r 0 ) and appears a region between x = 0 and x = r s ( t ) , with r 1 . Temperature distributions of the solid and liquid phases and the position of the two free boundaries ( x = r s ( t ) and x = s ( t ) ) in the solidification process are studied. For three different cases, changing the condition on the free boundary x = r s ( t ) (temperature boundary condition, heat flux boundary condition and convective boundary condition) an explicit solution is obtained. Moreover, the solution of each problem is given as a function of a parameter which is the unique solution of a transcendental equation and for two of the three cases a condition on the parameter must be verified by data of the problem in order to have an instantaneous phase-change process. In all the cases, the explicit solution is given by a representation of the similarity type.