A singular-degenerate free boundary problem arising from the moisture evaporation in a partially saturated porous medium

Summary We consider a moisture evaporation process in a porous medium which is partially saturated by a fluid. The mathematical model is a singular-degenerate nonlinear parabolic free boundary problem. We first transform the problem into a weak form in a fixed domain and then derive some uniform estimates for the proper approximate solution. The existence of a weak solution is established by a compactness argument. Finally, the regularity of the solution and interfaces are investigated.     

A free boundary problem arising from DCIS mathematical model

Ductal carcinoma in situ – a special cancer – is confined within the breast ductal only. We derive the mathematical ductal carcinoma in situ model in a form of a nonlinear parabolic equation with initial, boundary, and free boundary conditions. Existence, uniqueness, and stability of problem are proved. Algorithm and illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. Copyright © 2016 John Wiley & Sons, Ltd.     

Free boundary regularity close to initial state for parabolic obstacle problem

In this paper we study the behavior of the free boundary , arising in the following complementary problem:

Here denotes the parabolic boundary, is a parabolic operator with certain properties, is the upper half of the unit cylinder in , and the equation is satisfied in the viscosity sense. The obstacle is assumed to be continuous (with a certain smoothness at , ), and coincides with the boundary data at time zero. We also discuss applications in financial markets.

     

Stability and instability of Liapunov-Schmidt and Hopf bifurcation for a free boundary problem arising in a tumor model

We consider a free boundary problem for a system of partial differential equations, which arises in a model of tumor growth. For any positive number there exists a radially symmetric stationary solution with free boundary . The system depends on a positive parameter , and for a sequence of values there also exist branches of symmetric-breaking stationary solutions, parameterized by , small, which bifurcate from these values. In particular, for near the free boundary has the form where is the spherical harmonic of mode . It was recently proved by the authors that the stationary solution is asymptotically stable for any , but linearly unstable if , where if and if ; . In this paper we prove that for each of the stationary solutions which bifurcates from is linearly stable if and linearly unstable if . We also prove, for , that the point is a Hopf bifurcation, in the sense that the linearized time-dependent problem has a family of solutions which are asymptotically periodic in .

     

Free boundary problem for the equation of one-dimensional motion of compressible gas with density-dependent viscosity

We consider a free boundary problem for the equation of the one-dimensional isentropic motion with density-dependent viscosity μ =b _ϱ ^β, whereb and β are positive constants. We prove that there exists an unique weak solution globally in time, provided that β<1/3.

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Sunto Si considera un problema di frontiera libera per l’equazione del moto unidimensionale isoentropico con viscosità dipendente dalla densità secondo la legge μ =b _ϱ ^β, doveb e β sono costanti positive. Si dimostra che esiste un’unica soluzione debole globale nel tempo, purché β<1/3.

     

On a certain boundary value problem arising in shrinking sheet flows

This paper presents an analysis of the boundary value problem resulting from the magnetohydrodynamic (MHD) viscous flow influenced by a shrinking sheet with suction for the cases of two-dimensional (m = 1) and axisymmetric (m = 2) shrinking. The influences of the parameter m as well as the effects of suction parameter s and Hartmann number M² on similar entrainment velocity f(∞) and flow characteristics are studied. To this purpose, the resulting nonlinear ordinary differential equation is solved numerically using the 4th order Runge-Kutta method in combination with a shooting procedure. The obtained results elucidate reliability and efficiency of the technique from which interesting features between the skin friction coefficient f″(0) and the entrainment velocity f(∞) as function of the mass transfer parameter s can also be obtained.     

A free boundary problem arising from pricing convertible bond

In this article we study the behaviours of the optimal conversion boundary (i.e. free boundary) of an American-style convertible bond with finite horizon (i.e. parabolic case). We prove the existence and the uniqueness of the strong solution of the problem and the boundedness and smoothness of the free boundary. Moreover, we characterize the free boundary's start point and present two numerical results.     

连续铸钢中的一类周期自由边界问题

本文讨论了连续铸钢中的一类周期自由边界问题,证明了周期古典解的存在性,唯一性和稳定性。     

The regularity of the free boundary in a multidimensional filtration problem

We prove the regularity of the free boundary for a filtration problem with capillarity in more than one space dimension. The free boundary is the interface between the saturated region (in which the governing equation is elliptic) and the unsaturated region (where a degenerate parabolic equation is to be solved).This work was partially supported by National Project Equazioni di Evoluzione e Applicazioni Fisico Matematiche (M.U.R.S.T.).     

Free boundary problem for a viscous compressible flow associated with oxidation of silicon

This paper is concerned with a free boundary problem describing the oxidation process of silicon. Its mathematical model is a compressible Navier-Stokes equations coupling a parabolic equation and a hyperbolic one. Surface tension is involved at the free boundary and density equation is non-homogeneous. It is proved that for arbitrary data satisfying only natural consistency conditions the problem is uniquely solvable on some finite time interval. Supported by National Natural Science Foundation of China     

On a free boundary problem in ground freezing

We consider a boundary value problem describing the stationary flow of a non‐Newtonian fluid through the frozen ground, with a free interface between the liquid and the solid phases. We prove the existence of at least one weak solution of the problem. Copyright © 2000 John Wiley & Sons, Ltd.     

Free convection boundary layers on a vertical surface in a heat-generating porous medium

The natural convection boundary-layer flow on a solid verticalsurface with heat generated within the boundary layer at a rateproportional to (T – T)^p (p 1) is considered. The surfaceis held at the ambient temperature T except near the leadingedge where it is held at a temperature above ambient. The behaviourof the flow as it develops from the leading edge is examinedand is seen to become independent of the initial heat input;however, it does depend strongly on the exponent p. For 1 p 2, the local heating eventually dominates at large distancesand there is a convective flow driven by this mechanism. Forp 4, the local heating does not have a significant effect,the fluid temperature remains relatively small throughout andthe heat transfer dies out through a wall jet flow. For 2 <p < 4, the local heating has a significant effect at relativelysmall distances, with a thermal runaway developing at a finitedistance along the surface.     

A free boundary problem for a fluid flow in a heterogeneous porous medium

In this paper we study the problem of seepage of a fluid through a porous medium, assuming the flow governed by a nonlinear Darcy law and nonlinear leaky boundary conditions. We prove the continuity of the free boundary and the existence and uniqueness of minimal and maximal solutions. We also prove the uniqueness of theS ₃-connected solution in various situations.     

Analysis of an elliptic-parabolic free boundary problem modelling the growth of non-necrotic tumor cord

We study a free boundary problem modelling the growth of a tumor cord in which tumor cells live around and receive nutrient from a central blood vessel. The evolution of the tumor cord surface is governed by Darcy's law together with a surface tension equation. The concentration of nutrient in the tumor cord satisfies a reaction-diffusion equation. In this paper we first establish a well-posedness result for this free boundary problem in some Sobolev-Besov spaces with low regularity by using the analytic semigroup theory. We next study asymptotic stability of the unique radially symmetric stationary solution. By making delicate spectrum analysis for the linearized problem, we prove that this stationary solution is locally asymptotically stable provided that the constant c representing the ratio between the diffusion time of nutrient and the birth time of new cells is sufficiently small.     

基于美式障碍期权定价的非线性变分不等式问题

研究了一类基于美式障碍期权定价的非线性变分不等式问题.首先定义了变分不等式问题的弱解.其次利用惩罚方法和Schaefer不动点定理证明了该变分不等式在弱意义下的解是存在且唯一的.