Global classical solution of Muskat free boundary problem

In this paper the Muskat problem which describes a two-phase flow of two fluids, for example, oil and water, in porous media is discussed. The problem involves in seeking two time-dependent harmonic functions u₁(x,y,t) and u₂(x,y,t) in oil and water regions, respectively, and the interface between oil and water, i.e., the free boundary Γ:y=ρ(x,t), such that on the free boundary

     

Numerical algorithms for a free boundary problem model of DCIS and a related inverse problem

ABSTRACT

The Ductal carcinoma in situ (DCIS) model has wide applications in the diagnosis of breast cancer, and has been attracted much attention in recent years. In this paper, an effective method has been provided for solving the direct problem of the DCIS model. Moreover, the uniqueness result is established for a kind of inverse problems of DCIS model. Based on the uniqueness theorem, an optimization method has been developed for dealing with the related inverse problem of the DCIS model. The numerical experiments show that the algorithms in this paper are efficient, accurate, robust against noise and fast.     

Regularity of solutions to a free boundary problem modeling tumor growth by Stokes equation

In this paper we investigate regularity of solutions to a free boundary problem modeling tumor growth in fluid-like tissues. The model equations include a quasi-stationary diffusion equation for the nutrient concentration, and a Stokes equation with a source representing the proliferation density of the tumor cells, subject to a boundary condition with stress tensor effected by surface tension. This problem is a fully nonlinear problem involving nonlocal terms. Based on the employment of the functional analytic method and the theory of maximal regularity, we prove that the free boundary of this problem is real analytic in temporal and spatial variables for initial data of less regularity.     

A critical case of stability in a free boundary problem

We study the stability of the planar travelling wave solution to a free boundary problem for the heat equation in the whole . We turn the problem into a fully nonlinear parabolic system and establish a stability result which is the proper generalization of the one-dimensional case. The curvature terms contribute a gradient squared corresponding to critical growth. The latter is eliminated by means of the Hopf-Cole transformation. Received August 18, 2000, accepted September 27, 2000.     

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.     

A coefficient identification problem for a mathematical model related to ductal carcinoma in situ

We consider a coefficient identification problem for a mathematical model with free boundary related to ductal carcinoma in situ (DCIS). This inverse problem aims to determine the nutrient consumption rate from additional measurement data at a boundary point. We first obtain a global‐in‐time uniqueness of our inverse problem. Then based on the optimization method, we present a regularization algorithm to recover the nutrient consumption rate. Finally, our numerical experiment shows the effectiveness of the proposed numerical method.     

A uniqueness theorem for a free boundary problem

In this paper, we prove a uniqueness theorem for a free boundary problem which is given in the form of a variational inequality. This free boundary problem arises as the limit of an equation that serves as a basic model in population biology. Apart from the interest in the problem itself, the techniques used in this paper, which are based on the regularity theory of variational inequalities and of harmonic functions, are of independent interest, and may have other applications.

     

Global classical solution of quasi-stationary Stefan free boundary problem

We consider an one-phase quasi-stationary Stefan problem (Hele–Shaw problem) in multidimensional case. Under some reasonable conditions we prove that the problem has a classical solution globally in time. The method can be used in two-phase problem as well. We also discuss asymptotic behavior of solution as t→+∞. The method developed here can be extended to a general class of free boundary problems.     

Bifurcation for a free boundary problem modeling the growth of a tumor with a necrotic core

We consider a free boundary problem for a system of partial differential equations, which arises in a model of tumor growth with a necrotic core. For any positive numbers ρ<R, there exists a radially symmetric stationary solution with tumor boundary r=R and necrotic core boundary r=ρ. The system depends on a positive parameter μ, which describes the tumor aggressiveness. There also exists a sequence of values μ₂<μ₃<? for which branches of symmetry-breaking stationary solutions bifurcate from the radially symmetric solution branch.     

Asymptotic behavior of solutions of a free boundary problem modeling multi-layer tumor growth in presence of inhibitor

In this paper we study well-posedness and asymptotic behavior of solution of a free boundary problem modeling the growth of multi-layer tumors under the action of an external inhibitor. We first prove that this problem is locally well-posed in little Hölder spaces. Next we investigate asymptotic behavior of the solution. By computing the spectrum of the linearized problem and using the linearized stability theorem, we give the rigorous analysis of stability and instability of all stationary flat solutions under the non-flat perturbations. The method used in proving these results is first to reduce the free boundary problem to a differential equation in a Banach space, and next use the abstract well-posedness and geometric theory for parabolic differential equations in Banach spaces to make the analysis.     

On the free boundary problem to the two viscous immiscible fluids

In this paper, we prove the local solvability of the free boundary problem describing the motion of two layers of immiscible, heavy, viscous, incompressible fluid lying above an infinite rigid bottom and with surface tension on the interfaces, and global solvability near the equilibrium state.     

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.).     

Linearization of a free boundary problem in corrosion detection

We consider a boundary identification problem arising in nondestructive testing of materials. The problem is to recover a part ΓI⊂∂Ω of the boundary of a bounded, planar domain Ω from one Cauchy data pair (u,∂u/∂ν) of a harmonic potential u in Ω collected on an accessible boundary subset ΓA⊂∂Ω. We prove Fréchet differentiability of a suitably defined forward map, and discuss local uniqueness and Lipschitz stability results for the linearized problem.     

The saddle point property for focusing selfsimilar solutions in a free boundary problem

We establish the saddle point property of the focusing selfsimilar solution of a free boundary problem for the heat equation with free boundary conditions given by and .