Qualitative analysis and simulations of a free boundary problem for multispecies biofilm models

The work presents an analysis of solutions to a free boundary value problem for a multispecies biofilm growth model in one space dimension. The mathematical model consists of a system of nonlinear partial differential equations and a free boundary. It is quite general and can include a large variety of special situations. An existence and uniqueness theorem is discussed and properties of solutions are given. As a numerical application, simulations for a heterotrophic–autotrophic competition are developed by the method of characteristics.     

A free boundary problem for Aedes aegypti mosquito invasion

An advection–reaction–diffusion model with free boundary is proposed to investigate the invasive process of Aedes aegypti mosquitoes. By analyzing the free boundary problem, we show that there are two main scenarios of invasive regime: vanishing regime or spreading regime, depending on a threshold in terms of model parameters. Once the mortality rate of the mosquito becomes large with a small specific rate of maturation, the invasive mosquito will go extinct. By introducing the definition of asymptotic spreading speed to describe the spreading front, we provide an estimate to show that the boundary moving speed cannot be faster than the minimal traveling wave speed. By numerical simulations, we consider that the mosquitoes invasive ability and wind driven advection effect on the boundary moving speed. The greater the mosquito invasive ability or advection, the larger the boundary moving speed. Our results indicate that the mosquitoes asymptotic spreading speed can be controlled by modulating the invasive ability of winged mosquitoes.     

Qualitative analysis of a delayed free boundary problem for tumor growth under the effect of inhibitors

In this paper we study a delayed free boundary problem for the growth of tumors under the effect of inhibitors. The establishing of the model is based on the diffusion of nutrient and inhibitors, and mass conservation for the two processes proliferation and apoptosis. It is assumed that the process of proliferation is delayed compared to apoptosis. We mainly study the asymptotic behavior of the solution, and prove that under some assumptions, in the case where c₁ and c₂ are sufficiently small, the volume of the tumor cannot expand without limit; it will either disappear or evolve to a dormant state as t→∞.     

Variational analysis of a perturbed free boundary problem

We consider maps u:X→Y of Riemannian manifolds which are locally energy minimizing subject to a constraint of the form Im(u) where M denotes a smooth bounded domain contained in a regular ball B of the target manifold. In general local minima are regular up to a set of vanishing Hausdorff measure, here we show for star-shaped obstacles M singular points can be exluded.     

Moving boundary problem for the detachment in multispecies biofilms

The work presents the qualitative analysis of the free boundary value problem related to the detachment process in multispecies biofilms. In the framework of continuum approach to one-dimensional mathematical modelling of multispecies biofilm growth, we consider the system of nonlinear hyperbolic partial differential equations governing the microbial species growth, the differential equation for the biomass velocity, the differential equation that governs the free boundary evolution and also accounts for detachment, and the elliptic system for substrate dynamics. The characteristics are used to convert the original moving boundary equation into a suitable differential equation useful to solve the mathematical problem. We also provide another form of the same equation that could be used in numerical applications. Several properties of the solutions to the free boundary problem are shown, such as positiveness of the functions that describe the microbial concentrations and estimates on the characteristic functions. Uniqueness and existence of solutions are proved by introducing a suitable system of Volterra integral equations and using the fixed point theorem.

     

Qualitative analysis of a time‐delayed free boundary problem for tumor growth under the action of external inhibitors

In this paper, a time‐delayed free boundary problem for tumor growth under the action of external inhibitors is studied. It is assumed that the process of proliferation is delayed compared with apoptosis. By L^p theory of parabolic equations, the Banach fixed point theorem and the continuation theorem, the existence and uniqueness of a global solution is proved. The asymptotic behavior of the solution is also studied. The proof uses the comparison principle and the iteration method. Copyright © 2014 John Wiley & Sons, Ltd.     

A problem with free boundary

One considers the Dirichlet problem for the equation u=(u), where is the Heaviside function. Under special assumptions one constructs the solution of this problem with convex and smooth level surfaces and, in particular, with a regular free surface, which coincides with the set of level zero. One proves the solvability in the small of the problem in the neighborhood of the constructed regular solution under perturbations of the boundary condition and a smooth boundary of the domain .Translated from Problemy Matematicheskogo Analiza, No. 10, pp. 72–83, 1986.     

On the free boundary problem of magnetohydrodynamics

In the paper, the solvability of the free boundary problem of magnetohydrodynamics for a viscous incompressible fluid in a simply connected domain is proved. The solution is obtained in the Sobolev–Slobodetskii spaces W₂^{2 + l,1 + l/2},1/2 < l < 1 W_2^{2 + l,1 + l/2},1/2 < l < 1 . Bibliography: 15 titles.     

Convergence analysis of trial free boundary methods for the two-dimensional filtration problem

Summary In this paper, we present a scheme of convergence analysis of trial free boundary methods for the two-dimensional filtration (or dam) problem. For the purpose we present a new variational principle of the filtration problem. This variational principle is defined on the set of admissible domains (candidates of the solution) in the dam. Under mild assumptions on the configuration of the dam, we may assume that all admissible domains are mapped from the unit disk by conformal mappings. Thus, proving convergence of trial free boundaries is reduced to proving convergence of the conformal mappings on the unit disk, and it is done using a method in the theory of minimal surfaces. Numerical examples are given.     

Bifurcation analysis for a free boundary problem modeling tumor growth

In this paper we deal with a free boundary problem modeling the growth of nonnecrotic tumors. The tumor is treated as an incompressible fluid, the tissue elasticity is neglected and no chemical inhibitor species are present. We re-express the mathematical model as an operator equation and by using a bifurcation argument we prove that there exist smooth stationary solutions of the problem which are not radially symmetric.     

Regularity of a free boundary problem

Let u be the Newtonian potential of a real analytic distribution in an open set Ω. In this paper we assume u is analytically continuable from the complement of Ω into some neighborhood of a point x₀ ∈ ∂Ω, and we study conditions under which the analytic continuation implies that ∂Ω is a real analytic hypersurface in some neighborhood of x₀.     

Well-posedness of the free boundary problem in compressible elastodynamics

We study the free boundary problem for the flow of a compressible isentropic inviscid elastic fluid. At the free boundary moving with the velocity of the fluid particles the columns of the deformation gradient are tangent to the boundary and the pressure vanishes outside the flow domain. We prove the local-in-time existence of a unique smooth solution of the free boundary problem provided that among three columns of the deformation gradient there are two which are non-collinear vectors at each point of the initial free boundary. If this non-collinearity condition fails, the local-in-time existence is proved under the classical Rayleigh–Taylor sign condition satisfied at the first moment. By constructing an Hadamard-type ill-posedness example for the frozen coefficients linearized problem we show that the simultaneous failure of the non-collinearity condition and the Rayleigh–Taylor sign condition leads to Rayleigh–Taylor instability.     

On a free boundary problem in electrostatics

Let Ω_i ? ?^N, i = 0, 1, be two bounded separately star-shaped domains such that $ \Omega _0 \supset \bar \Omega _1 $. We consider the electrostatic potential u defined in $ \Omega : = \Omega _0 \backslash \bar \Omega _1 $: The geometry of the two boundary components Γ₀ and Γ₁ is not given, but instead the electrostatic potential u is supposed to satisfy the further boundary conditions Using a best possible maximum principle, we show that this free boundary problem has a unique solution which is radially symmetric.     

Almost minimizers of the one-phase free boundary problem

Abstract

We consider almost minimizers to the one-phase energy functional and we prove their optimal Lipschitz regularity and partial regularity of their free boundary. These results were recently obtained by David and Toro, and David, Engelstein, and Toro. Our proofs provide a different method based on a non-infinitesimal notion of viscosity solutions that we introduced.     

A nonlinear parabolic free boundary problem

Summary When two immiscible fluids in a porous medium are in contact with one another, an interface is formed and the movement of the fluids results in a free boundary problem for determining the location of the interface along with the pressure distribution throughout the medium. The pressure satisfies a nonlinear parabolic partial differential equation on each side of the interface while the pressure and the volumetric velocity are continuous across the interface. The movement of the interface is related to the pressure through Darcy’s law. Two kinds of boundary conditions are considered. In Part I the pressure is prescribed on the known boundary. A weak formulation of the classical problem is obtained and the existence of a weak solution is demonstrated as a limit of a sequence of classical solutions to certain parabolic boundary value problems. In Part II the same analysis is carried out when the flux is specified on the known boundary, employing special techniques to obtain the uniform parabolicity of the sequence of approximating problems. Entrata in Redazione il 29 novembre 1975. This research was supported in part by the National Science Foundation, the Senior Fellowship Program of the North Atlantic Treaty Organization, the Italian Consiglio Nazionale delle Ricerche, and the Texas Tech. University.