Free Boundary Value Problem Research Articles

Rearrangement minimization problems with indefinite external forces

This paper discusses a rearrangement minimization problem related to a boundary value problem where the differential operator is the p-Laplacian and the external force is indefinite. By applying the rearrangement theory established by G. R. Burton, we show that the minimization problem has a unique solution in the weak closure of the admissible set in an appropriate function space. What makes this paper different from many others in the literature is that the admissible set here is allowed to be indefinite (takes on both positive and negative values). We show that in this case the optimal solution will be a weak limit point of the admissible set and cannot be a rearrangement element itself. Optimality conditions are derived based on the mean value of the generator. In case the generator is a three valued function, we shall show that the optimality conditions lead to free boundary value problems that are of interest independently. We also discuss a minimization problem where the admissible set is not seemingly related to rearrangement classes but we shall prove otherwise, and apply the earlier results of the current paper to draw interesting conclusions regarding the optimal solution. We have also discussed the case where the domain of interest is radial, and shown that the optimal solutions are radial as well.

Optimal Consumption Until Ruin for an Endowment Described by an Autonomous ODE for an Infinite Time Horizon

We give an algorithmic solution of the optimal consumption problem [Formula: see text], where Ct denotes the accumulated consumption until time t, and τ denotes the time of ruin. Moreover, the endowment process Xt is modeled by [Formula: see text]. We solve the problem by showing that the function provided by the algorithm solves the Hamilton-Jacobi (HJ) equation in a viscosity sense and that the same is true for the value function of the problem. The argument is finished by a uniqueness result. It turns out that one has to change the optimal strategy at a sequence of endowment values, described by a free boundary value problem. Finally we give an illustrative example.

A blow-up criterion of spherically symmetric strong solutions to 3d compressible navier-stokes equations with free boundary

In this article, we consider the free boundary value problem of 3D isentropic compressible Navier-Stokes equations. A blow-up criterion in terms of the maximum norm of gradients of velocity is obtained for the spherically symmetric strong solution in terms of the regularity estimates near the symmetric center and the free boundary respectively.

Qualitative analysis of the invasion free boundary problem in biofilms

The work presents the qualitative analysis of the free boundary value problem related to the invasion model for multispecies biofilms. This model is based on the continuum approach for biofilm modeling and consists of a system of nonlinear hyperbolic partial differential equations for microbial species growth and spreading, a system of semilinear elliptic partial differential equations describing the substrate trends and a system of semilinear elliptic partial differential equations accounting for the diffusion and reaction of motile species within the biofilm. The free boundary evolution is regulated by a nonlinear ordinary differential equation. Overall, this leads to a free boundary value problem essentially hyperbolic. By using the method of characteristics, the partial differential equations constituting the invasion model are converted to Volterra integral equations. Then, the fixed point theorem is used for the uniqueness and existence result. The work is completed with numerical simulations describing the invasion of nitrite oxidizing bacteria in a biofilm initially constituted by ammonium oxidizing bacteria.

English

Fil: Ceretani, Andrea Noemi. Consejo Nacional de Investigaciones Cientificas y Tecnicas; Argentina. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matematicas; Argentina. Universidad Nacional de Rosario; Argentina

Continuum approach to mathematical modelling of multispecies biofilms

The work presents a contribution to the mathematical modelling of formation and growth of multispecies biofilms in the framework of continuum approach, without claiming to be complete. Mathematical models for biofilms often lead to consider free boundary value problems for nonlinear PDEs. The emphasis is on the qualitative analysis, uniqueness and existence of solutions and their main properties. Biofilm life is a complex biological process formed by several phases from the formation, development of colonies, attachment and detachment of microbial mass from (to) biofilm to (from) bulk liquid. Most of these processes are modelled and discussed. Moreover, some problems of interest for engineering and biological applications are considered. Indeed, we discuss the free boundary value problem related to biofilm reactors extensively used in wastewater treatment, and the invasion of new species into an already constituted biofilm with the successive colonizations. The main mathematical methodology used is the method of characteristics. The original differential problem is converted to integral equations. Then, the fixed point theorem is applied.

Qualitative analysis of the moving boundary problem for a biofilm reactor model

The work presents the qualitative analysis of the free boundary value problem related to a biofilm reactor model. In the framework of continuum approach to mathematical modelling of biofilm growth, the problem consists of a system of nonlinear hyperbolic partial differential equations governing the microbial species growth, a system of semilinear elliptic partial differential equations describing the substrate trends, and a system of nonlinear differential equations for the mass balance in the reactor. The free boundary evolution is governed by a differential equation that also accounts for detachment and attachment. The main result is a uniqueness and existence theorem. By using the method of characteristics, the original differential system is converted to Volterra integral equations and then the fixed point theorem is used. The work is completed with numerical simulations describing the free boundary evolution and mainly focused on attachment/detachment process.

Finite-Time Singularities of Solutions to Microelectromechanical Systems with General Permittivity

Of concern is a study of qualitative properties of solutions to the evolution problem modelling microelectromechanical systems with general permittivity. The system couples a quasilinear parabolic evolution problem for a membrane’s displacement with an elliptic free boundary value problem for the electrostatic potential in the region between the membrane and a rigid ground plate. It is shown that, under a structural condition on the permittivity profile, non-positive solutions develop a singularity in finite time, provided that the applied voltage is large enough and the aspect ratio of the system is small enough.

Free Boundary Value Problem for the Spherically Symmetric Compressible Navier-Stokes Equations with a Nonconstant Exterior Pressure

This paper is concerned with the free boundary value problem (FBVP) for the spherically symmetric barotropic compressible Navier-Stokes equations (CNS) with density-dependent viscosity coefficients in the case that across the free surface stress tensor is balanced by a nonconstant exterior pressure. Under certain assumptions imposed on the initial data, we show that there exists a unique global strong solution which is strictly positive for any finite time and decays pointwise to zero time-asymptotically.

Global Transonic Solutions of Planetary Atmospheres in a Hydrodynamic Region---Hydrodynamic Escape Problem Due to Gravity and Heat

The hydrodynamic escape problem (HEP), which is characterized by a free boundary value problem of Euler equation with gravity and heat, is crucial for investigating the evolution of planetary atmospheres. In this paper, the global existence of transonic solutions to the HEP is established using the generalized Glimm method. New versions of Riemann and boundary-Riemann solvers are provided as building blocks of the generalized Glimm method by applying the contraction matrices to the homogeneous Riemann (or boundary-Riemann) solutions. The extended Glimm--Goodman wave interaction estimates are investigated for obtaining a stable scheme and positive gas velocity, which matches the physical observation. The limit of approximation solutions serves as an entropy solution of bounded variations. Moreover, the range of the feasible hydrodynamical region is also obtained.

Mathematical Modeling of Heavy Metal Biosorption in Multispecies Biofilms

AbstractThe biofilm matrix is a complex of secreted polymers, absorbed nutrients and metabolites, cell lysis products, and even particulate material. Being polyanionic in nature, this matrix plays a crucial role in the biosorption of metal cations. In this work, the adsorption process of heavy metals in biofilms is modeled in one space dimension. The mathematical model is a free-boundary value problem for nonlinear hyperbolic and parabolic partial differential equations. Biomass and extracellular polymeric substances (EPS) growth is governed by hyperbolic equations, and substrate evolution by parabolic equations. All equations are mutually connected. The model is general and can work for any number of microbial species, EPS, and substrates. In numerical analysis, heterotrophic–autotrophic competition for space with oxygen as common substrate is considered. The model can describe biofilm growth dynamics including spatial distribution of microbial species, substrate concentrations, EPS formation, and, in pa...

Free boundary value problem for the cylindrically symmetric compressible navier-stokes equations with density-dependent viscosity

In this paper, we investigate the free boundary value problem (FBVP) for the cylindrically symmetric isentropic compressible Navier-Stokes equations (CNS) with density-dependent viscosity coefficients in the case that across the free surface stress tensor is balanced by a constant exterior pressure. Under certain assumptions imposed on the initial data, we prove that there exists a unique global strong solution which tends pointwise to a non-vacuum equilibrium state at an exponential time-rate as the time tends to infinity.

Stefan problem, traveling fronts, and epidemic spread

The scalar reaction diffusion equation with a nonlinearity of logistic type has a minimal speed $c_0$ for standard traveling fronts. It is shown that also for speeds $0 < c < c_0$ there are traveling fronts but these are solutions to free boundary value (Stefan) problems. Furthermore, these speeds depend in a monotone way on the Stefan coefficient which links the loss of matter at the free boundary to the displacement per time. The results are extended to correlated random walks, Cattaneo systems and, in particular, to models for epidemic spread. In the epidemic problems a dichotomy phenomenon shows up: For small values of the Stefan coefficient there are no fronts indicating that for such values and certain data the free boundary stays bounded.

A conformal mapping algorithm for the Bernoulli free boundary value problem

We propose a new numerical method for the solution of the Bernoulli free boundary value problem for harmonic functions in a doubly connected domain D in where an unknown free boundary Γ0 is determined by prescribed Cauchy data on Γ0 in addition to a Dirichlet condition on the known boundary Γ1. Our main idea is to involve the conformal mapping method as proposed and analyzed by Akduman, Haddar, and Kress for the solution of a related inverse boundary value problem. For this, we interpret the free boundary Γ0 as the unknown boundary in the inverse problem to construct Γ0 from the Dirichlet condition on Γ0 and Cauchy data on the known boundary Γ1. Our method for the Bernoulli problem iterates on the missing normal derivative on Γ1 by alternating between the application of the conformal mapping method for the inverse problem and solving a mixed Dirichlet–Neumann boundary value problem in D. We present the mathematical foundations of our algorithm and prove a convergence result. Some numerical examples will serve as proof of concept of our approach. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Global existence of weak solution for the compressible Navier–Stokes–Poisson system for gaseous stars

We study the multi-dimensional compressible Navier–Stokes–Poisson system with γ-law pressure and density-dependent viscosity coefficients in the simulation of the motion of gaseous stars. The global existence of spherically symmetric weak solutions to the free boundary value problem for the Navier–Stokes–Poisson system for γ∈(65,43] is shown for arbitrarily large initial data with compact support.

A free boundary value problem modelling microelectromechanical systems with general permittivity

An investigation of the free boundary value problem arising from the modelling of electrostatically actuated microelectromechanical systems with general permittivity is presented. Consisting of a parabolic evolution problem for the displacement of a membrane as well as of an elliptic moving boundary problem for the electric potential between the membrane and a rigid ground plate, the system is shown to be well-posed locally in time for all values λ of the applied voltage. It is in addition verified that the solution exists even globally in time, provided that the applied voltage does not exceed a certain critical value λ∗. Furthermore, we establish the convergence of the solution of the free boundary problem towards the solution of the small-aspect ratio model, as the aspect ratio tends to zero.

On the smoothness of the free boundary in a nonlocal one-dimensional parabolic free boundary value problem

Abstract We consider one-dimensional parabolic free boundary value problem with a nonlocal (integro-differential) condition on the free boundary. Results on Cm-smoothness of the free boundary are obtained. In particular, a necessary and sufficient condition for infinite differentiability of the free boundary is given.

Modelling microbial population dynamics in multispecies biofilms including Anammox bacteria

A 1D mathematical model for analysis and prediction of microbial interactions within multispecies biofilms including Anammox pathway is presented. The model combines the related processes of organic carbon oxidation, denitrification, nitrification and Anammox and phenomena of substrate reaction and diffusion, biomass growth and advection, detachment. The biofilm growth process is governed by nonlinear hyperbolic PDEs and substrate dynamics are dominated by semilinear parabolic PDEs. It follows a complex system of PDEs on a free boundary domain. Equations are integrated numerically by using the method of characteristics as strongly suggested by the qualitative analysis of the free boundary value problem. Mass conservation equation plays an important role in checking the accuracy of simulations. The model has been applied to simulate Anammox competition and to evaluate the influence of substrate diffusion on microbial stratification. Specific scenarios are analyzed. The results reveal that in a thick multispecies biofilm, including heterotrophic, aerobic autotrophic nitrifying and Anammox bacteria, oxygen diffusion limitation determines the formation of both aerobic and anoxic microenvironments favouring interspecies competition. In contrast, oxygen excess causes a disturbance on microbial interactions leading to Anammox bacteria loss. The model predictions may help engineers or operators to have a better insight into biofilm dynamics in order to optimize process design or practical operation.

On qualitative properties of solutions to microelectromechanical systems with general permittivity

Qualitative properties of solutions to the evolution problem modelling microelectromechanical systems with general permittivity profile are investigated. The system couples a parabolic evolution problem for the displacement of a membrane with an elliptic free boundary value problem for the electric potential in the region between the membrane and a rigid ground plate. We briefly allude to results concerning local and global well-posedness and the small-apect ratio limit. However, the focus is here on proving non-positivity of the membrane displacement for the full moving boundary problem under certain boundary conditions on the potential, as well as the existence of finite-time singularities assuming to have a non-positive solution.

Simulation of a pending drop at a capillary tip

We derive four different numerical schemes to compute the shape of axisymmetric drops at the tip of a capillary for a given drop volume and a given Bond number. Small drop volumes show a convex shape, however increasing the drop volume a point of inflection appears and at a critical drop volume (depending from the Bond number) the drop detaches from its capillary. One of the four schemes produces drop shapes completely different from the others. We show that these shapes are unstable by solving numerically the free boundary value problem for the transient Navier–Stokes equations.